Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems: Regularization Approach 303107050X, 9783031070501

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Static & Dynamic Game Theory: Foundations & Applications Series Editor Tamer Ba¸sar, University of Illinois, Urbana-Champaign, IL, USA Editorial Board Daron Acemoglu, Massachusetts Institute of Technology, Cambridge, MA, USA Pierre Bernhard, INRIA, Sophia-Antipolis, France Maurizio Falcone , Università degli Studi di Roma “La Sapienza”, Roma, Italy Alexander Kurzhanski, University of California, Berkeley, CA, USA Ariel Rubinstein, Tel Aviv University, Ramat Aviv, Israel Yoav Shoham, Stanford University, Stanford, CA, USA Georges Zaccour, GERAD, HEC Montréal, QC, Canada

Valery Y. Glizer · Oleg Kelis

Singular Linear-Quadratic Zero-Sum Differential Games and H ∞ Control Problems Regularization Approach

Valery Y. Glizer The Galilee Research Center for Applied Mathematics ORT Braude College of Engineering Karmiel, Israel

Oleg Kelis The Galilee Research Center for Applied Mathematics ORT Braude College of Engineering Karmiel, Israel Faculty of Mathematics Technion-Israel Institute of Technology Haifa, Israel

ISSN 2363-8516 ISSN 2363-8524 (electronic) Static & Dynamic Game Theory: Foundations & Applications ISBN 978-3-031-07050-1 ISBN 978-3-031-07051-8 (eBook) https://doi.org/10.1007/978-3-031-07051-8 Mathematics Subject Classification: 49N70, 91A05, 91A10, 91A23, 93B36, 93C70, 93C73 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my family Elena, Evgeny and Svetlana V.Y.G. To my wife Tanya and my children Louisa, Nadav and Gavriel O.K.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Examples of Singular Extremal Problems and Some Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Academic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Minimization of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 The First Way of the Reformulation of (2.1)–(2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 The Second Way of the Reformulation of (2.1)–(2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1 The First Way of the Reformulation of (2.11)–(2.12), (2.13) . . . . . . . . . . . . . . . . . . . . . . 2.2.2.2 The Second Way of the Reformulation of (2.11)–(2.12), (2.13) . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Saddle Point of a Function of Two Variables . . . . . . . . . . . . 2.2.3.1 The First Way of the Reformulation of the Saddle-Point Problem for the Function (2.45)–(2.46) . . . . . . . . . . . . . . . . 2.2.3.2 The Second Way of the Reformulation of the Saddle-Point Problem for the Function (2.45)–(2.46) . . . . . . . . . . . . . . . . 2.2.4 Zero-Sum Differential Game . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4.1 The First Way of the Reformulation of the Game (2.57)–(2.58) . . . . . . . . . . . . . . . . . . . . 2.2.4.2 The Second Way of the Reformulation of the Game (2.57)–(2.58) . . . . . . . . . . . . . . . . . . . .

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2.3

Mathematical Models of Real-Life Problems . . . . . . . . . . . . . . . . . . 2.3.1 Planar Pursuit-Evasion Engagement: Zero-Order Dynamics of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Planar Pursuit-Evasion Engagement: First-Order Dynamics of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Three-Dimensional Pursuit-Evasion Engagement: Zero-Order Dynamics of Participants . . . . . . . . . . . . . . . . . . 2.3.4 Infinite-Horizon Robust Vibration Isolation . . . . . . . . . . . . . 2.4 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solvability Conditions of Regular Problems . . . . . . . . . . . . . . . . . . . 3.2.1 Finite-Horizon Differential Game . . . . . . . . . . . . . . . . . . . . . 3.2.2 Infinite-Horizon Differential Game . . . . . . . . . . . . . . . . . . . . 3.2.3 Finite-Horizon H∞ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Infinite-Horizon H∞ Problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 State Transformation in Linear System and Quadratic Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 52 56 60 62

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4 Singular Finite-Horizon Zero-Sum Differential Game . . . . . . . . . . . . . 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Initial Game Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.3 Transformation of the Initially Formulated Game . . . . . . . . . . . . . . 76 4.4 Regularization of the Singular Finite-Horizon Game . . . . . . . . . . . . 79 4.4.1 Partial Cheap Control Finite-Horizon Game . . . . . . . . . . . . 79 4.4.2 Saddle-Point Solution of the Game (4.8), (4.12) . . . . . . . . . 79 4.5 Asymptotic Analysis of the Game (4.8), (4.12) . . . . . . . . . . . . . . . . 81 4.5.1 Transformation of the Problem (4.14) . . . . . . . . . . . . . . . . . . 81 4.5.2 Asymptotic Solution of the Terminal-Value Problem (4.23)–(4.25) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5.3 Asymptotic Representation of the Value of the Game (4.8), (4.12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 Reduced Finite-Horizon Differential Game . . . . . . . . . . . . . . . . . . . . 89 4.7 Saddle-Point Sequence of the SFHG . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.7.1 Main Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.7.2 Proof of Lemma 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7.3 Proof of Lemma 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.7.3.1 Stage 1: Regularization of (4.85)–(4.86) . . . . . . . . 97 4.7.3.2 Stage 2: Asymptotic Analysis of the Problem (4.85), (4.90) . . . . . . . . . . . . . . . . . 97 4.7.3.3 Stage 3: Deriving the Expression for the Optimal Value of the Functional in the Problem (4.85)–(4.86) . . . . . . . . . . . . . . . . . . 100

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Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Example 2: Solution of Planar Pursuit-Evasion Game with Zero-Order Dynamics of Players . . . . . . . . . . . . 4.8.3 Example 3: Solution of Three-Dimensional Pursuit-Evasion Game with Zero-Order Dynamics of Players . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3.1 Pursuit-Evasion Game with One “Regular” and One “Singular” Coordinates of the Pursuer’s Control . . . . . . . . . . . . . . . . . . . . . . 4.8.3.2 Pursuit-Evasion Game with Both “Singular” Coordinates of the Pursuer’s Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101

5 Singular Infinite-Horizon Zero-Sum Differential Game . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Initial Game Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Transformation of the Initially Formulated Game . . . . . . . . . . . . . . 5.4 Auxiliary Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Regularization of the Singular Infinite-Horizon Game . . . . . . . . . . 5.5.1 Partial Cheap Control Infinite-Horizon Game . . . . . . . . . . . 5.5.2 Saddle-Point Solution of the Game (5.9), (5.29) . . . . . . . . . 5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29) . . . 5.6.1 Transformation of the Eq. (5.33) . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Asymptotic Solution of the Set (5.44)–(5.46) . . . . . . . . . . . . 5.6.3 Asymptotic Representation of the Value of the Game (5.9), (5.29) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Reduced Infinite-Horizon Differential Game . . . . . . . . . . . . . . . . . . 5.8 Saddle-Point Sequence of the SIHG . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Main Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Proof of Lemma 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Proof of Lemma 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3.1 Stage 1: Regularization of (5.95)–(5.96) . . . . . . . . 5.8.3.2 Stage 2: Asymptotic Analysis of the Problem (5.95), (5.100) . . . . . . . . . . . . . . . . 5.8.3.3 Stage 3: Deriving the Expression for the Optimal Value of the Functional in the Problem (5.95)–(5.96) . . . . . . . . . . . . . . . . . . 5.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Example 2: Solution of Infinite-Horizon Vibration Isolation Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Singular Finite-Horizon H∞ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Initial Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Transformation of the Initially Formulated H∞ Problem . . . . . . . . 6.4 Regularization of the Singular Finite-Horizon H∞ Problem . . . . . . 6.4.1 Partial Cheap Control Finite-Horizon H∞ Problem . . . . . . 6.4.2 Solution of the H∞ Problem (6.9), (6.11) . . . . . . . . . . . . . . . 6.5 Asymptotic Analysis of the H∞ Problem (6.9), (6.11) . . . . . . . . . . 6.5.1 Transformation of the Problem (6.13) . . . . . . . . . . . . . . . . . . 6.5.2 Asymptotic Solution of the Problem (6.20)–(6.22) . . . . . . . 6.6 Reduced Finite-Horizon H∞ Problem . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Controller for the SFHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Formal Design of the Controller . . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Properties of the Simplified Controller (6.39) . . . . . . . . . . . 6.7.3 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Proof of Theorem 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4.1 Auxiliary Proposition . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4.2 Main Part of the Proof . . . . . . . . . . . . . . . . . . . . . . . 6.8 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 149 150 151 153 153 154 154 155 156 159 160 160 162 163 168 168 169 171 172 173

7 Singular Infinite-Horizon H∞ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Initial Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Transformation of the Initially Formulated H∞ Problem . . . . . . . . 7.4 Regularization of the Singular Infinite-Horizon H∞ Problem . . . . 7.4.1 Partial Cheap Control Infinite-Horizon H∞ Problem . . . . . 7.4.2 Solution of the H∞ Problem (7.9), (7.15) . . . . . . . . . . . . . . . 7.5 Asymptotic Analysis of the H∞ Problem (7.9), (7.15) . . . . . . . . . . 7.5.1 Transformation of Eq. (7.17) . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Asymptotic Solution of the Set (7.24) . . . . . . . . . . . . . . . . . . 7.6 Reduced Infinite-Horizon H∞ Problem . . . . . . . . . . . . . . . . . . . . . . . 7.7 Controller for the SIHP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Formal Design of the Controller . . . . . . . . . . . . . . . . . . . . . . . 7.7.2 Properties of the Simplified Controller (7.45) . . . . . . . . . . . 7.7.3 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.4 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Example 2: H∞ Control in Infinite-Horizon Vibration Isolation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Concluding Remarks and Literature Review . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175 175 176 177 179 179 179 180 180 181 184 185 185 187 188 191 195 195 196 199 200

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Chapter 1

Introduction

A singular differential game is the game which cannot be solved by the application of the first-order solvability conditions. In particular, a singular zero-sum differential game can be solved neither by the application of the Isaacs MinMax principle, nor by the application of the Bellman-Isaacs equation, [3, 4, 21, 22]. This occurs because the problem of minimization (maximization) of the game’s Variational Hamiltonian with respect to control of at least one player either having no solution or having infinitely many solutions. Singular differential games appear in many applications: for example, in pursuitevasion problems [9, 18, 26, 27, 29, 30], robust controllability problems [36], robust interception of a manoeuvring target [28, 37], robust tracking [38], biology processes [19] and robust investment [20]. In some cases, higher order control optimality conditions can be helpful in solving singular zero-sum differential games (see, e.g., [9, 29]). However, such conditions are useless for analysis of such games not having an optimal control of at least one of the players in the class of regular (non-generalized) functions, even if the corresponding infsup/supinf of its cost functional is finite in this class of controls. To the best of our knowledge, this case was analysed only in several works in the literature. Thus, in the works [31, 33], the definition of an almost equilibrium in an infinite-horizon differential game with a singular weight matrix of the minimizer’s control cost in the cost functional was proposed. Using the Riccati matrix inequality approach, conditions for the existence of such an equilibrium were derived. In [1], a finite-horizon differential game with a singular minimizer’s control cost and no terminal state cost in the cost functional was solved in a class of generalized functions. Regularization approach to analysis and solution of various types of finite/infinitehorizon singular zero-sum differential games was proposed and developed in [11–15, 28].

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_1

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1 Introduction

Systems with uncertainties in dynamics are extensively studied in the literature. The most studied classes of the uncertainties are (1) the uncertainties belonging to a known bounded set of an Euclidean space; (2) the uncertainties, quadratically integrable in a given finite/infinite interval. For controlled systems with quadratically integrable uncertainties (disturbances), the H∞ control problem is frequently considered (see, e.g., [2, 5, 8, 25]). If the rank of the matrix of coefficients for the control variable in the output equation equals the dimension of the control, then the solution of the H∞ control problem can be reduced to a solution of a gametheoretic matrix Riccati equation. This equation is an algebraic equation in the case of the infinite-horizon problem, and it is a differential equation in the case of the finite-horizon problem. If the rank of the matrix of coefficients for the control in the output is smaller than the dimension of the control, then the above-mentioned Riccati equation does not exist. Such H∞ control problems are called singular or nonstandard. Various types of singular H∞ control problems were studied in the literature, mainly in the infinite-horizon case. For these problems, different approaches to their analysis were proposed. Thus, in [24], the H∞ problem with no control in the output equation was considered. For this problem, an “extended” game-theoretic matrix Riccati algebraic equation was constructed. Based on the assumption of the existence of a proper solution to this equation, the solution of the considered H∞ problem was derived. In [32], an H∞ problem with the non-injective direct feedthrough matrix is considered. The minimum entropy of this problem was studied subject to the requirement on the internal stability of the problem’s dynamics and the requirement that the closed-loop transfer matrix has the H∞ norm smaller than a given number. In [7], a zero compensation approach to solution of some singular H∞ problems was proposed. In [39], H∞ problems were studied in the case where the direct feedthrough matrices from the input to the error and from the exogenous signal to the output are not injective. The linear matrix inequality approach to the design of controllers for singular H∞ problems was proposed. In [10], a numerical algorithm for computing the optimal performance level in an H∞ problem with rank-deficient feedthrough matrices was developed. In [23], an H∞ problem with no control in the output equation was considered. The solution of this problem is based on its approximate transformation to a regular H∞ problem. In [33, 34], a matrix Riccati inequality approach was used to solve some singular H∞ problems. In [6], the controller design was based on a physical model of the Atomic Force Microscope studied in the paper. In [16], an H∞ problem with a non-injective (but non-zero) matrix of the coefficients for the control in the output equation was studied. This problem was solved by regularization and asymptotic analysis approaches. In contrast with the infinite-horizon case in the topic of singular H∞ control problems, the finite-horizon case was studied much less. Thus in [33, 35], an H∞ problem with non-injective matrices of the coefficients for the control and the disturbance in the output equations was considered. Necessary and sufficient solvability conditions in the terms of Riccati differential inequalities were derived. In [17], an H∞ problem was studied in the case where the matrix of coefficients for the control in the output equation is non-injective

1 Introduction

3

(but non-zero). The regularization and asymptotic study of the regularized problem were applied to analysis and solution of the original problem. In the present book, a unified approach to analysis and solution of singular linearquadratic zero-sum differential games and singular H∞ control problems is proposed. Namely, we propose the regularization approach to study these problems. Due to this approach, the original singular problem is associated with a new regular problem. The new problem depends on a small positive parameter ε, and it becomes the original problem if this parameter is formally replaced with zero. By the solvability conditions, valid for a regular problem, the solution of the new problem is reduced to solution of a Riccati matrix algebraic/differential equation perturbed by the parameter ε > 0. For this equation, an asymptotic solution (with respect to the small parameter ε > 0) is formally constructed and justified. Based on this asymptotic solution, the solution of the original singular problem is derived. The book is the extension and generalization of the authors’ results published in the journal and conference papers [13–17, 28]. The book consists of the introduction chapter (the present one) and six body chapters (Chaps. 2–7). In Chap. 2, examples of singular extremal problems (academic and real-life) are presented. Basic notions, arising in the topic of singular extremal problems, as well as methods of their solution, are discussed. In Chap. 3, some preliminary results are given. Namely, solvability conditions of regular finite/infinitehorizon linear-quadratic zero-sum differential games and H∞ control problems are presented. Also, properties of a state transformation in a linear controlled system and quadratic functional are studied. In Chap. 4, the singular finite-horizon linearquadratic differential game is solved by the regularization method. An application of such a method to the solution of the singular infinite-horizon linear-quadratic differential game is given in Chap. 5. Chapters 6 and 7 are devoted to solution of singular finite-horizon and infinite-horizon linear-quadratic H∞ control problems. Each of Chaps. 4–7, devoted to the theoretical analysis of the singular games/H∞ problems is provided with academic/real-life examples illustrating the theoretical results and their applicability. Technically complicated proofs are placed into separate sections/subsections. For the sake of better readability, each chapter is organized to be self-contained as much as possible. Moreover, each of Chaps. 2–7 contains an introduction, a list of notations used in the chapter, concluding remarks with a brief literature review on the topic of the chapter and a corresponding bibliography. Due to such an organization of the book’s chapters, each chapter can be studied independently of the others. The book can be helpful for researchers and engineers, working in such areas as Applied Mathematics, Control Engineering, Mechanical and Aerospace Engineering, Electrical Engineering and even Biology. Also, the book can be useful for graduate students in these areas. Moreover, due to the detailed and systematic presentation of the theoretical material accompanied by the illustrative examples, the book can be used as a textbook in various courses on Control Theory and Dynamic Games Theory.

4

1 Introduction

References 1. Amato, F., Pironti, A.: A note on singular zero-sum linear quadratic differential games. In: Proceedings of the 33rd Conference on Decision and Control, pp. 1533–1535. Lake Buena Vista, FL, USA (1994) 2. Basar, T., Bernhard, P.: H ∞ -Optimal Control and Related Minimax Design Problems: A Dynamic Games Approach, 2nd edn. Birkhauser, Boston, MA, USA (1995) 3. Basar, T., Olsder, G.J.: Dynamic Noncooparative Game Theory, 2nd edn. SIAM Books, Philadelphia, PA, USA (1999) 4. Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization. Estimation and Control, Hemisphere, New York, NY, USA (1975) 5. Chang, X.H.: Robust Output Feedback H∞ Control and Filtering for Uncertain Linear Systems. Springer, Berlin, Germany (2014) 6. Chuang, N., Petersen, I. R., Pota, H.R.: Robust H ∞ control in fast atomic force microscopy. In: Proceedings of the 2011 American Control Conference, pp. 2258–2265. San Francisco, CA, USA (2011) 7. Copeland, B.R., Safonov, M.G.: A zero compensation approach to singular H 2 and H ∞ problems. Internat. J. Robust Nonlinear Control 5, 71–106 (1995) 8. Doyle, J.C., Glover, K., Khargonekar, P.P., Francis, B.: State-space solution to standard H2 and H∞ control problems. IEEE Trans. Automat. Control 34, 831–847 (1989) 9. Forouhar, K.: Singular differential game techniques and closed-loop strategies. In: Leondes, C.T. (ed.) Control and Dynamic Systems: Advances in Theory and Applications, vol. 17, pp. 379–419 (1981) 10. Gahinet, P., Laub, A.J.: Numerically reliable computation of optimal performance in singular H∞ control. SIAM J. Control Optim. 35, 1690–1710 (1997) 11. Glizer, V.Y.: Saddle-point equilibrium sequence in one class of singular infinite horizon zerosum linear-quadratic differential games with state delays. Optimization 68, 349–384 (2019) 12. Glizer, V.Y.: Saddle-point equilibrium sequence in a singular finite horizon zero-sum linearquadratic differential game with delayed dynamics. Pure Appl. Funct. Anal. 6, 1227–1260 (2021) 13. Glizer, V.Y., Kelis, O.: Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser. J. Control Decis. 2, 155–184 (2015) 14. Glizer, V.Y., Kelis, O.: Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence. Numer. Algebr. Control Optim. 7, 1–20 (2017) 15. Glizer, V.Y., Kelis, O.: Upper value of a singular infinite horizon zero-sum linear-quadratic differential game. Pure Appl. Funct. Anal. 2, 511–534 (2017) 16. Glizer, V.Y., Kelis, O.: Solution of a singular H∞ control problem: a regularization approach. In: Proceedings of the 14th International Conference on Informatics in Control. Automation and Robotics, pp. 25–36. Madrid, Spain (2017) 17. Glizer, V.Y., Kelis, O.: Finite-horizon H∞ control problem with singular control cost. In: Gusikhin, O., Madani, K. (eds.) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol. 495, pp. 23–46. Springer Nature, Switzerland (2020) 18. Glizer, V.Y., Shinar, J.: On the structure of a class of time-optimal trajectories. Optimal Control Appl. Methods 14, 271–279 (1993) 19. Hamelin, F.M., Lewis, M.A.: A differential game theoretical analysis of mechanistic models for territoriality. J. Math. Biol. 61, 665–694 (2010) 20. Hu, Y., Øksendal, B., Sulem, A.: Singular mean-field control games. Stoch. Anal. Appl. 35, 823–851 (2017) 21. Isaacs, R.: Differential Games. Wiley, New York, NY, USA (1967) 22. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York, NY, USA (1988) 23. Orlov, Y.V.: Regularization of singular H2 and H∞ control problems. In: Proceedings of the 36th IEEE Conference on Decision and Control, pp. 4140–4144. San Diego, CA, USA (1997)

References

5

24. Petersen, I.R.: Disturbance attenuation and H∞ optimization: a design method based on the algebraic Riccati equation. IEEE Trans. Automat. Control 32, 427–429 (1987) 25. Petersen, I., Ugrinovski, V., Savkin, A.V.: Robust Control Design using H∞ Methods. Springer, London, UK (2000) 26. Shinar, J.: Solution techniques for realistic pursuit-evasion games. In: Leondes, C. (ed.) Advances in Control and Dynamic Systems, pp. 63–124. Academic Press, New York, NY, USA (1981) 27. Shinar, J., Glizer, V.Y.: Application of receding horizon control strategy to pursuit-evasion problems. Optimal Control Appl. Methods 16, 127–141 (1995) 28. Shinar, J., Glizer, V.Y., Turetsky, V.: Solution of a singular zero-sum linear-quadratic differential game by regularization. Int. Game Theory Rev. 16, 1440007-1–1440007-32 (2014) 29. Simakova, E.N.: Differential pursuit game. Autom. Remote. Control 28, 173–181 (1967) 30. Starr, A.W., Ho, Y.C.: Nonzero-sum differential games. J. Optim. Theory Appl. 3, 184–206 (1969) 31. Stoorvogel, A.A.: The singular zero-sum differential game with stability using H∞ control theory. Math. Control Signals Syst. 4, 121–138 (1991) 32. Stoorvogel, A.A.: The singular minimum entropy H∞ control problem. Syst. Control Lett. 16, 411–422 (1991) 33. Stoorvogel, A.A.: The H∞ Control Problem: A State Space Approach (e-book). University of Michigan, Ann-Arbor, MI, USA (2000) 34. Stoorvogel, A.A., Trentelman, H.L.: The quadratic matrix inequality in singular H∞ control with state feedback. SIAM J. Control Optim. 28, 1190–1208 (1990) 35. Stoorvogel, A.A., Trentelman, H.L.: The finite-horizon singular H∞ control problem with dynamic measurement feedback. Linear Algebr. Appl. 187, 113–161 (1993) 36. Turetsky, V., Glizer, V.Y.: Robust state-feedback controllability of linear systems to a hyperplane in a class of bounded controls. J. Optim. Theory Appl. 123, 639–667 (2004) 37. Turetsky, V., Glizer, V.Y.: Robust solution of a time-variable interception problem: a cheap control approach. Int. Game Theory Rev. 9, 637–655 (2007) 38. Turetsky, V., Glizer, V.Y., Shinar, J.: Robust trajectory tracking: differential game/cheap control approach. Int. J. Syst. Sci. 45, 2260–2274 (2014) 39. Xin, X., Guo, L., Feng, C.: Reduced-order controllers for continuous and discrete-time singular H ∞ control problems based on LMI. Autom. J. IFAC 32, 1581–1585 (1996)

Chapter 2

Examples of Singular Extremal Problems and Some Basic Notions

2.1 Introduction In this chapter, we present several examples of singular extremal problems. Some of these examples are academic ones. Namely, we consider the examples of the following problems: the singular problem of the minimization of one variable function; the singular optimal control problem; the singular problem of searching a saddle point of a function of two variables; the singular zero-sum differential game. By analysis of these examples, we discuss basic notions, arising in the topic of singular extremal problems, as well as methods of their solution. The other examples, considered in this chapter, represent singular mathematical models of real-life problems. These examples are the following: the mathematical model of the planar pursuit-evasion engagement between two flying vehicles with the zero-order dynamics of the participants; the mathematical model of the planar pursuit-evasion engagement between two flying vehicles with the first-order dynamics of the participants; the mathematical model of the three-dimensional pursuit-evasion engagement with the zero-order dynamics of the participants; the mathematical model of the infinite-horizon robust vibration isolation. Here, these mathematical models are considered in a brief form, while their more detailed analysis is presented in subsequent chapters, based on theoretical results obtained in these chapters. The following main notations are applied in this chapter: 1. R is the set of all real numbers. 2. δ(t), t ≥ 0, is the right-hand side Dirac delta-function, which is defined by the following integral equality valid for any positive number t ∗ and any continuous function ϕ(t), t ∈ [0, t ∗ ]:  0

t



ϕ(s)δ(s)ds =



ϕ(0), t ∈ (0, t ∗ ] 0, t = 0.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_2

7

8

2 Examples of Singular Extremal Problems and Some Basic Notions

3. 1 × 2 , where 1 and 2 are some sets either of numbers or of functions, denotes the direct product of these sets, i.e., the set of all pairs (θ1 , θ2 ) with any entries θ1 ∈ 1 and θ2 ∈ 2 . 4. col(x, y), where x is an n-dimensional vector and y is an m-dimensional vector, denotes the column block-vector of the dimension n + m with the upper block x and the lower block y.

2.2 Academic Examples 2.2.1 Minimization of a Function Consider the following problem: f (t) → min, t∈Ω

(2.1)

where f (t) =

1 , t +1

Ω = {t : t ≥ 0}.

(2.2)

It is quite obvious that inf f (t) = 0,

t∈Ω

but there is not a number t ∈ Ω, for which this infimum value is attained. This means that the problem (2.1)–(2.2) does not have a solution, i.e., this problem is ill- posed, and we call it a singular problem. Nevertheless, we would like to solve this problem in some sense. Therefore, the following question arises. Can we change the formulation of this problem in such a way that the new problem formulation allows the existence of a solution t ∗ such that 

f ∗ = f (t ∗ ) = inf f (t) = 0. t∈Ω

(2.3)

Here, we present two ways of the reformulation of the problem (2.1)–(2.2).

2.2.1.1

The First Way of the Reformulation of (2.1)–(2.2)

This way consists in some extension of the set Ω of possible solutions of the minimization problem. Namely, instead of Ω, let us consider the following set of possible   = Ω {+∞}. Thus, instead of the problem (2.1)–(2.2), we consider solutions: Ω the problem f (t) → min .  t∈Ω

2.2 Academic Examples

9

This new problem has the solution arg min f (t) = +∞,  t∈Ω

i.e., t ∗ = +∞, and the condition (2.3) is satisfied. This elegant from the theoretical viewpoint solution is, however, inconvenient for practical implementation. Below, we present the second way of the reformulation of the problem (2.1)–(2.2), which yields the solution more convenient for the implementation.

2.2.1.2

The Second Way of the Reformulation of (2.1)–(2.2)

This way consists in an introducing a new definition of the solution to this problem. Namely, let us consider a sequence of numbers {tk }+∞ k=1 , satisfying the condition tk ∈ Ω, (k = 1, 2, . . .). Definition 2.1 The sequence {tk }+∞ k=1 is called a solution (or a minimizing sequence) of the problem (2.1)–(2.2), if the following limit equality is valid: lim f (tk ) = inf f (t) = 0.

k→+∞

t∈Ω

(2.4)

Remark 2.1 Due to Definition 2.1, the solution of the problem (2.1)–(2.2) is not a single number t from the set Ω, but it is a sequence of numbers from this set, i.e., 

t ∗ = {tk }+∞ k=1 . Moreover, due to Eq. (2.4), the minimum value of the function f (·) in the set Ω is defined as   (2.5) f ∗ = f (t ∗ ) = f {tk }+∞ k=1 = lim f (tk ). k→+∞

Therefore, by virtue of (2.4)–(2.5), the value f ∗ satisfies the condition (2.3). Remark 2.2 It should be noted that the solution of the problem (2.1)–(2.2) in the form of a minimizing sequence of numbers is more convenient for practical implementation than the solution in the form of the abstract element +∞. Now, let us consider how to derive a minimizing sequence in the problem (2.1)– (2.2). First of all, we would like to note that in this simple problem it is quite easy to find such a sequence. Namely, the sequence {tk } = {k}, (k = 1, 2, . . .), is a minimizing sequence in the problem (2.1)–(2.2). However, this sequence has been obtained by rather a guess, but not by a rigorous constructive method. Here, we present such a method. It is called the regularization method. This method consists in approximately replacing the singular minimization problem (2.1)–(2.2) with a new minimization problem which has a solution in the set Ω, satisfying the first-order extremum condition [5]. Thus, the new minimization problem is well posed (or regular). This new problem depends on a small parameter ε > 0, and it becomes the original problem (2.1)–(2.2) when ε is replaced with zero. In the new problem, we keep the set Ω to

10

2 Examples of Singular Extremal Problems and Some Basic Notions

be unchanged, while we chose the following new minimized function instead of the function f (·): f ε (t) = f (t) + ε2 t =

1 + ε2 t, t +1

t ≥ 0.

(2.6)

Let us solve the problem of the minimization of the function in (2.6). Due to the necessary condition for the unconstrained minimum, we have the equation with respect to t d f ε (t) 1 + ε2 = 0, =− dt (t + 1)2 which yields two solutions t1 (ε) = 1/ε − 1 and t2 (ε) = −1/ε − 1. Since the second solution is negative, only t1 (ε) for all sufficiently small ε > 0, (ε < 1), and the boundary point t = 0 can be considered as candidates with minimum points of the function f ε (t) in the interval  [0, +∞). Calculating the values of this function at these we have f ε t1 (ε) = 2ε − ε2 and f ε (0) = 1. Since, for all ε ∈ (0, 1),  points, f ε t1 (ε) < f ε (0), then, for these values of ε, t1 (ε) is the unique minimum point of the function f ε (t) in the interval [0, +∞). Let {εk }+∞ k=1 be the sequence of numbers such that εk ∈ (0, 1), (k = 1, 2, . . .), and limk→+∞ εk = 0. For this sequence, we have  lim f εk t1 (εk ) = 2εk − ε2k = 0.

(2.7)

k→+∞

Due to Definition 2.1 and Remark 2.1, Eq. (2.7) means that t ∗ = {t1 (εk )}+∞ k=1 =



+∞

1 −1 εk

(2.8) k=1

is the solution (the minimizing sequence) of the problem (2.1)–(2.2). Remark 2.3 It should be noted that the minimization of the function f ε (t), given in (2.6), is not a single form of the regularization of the problem (2.1)–(2.2). For instance, we can choose f ε (t) as f ε (t) = f (t) +

ε2 t 3 1 ε2 t 3 = + , 3 t +1 3

t ≥ 0.

(2.9)

For such a chosen regularization of the problem (2.1)–(2.2), its solution (the minimizing sequence) is t∗ =

⎧ ⎨ ⎩

εk +



2 ε2k + 4εk

⎫+∞ ⎬ ⎭ k=1

.

(2.10)

2.2 Academic Examples

11

It is seen that the solution (2.10) of the problem (2.1)–(2.2), obtained by its regularization (2.9), is much more complicated than the solution (2.8) of (2.1)– (2.2), obtained by its regularization (2.6). This observation means the importance of choosing the regularization in a form, which provides a simple form of the solution (the minimizing sequence) of the minimization problem.

2.2.2 Optimal Control Problem Consider the scalar differential equation d x(t) = x(t) + u(t), dt

t ∈ [0, 1],

x(0) = x0 = 0,

(2.11)

where x(t), t ∈ [0, 1] is a state variable; u(t), t ∈ [0, 1] is a control function. Along with Eq. (2.11), we consider the functional  J u(·) =



1

x 2 (t)dt.

(2.12)

0

Let U be the set of all functions u(·), given for t ∈ [0, 1] and such that the Lebesgue 1 integral 0 u 2 (t)dt exists. The problem is to find a function u ∗ (·) ∈ U , such that   J u ∗ (·) ≤ J u(·) ∀u(·) ∈ U

(2.13)

along trajectories of Eq. (2.11). The problem (2.11)–(2.12), (2.13) is called an optimal control problem. The set U is called a set of all admissible control functions in this problem, and any function u(·) ∈ U is called an admissible control of this problem. The function u ∗ (·) ∈ U is called a solution (an optimal control) of the problem (2.11)–(2.12), (2.13). The solution of Eq. (2.11), generated by the optimal control u ∗ (·) ∈ U , is an optimal  called trajectory of the problem (2.11)–(2.12), (2.13), while the value J u ∗ (·) is called an optimal value of the functional in this problem. Let us show that the problem (2.11)–(2.12), (2.13) does not have a solution. Let u(·) be any given admissible control of this problem. The solution of Eq. (2.11), generated by this control, has the form 

t

x(t) = x0 exp(t) +

exp(t − s)u(s)ds,

0

Due to (2.14), the value of the functional (2.12) is

t ∈ [0, 1].

(2.14)

12

2 Examples of Singular Extremal Problems and Some Basic Notions







1

J u(·) =





t

x0 exp(t) +

0

2 exp(t − s)u(s)ds

dt.

(2.15)

0

t Since u(·) ∈ U , then, for any t ∈ [0, 1], the Lebesgue integral 0 exp(−s)u(s)ds exists. Therefore, this integral, as a function of t, is an absolutely continuous function in the interval [0, 1]. Therefore, x(t), given by (2.14), also is an absolutely continuous function in the interval [0, 1]. Due to this observation and the condition x0 = 0, we directly can conclude the existence of the interval [0, t¯](0 < t¯ ≤ 1), such that x(t) = 0 for all t ∈ [0, t¯]. The latter means that the value J u(·) , given by (2.15), satisfies the inequality  J u(·) ≥ au ,

(2.16)

where au is some positive number depending on u(·). Now, let us consider the following control function, depending not only on time t but also on the parameter ε > 0 (ε  1):  u (t, ε) = −

  x0 t , exp − ε ε

t ∈ [0, 1].

(2.17)

For any ε > 0, this function is an admissible control of the problem (2.11)–(2.12), (2.13), i.e.,  u (·, ε) ∈ U . The solution  x (t, ε) of Eq. (2.11), generated by the control  u (t, ε), has the form    x0 t  x (t, ε) = ε exp(t) + exp − , 1+ε ε

t ∈ [0, 1].

(2.18)

Using (2.18), we obtain the value of the functional (2.12) for the control  u (t, ε) as  J  u (·, ε) =

x02 ε (1 + ε)2



    2ε 1 ε exp(2) − 1 + exp 1 − −1 2 ε−1 ε    

1 2 − exp − − 1 . (2.19) 2 ε

Calculating the limit of (2.19) for ε → +0, we obtain  u (·, ε) = +0. lim J 

ε→+0

(2.20)

˜ u )], the Therefore, there exists a number 0 < ε(a ˜ u )  1 such that, for all ε ∈ (0, ε(a following inequality is valid: J  u (·, ε) < au . The latter, along with the inequality (2.16), implies   J  u (·, ε) < J u(·) ∀ε ∈ (0, ε(a ˜ u )].

(2.21)

2.2 Academic Examples

13

Since u(·) is any given admissible control of the problem (2.11)–(2.12), (2.13), then the inequality (2.21) means the nonexistence of the solution (optimal control) to this problem. Remark 2.4 Quite similarly, it is shown the nonexistence of solution to the probalong trajectories of Eq. (2.11) by lem of the minimization of the functional (2.12)  a proper choice of control function u(·) ∈ u(t), t ∈ [0, 1] : the Lebesgue integral 1  0 u(t)dt exists ⊃ U .  Remark 2.5 It is important to note that, due to the form of the functional J u(·) (see Eq. (2.12)) and the limit equality (2.20),  inf J u(·) = 0.

(2.22)

u(·)∈U

However, this infimum value is not attained in the problem (2.11)–(2.12), (2.13), i.e., this optimal control problem is ill-posed or singular. Thus, similar to the previous subsection, the following question can be asked. Can we reformulate this problem in such a way that the reformulated problem will have a solution (an optimal control) u ∗ (·) such that    J ∗ = J u ∗ (·) = inf J u(·) . u(·)∈U

(2.23)

Like in Sect. 2.2.1, we present two ways of the reformulation of the problem (2.11)–(2.12), (2.13).

2.2.2.1

The First Way of the Reformulation of (2.11)–(2.12), (2.13)

In this way, we extend in some sense the set U of the admissible controls. Namely, = of this set, we consider the following set of the admissible controls: U instead    U c∈R cδ(t) .  satisfying the condition (2.23). Let us show the existence of the control u ∗ (·) ∈ U Namely, let us choose such a control as u ∗ (t) = −x0 δ(t),

t ∈ [0, 1].

(2.24)

Substitution of this control into (2.14) instead of u(t) yields ∗



x (t) = x0 exp(t) + 0

t

 exp(t − s) − x0 δ(s) ds

= x0 exp(t) − x0 exp(t) = 0,

t ∈ (0, 1].

Thus, the solution of Eq. (2.11), generated by the control (2.24), has the form

14

2 Examples of Singular Extremal Problems and Some Basic Notions

x ∗ (t) =



0, t ∈ (0, 1], x0 , t = 0,

meaning, due to (2.12) and (2.22), that the control (2.24) satisfies the condition (2.23), i.e., this control is the solution to the problem of the minimization of the functional (2.12) along trajectories of Eq. (2.11) within the extended set of the admissible . The control (2.24) is convenient rather for the theoretical analysis and controls U solution of the singular optimal control problem. Below, we present the second way of the reformulation of the problem (2.11)–(2.12), (2.13), which yields the solution more convenient for a practical implementation.

2.2.2.2

The Second Way of the Reformulation of (2.11)–(2.12), (2.13)

This way consists in introducing a new definition of solution to this problem. Namely, let us consider a sequence of functions {u k (·)}+∞ k=1 , satisfying the condition u k (·) ∈ U , (k = 1, 2, . . .). Definition 2.2 The sequence {u k (·)}+∞ k=1 is called a solution (or a minimizing sequence of controls) of the problem (2.11)–(2.12), (2.13), if the following limit equality is valid:   lim J u k (·)) = inf J u(·) = 0

k→+∞

u(·)∈U

(2.25)

along trajectories of Eq. (2.11). Remark 2.6 Due to Definition 2.2, the solution of the problem (2.11)–(2.12), (2.13) is not a single function u(·) ∈ U , but it is a sequence of functions from the set U , 

+∞ i.e., u ∗ (·) = {u k (·)} k=1 . Moreover, due to Definition 2.2, the minimum value of the functional J u(·) in the set U along trajectories of Eq. (2.11) is defined as

    J ∗ = J u ∗ (·) = J {u k (·)}+∞ k=1 = lim J u k (·) . k→+∞

(2.26)

Thus, due to Eq. (2.25), the value J ∗ satisfies the condition (2.23). Remark 2.7 It is important to note that the solution of the optimal control problem (2.11)–(2.12), (2.13) in the form of a minimizing sequence of controls, belonging to the set U , is more convenient for a practical implementation than the solution in the form of the abstract (generalized) Dirac delta-function δ(t). Similar to Sect. 2.2.1, let us consider the sequence of positive numbers {εk }+∞ k=1 such that limk→+∞ εk = 0. For this sequence, due to the limit equality (2.20), we have u (·, εk )}+∞ that the sequence of controls { k=1 satisfies the condition (2.25). Remember u (t, ε) is given by Eq. (2.17). Thus, the sequence { u (·, εk )}+∞ that the control  k=1 is a solution (a minimizing sequence of controls) of the problem (2.11)–(2.12), (2.13).

2.2 Academic Examples

15

It is important to note that this sequence has been obtained by rather a guess, but not by a rigorous constructive method. Here, we present the rigorous method (the regularization method) allowing to derive a minimizing sequence of controls of the problem (2.11)–(2.12), (2.13). This method consists in approximately replacing this singular optimal control problem with a new optimal control problem which has a solution in the set U , satisfying the first-order optimality conditions [4] (Sect. 5.2). Thus, the new optimal control problem is well posed (or regular). This new problem depends on a small parameter ε > 0, and it becomes the original problem (2.11)– (2.12), (2.13) when ε is replaced with zero. In the new (regularized) optimal control and the set of the admissible controls problem, we keep the differential equation  (2.11) U , while we replace the functional J u(·) , given by (2.12), with the functional   Jε u(·) = J u(·) + ε2



1

 u 2 (t)dt =

0

1



 x 2 (t) + ε2 u 2 (t) dt.

(2.27)

0

Also, we replace the definition (2.13) of the solution (the optimal control) in the problem (2.11)–(2.12), (2.13) with the following definition. For a given ε > 0, the control function u ∗ (·, ε) ∈ U is called a solution (an optimal control) to the problem, consisting of Eq. (2.11), the set U of the admissible controls and the functional (2.27), if u ∗ (·, ε) satisfies the following condition:   Jε u ∗ (·, ε) ≤ Jε u(·) ∀u(·) ∈ U

(2.28)

along trajectories of Eq. (2.11). Let us solve the optimal control problem (2.11), (2.27), (2.28). By virtue of the results of [4] (Sect. 5.2), the optimal control in this problem has the form u¯ ∗ (t, ε) = −

1 λ(t, ε), 2ε2

t ∈ [0, 1],

(2.29)

where the function λ(t, ε) is obtained from the solution of the boundary-value problem d x(t) 1 = x(t) − 2 λ(t), x(0) = x0 , dt 2ε dλ(t) = −2x(t) − λ(t), λ(1) = 0. dt Solving this problem and substituting the λ-component of the obtained solution into Eq. (2.29), we have (after a routine algebra) the optimal control of the problem (2.11), (2.27), (2.28) in the form     x0 t t u¯ (t, ν) = − exp − − x0 exp − + c(ν)u(t, ¯ ν), t ∈ [0, 1], (2.30) ν ν ν ∗

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2 Examples of Singular Extremal Problems and Some Basic Notions

where ν=√

c(ν) =  u(t, ¯ ν) =

ε 1 + ε2

,

(2.31)

x0 (1 + ν) exp(−2/ν) , 1 − ν + (1 + ν) exp(−2/ν)

       1 t 1 t − 1 exp + + 1 exp − , ν ν ν ν

(2.32)

t ∈ [0, 1]. (2.33)

Equation (2.31) yields that ν > 0 for all ε > 0 and lim ν = +0.

ε→+0

Moreover, using Eq. (2.32), we obtain by direct calculation that the value c(ν) satisfies the inequality   2 0 < c(ν) < 2.4x0 exp − ∀ν ∈ (0, 0.4]. ν

(2.34)

The later, along with Eq. (2.33), yields the following inequality for all t ∈ [0, 1] and ν ∈ (0, 0.4]:      1 1 0 < c(ν)u(t, ¯ ν) < 2.4x0 exp − + exp − . 2ν ν

(2.35)

Thus, the third addend on the right-hand side of Eq. (2.30) is exponentially neglected for ν → +0, yielding an exponentially neglected contribution to the solution of Eq. ¯ ν) from (2.11) with u(t) = u¯ ∗ (t, ν). Therefore, we can remove the addend cu(t, (2.30). Due to this removing, we obtain the control u ∗ (t, ν) = −

    x0 t t − x0 exp − , t ∈ [0, 1], ν ∈ (0, 0.4]. (2.36) exp − ν ν ν

Let {νk }+∞ (k = 1, 2, . . .), k=1 be a sequence of numbers such that νk ∈ (0, 0.4],  ∗ +∞ and limk→+∞ νk = 0. Let us show that the sequence of controls u (·, νk ) k=1 is a solution (a minimizing sequence of controls) of the problem (2.11)–(2.12), (2.13). It is clear that u ∗ (·, νk ) ∈ U , (k = 1, 2, . . .). Substitution of u ∗ (t, νk ) into (2.14) instead of u(t) yields the following solution of Eq. (2.11), generated by the control u ∗ (t, νk ):

2.2 Academic Examples

17

Fig. 2.1 Graph of the control function u ∗ (t, ν)



∗

x (t, νk ) = x0 exp(t) − x0



1 +1 νk



 s exp(t − s) exp − ds νk 0   t , t ∈ [0, 1]. = x0 exp − νk t

Furthermore, substituting x ∗ (t, νk ) into the functional (2.12) instead of x(t), we directly have  J u ∗ (·, νk ) = x02

 0

1



2t exp − νk



   2 x02 νk 1 − exp − dt = , 2 νk

which yields  lim J u ∗ (·, νk ) = 0.

k→+∞

 +∞ The later, along with Definition 2.2 (see Eq. (2.25)), means that u ∗ (t, νk ) k=1 is a solution (a minimizing sequence of controls) of the problem (2.11)–(2.12), (2.13). In Fig. 2.1, the graphs of the control function u ∗ (t, ν) are depicted for x0 = 2 and various values of ν > 0. It is seen that, for decreasing ν, u ∗ (t, ν) approaches an impulse-like function. In Fig. 2.2, the graphs of the function x ∗ (t, ν) are depicted for x0 = 2 and various values of ν > 0. It is seen that, for decreasing ν, x ∗ (t, ν) as a function of t tends to zero point-wise in the interval (0, 1].  In Table 2.1, the values of J u ∗ (·, ν) are presented for x0 = 2 and various  values of ν > 0. It is seen that J u ∗ (·, ν) decreases for decreasing ν, and J u ∗ (·, ν) approaches zero for ν approaching zero.

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2 Examples of Singular Extremal Problems and Some Basic Notions

Fig. 2.2 Graph of the function x ∗ (t, ν)

 Table 2.1 Value of J u ∗ (·, ν) ν

 J u ∗ (·, ν)

0.4

0.2

0.1

0.05

0.01

0.001

0.7946

0.4000

0.2000

0.1000

0.0200

0.0020

2.2.3 Saddle Point of a Function of Two Variables 

Consider a function of two scalar variables f (t, τ ), defined on the set Ω = {(t, τ ) : t ∈ Ωt , τ ∈ Ωτ }. We consider the case where this function is simultaneously minimized with respect to t and maximized with respect to τ in the set Ω. Definition 2.3 For a given t¯ ∈ Ωt , the value f t (t¯) = sup f (t¯, τ ) τ ∈Ωτ

(2.37)

is called the guaranteed result of this t¯ for the function f (t, τ ), (t, τ ) ∈ Ω. Definition 2.4 For a given τ¯ ∈ Ωτ , the value f τ (τ¯ ) = inf f (t, τ¯ ) t∈Ωt

is called the guaranteed result of this τ¯ for the function f (t, τ ), (t, τ ) ∈ Ω. Consider a pair (t ∗ , τ ∗ ) ∈ Ω.

(2.38)

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19

Definition 2.5 The pair (t ∗ , τ ∗ ) is called a saddle point of the function f (·, ·) in the set Ω if the guaranteed results of t ∗ and τ ∗ for this function are equal to each other, i.e., f t (t ∗ ) = f τ (τ ∗ ).

(2.39)

If this equality is valid, then the value f ∗ = f t (t ∗ ) = f τ (τ ∗ )

(2.40)

is called a saddle-point value of the function f (·, ·) in the set Ω. We call the problem of searching a saddle point for the function f (t, τ ), (t, τ ) ∈ Ω, the saddle-point problem for this function. Proposition 2.1 If the saddle point (t ∗ , τ ∗ ) for the function f (t, τ ) in the set Ω exists, then the following inequality is valid: f (t ∗ , τ ) ≤ f ∗ ≤ f (t, τ ∗ ) ∀t ∈ Ωt , τ ∈ Ωτ .

(2.41)

f ∗ = f (t ∗ , τ ∗ ).

(2.42)

Moreover,

Proof Let us start with the proof of the inequality (2.41). The left-hand side of this inequality directly follows from Eq. (2.37) and the equality f ∗ = f t (t ∗ ), given in (2.40). The right-hand side of (2.41) follows immediately from Eq. (2.38) and the equality f ∗ = f τ (τ ∗ ), given in (2.40). Thus, the validity of the inequality (2.41) is proved. Proceed to the proof of the equality (2.42). The left-hand side of the inequality (2.41) yields f (t ∗ , τ ∗ ) ≤ f ∗ .

(2.43)

Similarly, the right-hand side of (2.41) yields f (t ∗ , τ ∗ ) ≥ f ∗ .

(2.44)

Now, the inequalities (2.43) and (2.44) directly imply the validity of the equality (2.42).   Consider a particular case of the function f (t, τ ), (t, τ ) ∈ Ω. Namely, f (t, τ ) = where

1 − τ 2, t +1

(t, τ ) ∈ Ω,

(2.45)

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2 Examples of Singular Extremal Problems and Some Basic Notions

Ω = Ωt × Ωτ , Ωt = {t : t ∈ [0, +∞)}, Ωτ = {τ : τ ∈ (−∞, +∞)}. (2.46) Due to Definition 2.3, we have the following equality for the function f (t, τ ) given by (2.45)–(2.46): f t (t¯) =

1 , t¯ + 1

(2.47)

where t¯ is any given point from the set Ωt . Similarly, due to Definition 2.4, we have for this function f τ (τ¯ ) = −τ¯ 2 ,

(2.48)

where τ¯ is any given point from the set Ωτ . Let us show that the function f (t, τ ), given by (2.45)–(2.46), does not have a saddle point. Assume the opposite, i.e., that this function has a saddle point (t ∗ , τ ∗ ) ∈ Ωt × Ωτ . In such a case, due to Definition 2.5 and Eqs. (2.47)–(2.48), we obtain  2 1 = − τ∗ , t∗ + 1

(2.49)

which is a contradictive equality, because its left-hand side is positive, while the righthand side is nonpositive. This contradiction implies the nonexistence of a saddle point for the function (2.45)–(2.46). Remark 2.8 It is important to note that, by virtue of Eqs. (2.47) and (2.48), we have inf f t (t¯) = 0 = sup f τ (τ¯ ) = f τ (0),

t¯∈Ωt

τ¯ ∈Ωτ

or equivalently lim f t (t¯) = f τ (0).

t¯→+∞

In this equality, the limit value is not attained for any t¯ ∈ Ωt . Thus, the saddle-point problem for the function (2.45)–(2.46) is ill-posed or singular. Like in Sects. 2.2.1 and 2.2.2, we consider two ways to improve this bad situation. 2.2.3.1

The First Way of the Reformulation of the Saddle-Point Problem for the Function (2.45)–(2.46)

This way consists in some extension of the set Ω of possible saddle points for the function. Namely, we do not change the set Ωτ of a possible τ -component in the saddle point, while we consider the following set instead of Ωt for a possible

2.2 Academic Examples

21

 t = Ωt {+∞}. Thus, we obtain the new set t-component in the saddle point: Ω =Ω t × Ωτ of possible saddle points for the function (2.45). Using this new set, Ω we replace the saddle-point problem for the function (2.45)–(2.46) with the new  For this new problem, the saddle-point problem for the function (2.45) in the set Ω.  satisfies the equality (2.49), i.e., this pair is a saddle pair (t ∗ , τ ∗ ) = (+∞, 0) ∈ Ω  and the corresponding saddle-point value point for the function (2.45) in the set Ω f ∗ equals zero. The obtained saddle point, being a simple and elegant solution from the theoretical viewpoint, is, however, inconvenient for a practical implementation. Below, we present the second way of the reformulation of the saddle-point problem for the function (2.45)–(2.46), which yields the solution more convenient for the implementation.

2.2.3.2

The Second Way of the Reformulation of the Saddle-Point Problem for the Function (2.45)–(2.46)

This way consists in introducing a new definition of solution to this problem. Namely, τ }+∞ let us consider a sequence of the pairs of numbers { tk , k=1 , satisfying the condition  τ ∈ Ωτ , i.e., ( tk , τ ) ∈ Ω, (k = 1, 2, . . .). tk ∈ Ωt , (k = 1, 2, . . .),  τ }+∞ Definition 2.6 The sequence { tk , k=1 is called a solution (or a saddle-point sequence) of the saddle-point problem for the function (2.45)–(2.46) if (a) there tk ); exists limk→+∞ f t ( (b) the following equality is valid: tk ) = f τ ( τ ). lim f t (

k→+∞

(2.50)

In this case, the value  tk ) = f τ ( τ) f = lim f t ( k→+∞

(2.51)

is called a saddle-point value of the function (2.45)–(2.46). Remark 2.9 Due to Definition 2.6, the solution of the saddle-point problem for the function (2.45)–(2.46) is not a single pair (t, τ ) from the set Ω, but it is a sequence  tk , τ }+∞ of pairs from this set, i.e., (t ∗ , τ ∗ ) = { k=1 . Moreover, due to Definition 2.6 and Proposition 2.1, the saddle-point value of the function (2.45)–(2.46) can be formally represented in the form    tk , τ }+∞ f ∗ = f (t ∗ , τ ∗ ) = f { k=1 = f .

(2.52)

Remark 2.10 It should be noted that the solution of the saddle-point problem for the function (2.45)–(2.46) in the form of a saddle-point sequence (a sequence of pairs of numbers) is more convenient for a practical implementation than the solution in the form of the pair, in which one entry is the abstract element +∞.

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2 Examples of Singular Extremal Problems and Some Basic Notions

Now, let us discuss how to obtain a saddle-point sequence for the function (2.45)– (2.46). First of all, it should be noted that, for this simple function, it is quite easy τ } = {k, 0}, (k = 1, 2, . . .) is a to find such a sequence. Namely, the sequence { tk , saddle-point sequence for the function (2.45)–(2.46). However, this sequence has been derived by rather a guess, but not by a rigorous constructive method. Here, we present such a method, called the regularization method. Due to this method, we approximately replace the singular saddle-point problem for the function (2.45)– (2.46) with the saddle-point problem for a new function which has a solution (a saddle point) in the set Ω, satisfying the first-order extremum condition with respect to both independent variables [6]. Thus, the new saddle-point problem is well posed (or regular). This new problem depends on a small parameter ε > 0, and it becomes the original saddle-point problem for the function (2.45)–(2.46) when ε is replaced with zero. In the new saddle-point problem, we keep the set Ω to be unchanged, while we choose the following new function instead of the function f (·, ·): f ε (t, τ ) = f (t, τ ) + ε2 t =

1 + ε2 t − τ 2 , t +1

(t, τ ) ∈ Ω.

(2.53)

For any given t¯ ∈ Ωt and τ¯ ∈ Ωτ , let us calculate the guaranteeing results f ε,t (t¯) and f ε,τ (τ¯ ) for the function f ε (t, τ ), (t, τ ) ∈ Ω. Due to Definition 2.3 and the form of this function, we immediately have f ε,t (t¯) =

1 + ε2 t¯. t¯ + 1

(2.54)

It should be noted that f ε,t (t¯) is achieved for τ = 0, and this maximum point of the function f ε (t¯, τ ) satisfies the first-order extremum condition. Proceed to obtaining f ε,τ (τ¯ ). This value is obtained similar to the minimum value of the function (2.6). Thus, we obtain the following minimum point of the function f ε (t, τ¯ ) in the set Ωt : tmin (ε) = 1/ε − 1, ε ∈ (0, 1). This point satisfies the first-order extremum condition. Substitution of t = tmin (ε) into f ε (t,τ¯ ) yields the minimum value (and, therefore, the infimum value) of this function f tmin (ε), τ¯ = f ε,τ (τ¯ ) = 2ε − ε2 − τ¯ 2 , ε ∈ (0, 1). Now, equating f ε,t (t¯) and f ε,τ (τ¯ ) with each other, we obtain (due to Definition 2.5) the following equation for obtaining a saddle point for the function (2.53): 1 + ε2 t¯ = 2ε − ε2 − τ¯ 2 , t¯ + 1

ε ∈ (0, 1).

(2.55)

For any given ε ∈ (0, 1), (2.55) is a scalar equation with two unknowns t¯ and τ¯ . ¯ We seek a solution of this equation t (ε), τ¯ (ε) ∈ Ω, valid for all ε ∈ (0, 1). Since the left-hand side of (2.55) is positive and the right-hand side becomes negative for any given τ¯  = 0 and all sufficiently small ε ∈ (0, 1), then the necessary condition for the existence of such a solution is τ¯ (ε) ≡ 0 for all ε ∈ (0, 1). Substituting τ¯ = τ¯ (ε) = 0 into Eq. (2.55) and solving the obtained equation with respect to t¯ yields the

2.2 Academic Examples

23

unique solution t¯ = t¯(ε) = 1/ε − 1 ∈ Ωt for all ε ∈ (0, 1). Thus, the unique solution of Eq. (2.55), valid for all ε ∈ (0, 1), is t¯ = t¯(ε) = 1/ε − 1, τ¯ = τ¯ (ε) ≡ 0 ∈ Ω. Therefore, for any given ε ∈ (0, 1), the point t¯(ε), τ¯ = (1/ε − 1, 0) ∈ Ω is the unique saddle point of the function (2.53). Let {εk }+∞ k=1 be the sequence of numbers such that εk ∈ (0, 1), (k = 1, 2, . . .), and limk→+∞ εk = 0. Using (2.47)–(2.48), we have for this sequence  lim f t t¯(εk ) = lim εk = 0 = f τ (τ¯ ) = f τ (0).

k→+∞

k→+∞

(2.56)

τ }+∞ Thus, due to Definition 2.6 and Eq. (2.56), the sequence { tk , k=1 = {1/εk − +∞ 1, 0}k=1 is a solution (a saddle-point sequence) of the saddle-point problem for the function (2.45)–(2.46). The saddle-point value of this function is zero.

2.2.4 Zero-Sum Differential Game Consider the scalar differential equation d x(t) = x(t) + u(t) + v(t), dt

t ∈ [0, 1],

x(0) = x0  = 0,

(2.57)

where x(t), t ∈ [0, 1] is a state variable; u(t) and v(t), t ∈ [0, 1] are control functions. Here, in contrast with the example of Sect. 2.2.2, the differential equation (2.57) is controlled by two decision makers. The first decision maker has the control function u(t) on its disposal, while the second decision maker has the control function v(t) on its disposal. Along with Eq. (2.57), we consider the functional  J u(·), v(·) =



1



 x 2 (t) − 2v 2 (t) dt.

(2.58)

0

Let U be the set of all functions u(·), given for t ∈ [0, 1] and such that the Lebesgue 1 integral 0 u 2 (t)dt exists. Similarly, let V be the set of all functions v(·), given for 1 t ∈ [0, 1] and such that the Lebesgue integral 0 v 2 (t)dt exists. The objective of the first decision maker is to minimize the functional (2.58) along trajectories of Eq. (2.57) by a proper choice of u(·) ∈ U . The objective of the second decision maker is to maximize the functional (2.58) along trajectories of Eq. (2.57) by a proper choice of v(·) ∈ V . In what follows of this subsection, we call the first decision maker and the second decision maker as the minimizer and the maximizer, respectively. The sets U and V are called the sets of the admissible controls of the minimizer and the maximizer, respectively. The problem, consisting of the differential equation (2.57), the functional (2.58), the sets of the admissible controls U and V , and the objectives of the decision makers,

24

2 Examples of Singular Extremal Problems and Some Basic Notions

is called a zero-sum differential game. In what follows, for the sake of briefness, we denote this game by the numbers of its differential equation and functional, i.e., (2.57)–(2.58). Also, we assume the following information pattern in this game. The minimizer knows about the set V , but it does not know about the choice of the control v(·) ∈ V by the maximizer. Similarly, the maximizer knows about the set U , but it does not know about the choice of the control u(·) ∈ U by the minimizer. Definition 2.7 For a given u(·) ¯ ∈ U , the value  Ju (u(·)) ¯ = sup J u(·), ¯ v(·) v(·)∈V

(2.59)

is called the guaranteed result of this minimizer’s control function u(·) ¯ in the game (2.57)–(2.58). ¯ ∈ V , the value Definition 2.8 For a given v(·)  Jv (v(·)) ¯ = inf J u(·), v(·) ¯ u(·)∈U

(2.60)

is called the guaranteed result of this maximizer’s control function v(·) ¯ in the game (2.57)–(2.58).  Consider a pair of functions u ∗ (·), v ∗ (·) ∈ U × V .  Definition 2.9 The pair u ∗ (·), v ∗ (·) is called a saddle-point solution of the game (2.57)–(2.58) if the guaranteed results of u ∗ (·) and v ∗ (·) in this game are equal to each other, i.e., Ju (u ∗ (·)) = Jv (v ∗ (·)).

(2.61)

If this equality is valid, then the value J ∗ = Ju (u ∗ (·)) = Jv (v ∗ (·))

(2.62)

is called a saddle-point value of the game (2.57)–(2.58). Let us show that the game (2.57)–(2.58) does not have a saddle-point solution.  Assume the opposite, i.e., this game has a saddle-point solution u ∗ (·), v ∗ (·) ∈ U × V . Now, let us calculate the values Ju (u ∗ (·)) and Jv (v ∗ (·)). Let us start with Ju (u ∗ (·)). The calculation of this value is equivalent to solution of the optimal control problem, consisting of the differential equation d x(t) = x(t) + u ∗ (t) + v(t), dt and the functional

t ∈ [0, 1],

x(0) = x0  = 0

(2.63)

2.2 Academic Examples

25



J u (·), v(·) = ∗



1



 x 2 (t) − 2v 2 (t) dt,

(2.64)

0

for which the supremum is sought with respect to v(·) ∈ V along trajectories of Eq. (2.63). Let us show that this supremum is larger than some positive number. Indeed, for v(t) ≡ 0 ∈ V , Eq. (2.63) and the functional (2.64) become d x(t) = x(t) + u ∗ (t), dt

t ∈ [0, 1],

x(0) = x0 = 0

(2.65)

and 





∗

1

J u (·), 0 =

x 2 (t)dt.

(2.66)

0

Since u ∗ (·) ∈ U , the solution x(t), t ∈ [0, 1] of Eq. (2.65) is an absolutely continuous function. Moreover, since x0  = 0, there exists t¯ ∈ (0, 1] such that x(t)  = 0 for all   t ∈ [0, t¯]. Therefore, a0∗ = J u ∗ (·), 0 > 0, yielding  1 Ju (u ∗ (·)) = sup J u ∗ (·), v(·) > a0∗ > 0. 2 v(·)∈V

(2.67)

Proceed to the calculation of Jv (v ∗ (·)). This value is the infimum with respect to u(·) ∈ U of the functional  J u(·), v ∗ (·) =



1



 2  x 2 (t) − 2 v ∗ (t) dt

(2.68)

0

along trajectories of the equation d x(t) = x(t) + u(t) + v ∗ (t), dt

t ∈ [0, 1],

x(0) = x0 = 0.

(2.69)

Let us consider the following control function, depending not only on time t but also on the parameter ε > 0 (ε  1):   x0 t  u (t, ε) = −v (t) − , exp − ε ε ∗

t ∈ [0, 1].

u (t, ε) ∈ U . For any ε > 0,  Substituting  u (t, ε) into (2.69) instead of u(t), we obtain the equation   d x(t) x0 t = x(t) − exp − , dt ε ε

t ∈ [0, 1],

x(0) = x0  = 0,

(2.70)

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2 Examples of Singular Extremal Problems and Some Basic Notions

which yields the solution    x0 t ε exp(t) + exp − ,  x (t, ε) = 1+ε ε

t ∈ [0, 1].

(2.71)

Using (2.68) and (2.71), we have  J  u (·, ε), v ∗ (·) =

x02 ε (1 + ε)2

    1 2ε ε exp 1 − −1 exp(2) − 1 + 2 ε−1 ε    

 1  ∗ 2 1 2 − exp − v (t) dt. (2.72) −1 −2 2 ε 0



Calculating the limit of (2.72) for ε → +0, we obtain  lim J  u (·, ε), v ∗ (·) = −2

ε→+0



1



2 v ∗ (t) dt ≤ 0.

(2.73)

0

˜ 0∗ )], Therefore, there exists a number 0 < ε(a ˜ 0∗ )  1 such that, for all ε ∈ (0, ε(a ∗ ∗ u (·, ε), v (·) < (1/2)a0 . Therefore, for all ε ∈ the following inequality is valid: J  (0, ε(a ˜ 0∗ )],   1 Jv (v ∗ (·)) = inf J u(·), v ∗ (·) ≤ J  u (·, ε), v ∗ (·) < a0∗ . u(·)∈U 2 ∗ ∗ The latter, along with the inequality (2.67), Jv (v (·)) < Ju (u (·)), i.e., the  ∗ implies ∗ equality (2.61) is not valid for the pair u (·), v (·) ∈ U × V . Thus, the assumption that this pair is a saddle-point solution of the game (2.57)–(2.58) is wrong, meaning that this game does not have a saddle-point solution.  Now, let us find out what is going on with the values inf u(·)∈U Ju u(·) and supv(·)∈V Jv v(·) . We start with the first value. To calculate this value, we choose u(·) ¯ ∈ U as a function, depending not only on time t but also on the small parameter ε > 0 (ε  1), i.e.,   x0 t u(t, ¯ ε) = − exp − , t ∈ [0, 1]. (2.74) ε ε

This function belongs to the set U for any ε > 0. Substituting (2.74) into Eq. (2.57) instead of u(t) and solving the resulting equa¯ ε) for any v(·) ∈ V and ε > 0 tion, we obtain its solution x(t) = x(t, x(t, ¯ ε) =

    t x0 t exp(t − s)v(s)ds, t ∈ [0, 1]. ε exp(t) + exp − + 1+ε ε 0

2.2 Academic Examples

27

Further, the substitution x(t) = x(t, ¯ ε) into the functional (2.58) yields after a routine algebra the following expression:  J u(·, ¯ ε), v(·) = b1 (ε) + b2 (ε) + b3 (ε) + b4 − 2b5 ,

(2.75)

where  x02 ε ε b1 (ε) = exp(2) − 1 2 (1 + ε) 2        

2ε 1 1 2 + exp 1 − −1 − exp − −1 , ε−1 ε 2 ε

b2 (ε) =

b3 (ε) =



2x0 ε 1+ε

2x0 1+ε



1

t

exp(t) 0



1 0



1

b4 = 0

 exp(t − s)v(s)ds dt,

(2.76)

(2.77)

0

   t  t exp − exp(t − s)v(s)ds dt, ε 0 

t

(2.78)

2 exp(t − s)v(s)ds

dt,

(2.79)

0

 b5 =

1

v 2 (t)dt.

(2.80)

0

Let us estimate the values bl (ε) (l = 1, 2, 3) and b4 . From the expression for b1 (ε) (see (2.76)), we directly have the existence of a positive number ε1 such that, for all ε ∈ (0, ε1 ], the following inequality is valid: 0 < b1 (ε) ≤ c1 x02 ε,

(2.81)

where c1 > 0 is some constant independent of ε. To estimate the value b2 (ε), first, we transform the double integral in its expression (see (2.77)). Using the integration-by-parts formula, we obtain 



1

exp(t) 0

t





exp(t − s)v(s)ds dt =

0

exp(2t) 0

=



1

1 2

t

 exp(−s)v(s)ds dt

0



1 0



exp(2 − t) − exp(t) v(t)dt.

28

2 Examples of Singular Extremal Problems and Some Basic Notions

Substituting the right-hand side expression of this equality into (2.77) instead of the double integral and applying the Cauchy-Bunyakovsky-Schwarz integral inequality [19] to the resulting expression for b2 (ε), we obtain    |x0 |ε  1  exp(2 − t) − exp(t) v(t)dt  |b2 (ε)| =  1+ε 0 1/2  1 1/2  1  2 |x0 |ε 2 exp(2 − t) − exp(t) dt v (t)dt ≤ 1+ε 0 0  1/2  1 1/2 |x0 | 0.5 exp(4) − 2 exp(2) − 0.5 ε = v 2 (t)dt . 1+ε 0 The latter directly yields the inequality  |b2 (ε)| ≤ c2 ε

1

1/2 v (t)dt 2

,

ε > 0,

(2.82)

0

 1/2 where c2 = |x0 | 0.5 exp(4) − 2 exp(2) − 0.5 . Proceed to b3 (ε). To estimate this value, first, we transform the double integral in its expression (2.78). Using the integration-by-parts formula, we obtain for all ε ∈ (0, 1)    t  t exp − exp(t − s)v(s)ds dt = ε 0 0      t  1 1 exp −t exp(−s)v(s)ds dt = −1 ε 0 0       1 ε 1 t − exp 1 − − t v(t)dt. exp − 1−ε 0 ε ε 

1

Substituting the right-hand side expression of this equality into (2.78) instead of the double integral and applying the Cauchy-Bunyakovsky-Schwarz integral inequality [19] to the resulting expression for b3 (ε), we obtain for all ε ∈ (0, 1)         2|x0 |ε  1 1 t |b3 (ε)| = − exp 1 − − t v(t)dt  exp −  2 1−ε ε ε 0   1/2 2 1/2  1    1 1 t 2|x0 |ε 2 dt v (t)dt − t − exp 1 − exp − ≤ 1 − ε2 ε ε 0 0  1 1/2 = |x0 |α(ε) v 2 (t)dt , (2.83) 0

where

2.2 Academic Examples

α(ε) =

2ε 1 − ε2

29

         2 2ε 2 1 ε 1 − exp − + exp − − exp 1 − 2 ε 1+ε ε ε      1 2 2 − exp − − exp 2 − . 2 ε ε

This expression for α(ε) yields the existence of a positive number ε3 < 1 such that, for all ε ∈ (0, ε3 ], the following inequality is valid: |α(ε)| ≤ c3 ε,

(2.84)

where c3 > 0 is some constant independent of ε. The inequalities (2.83) and (2.84) imply the estimate of b3 (ε) 

1

|b3 (ε)| ≤ c3 |x0 |ε

1/2 v (t)dt 2

,

ε ∈ (0, ε3 ].

(2.85)

0

Now, let us treat b4 . First, let us estimate the integrand of the outer integral in (2.79). Using the Cauchy-Bunyakovsky-Schwarz integral inequality [19], we obtain for any given t ∈ [0, 1] 

t

2



2

t

exp(t − s)v(s)ds = exp(2t) exp(−s)v(s)ds 0   t  t 2 1 v (s)ds. (2.86) ≤ exp(2t) exp(−2s)ds v 2 (s)ds = exp(2t) − 1 2 0 0 0 0 t

Replacing the integrand of the outer integral in (2.79) with the right-hand side expression in (2.86) and using the integration-by-parts formula, we have   t  1 1 2 0 ≤ b4 ≤ v (s)ds dt exp(2t) − 1 2 0 0   1     1 1 1 1 2 2 = v (t)dt − exp(2) − 1 exp(2t) − t v (t)dt 2 2 2 0 0  1  1 1 ≤ v 2 (t)dt. (2.87) exp(2) − 1 2 2 0 Using Eqs. (2.75), (2.80) and the inequalities (2.81), (2.82), (2.85), (2.87), we  obtain the following inequality for all ε ∈ 0, min{ε1 , ε3 } and v(·) ∈ V : 

J u(·, ¯ ε), v(·) ≤ c1 x02 ε + γ1 ε where



1

1/2 v (t)dt 2

0



1

− γ2 0

v 2 (t)dt,

(2.88)

30

2 Examples of Singular Extremal Problems and Some Basic Notions

γ1 = c2 + c3 |x0 | > 0,

  1 1 5 − exp(2) > 0. 2 2

γ2 =

(2.89)

Based on the latter, the inequality (2.88) can be rewritten as  γ 2 ε2 √ J u(·, ¯ ε), v(·) ≤ c1 x02 ε + 1 − γ2 4γ2 ≤ c1 x02 ε +

γ12 ε2 4γ2



1

1/2 v (t)dt) 2

0

γ1 ε − √ 2 γ2

!2

  ∀ε ∈ 0, min{ε1 , ε3 } , v(·) ∈ V . (2.90)

As it was aforementioned,  sup J u(·), v(·) > 0

v(·)∈V

∀u(·) ∈ U.

(2.91)

Using this inequality and the inequality (2.90), we have    γ 2 ε2 ∀ε ∈ 0, min{ε1 , ε3 } , 0 < sup J u(·, ¯ ε), v(·) ≤ c1 x02 ε + 1 4γ2 v(·)∈V

(2.92)

which yields   ¯ ε) = 0. ¯ ε), v(·) = lim Ju u(·, lim sup J u(·,

ε→+0 v(·)∈V

ε→+0

(2.93)

This limit equality, along with the inequality (2.91) and Definition 2.7, implies inf

  sup J u(·), v(·) = inf Ju u(·) = 0.

u(·)∈U v(·)∈V

u(·)∈U

(2.94)

However, due to (2.91), the infimum in this equality  is not attained for any u(·) ∈ U . Now, let us calculate the value supv(·)∈V Jv v(·) , which due to Definition 2.8  equals the value supv(·)∈V inf u(·)∈U J u(·), v(·) . Let u(·) ∈ U and v(·) ∈ V be any given controls of the minimizer and the maximizer, respectively, in the game (2.57)–(2.58). Let xu,v (t), t ∈ [0, 1], be the solution of Eq. (2.57), generated by these controls. Since x0  = 0, there exists tu,v ∈ (0, 1] such that xu,v (t)  = 0 for all t ∈ [0, tu,v ]. Therefore, 

1 0

2 xu,v (t)dt > 0,

and 

J u(·), v(·) =

 0

1



2 xu,v (t)

 − 2v (t) dt > −2



1

2

0

v 2 (t)dt.

2.2 Academic Examples

31

The latter yields 





Jv v(·) = inf J u(·), v(·) ≥ −2 u(·)∈U



1

v 2 (t)dt.

(2.95)

0

Choosing the control u(·) ∈ U in the form u(t) = u(t, ε) = −v(t) −

  x0 t exp − , ε ε

t ∈ [0, 1], ε > 0,

we obtain (similar to (2.73))  lim J u(·, ε), v(·) = −2

ε→+0



1

v 2 (t)dt.

0

This limit equality, along with the inequality (2.95), means that   Jv v(·) = inf J u(·), v(·) = −2 u(·)∈U



1

v 2 (t)dt.

(2.96)

0

Therefore,   sup Jv v(·) = sup inf J u(·), v(·) = 0 = Jv (0), v(·)∈V u(·)∈U

v(·)∈V

(2.97)

meaning that the supremum value with respect to v(·) ∈ V is attained for v(t) ≡ 0, t ∈ [0, 1]. From Eqs. (2.94) and (2.97), we have   inf Ju u(·) = sup Jv v(·) = Jv (0) = 0,

(2.98)

 ¯ ε) = Jv (0) = 0. lim Ju u(·,

(2.99)

u(·)∈U

v(·)∈V

or equivalently, ε→+∞

It is seen that in the equality (2.98) the supremum value is attained for v(·) ∈ V , namely for v(t) ≡ 0, while the infimum value is not attained for any u(·) ∈ U . Similarly, in the equality (2.99) the limit value is not attained for any u(·) ∈ U . This observation, along with the above-established absence of the saddle-point solution to the game (2.57)–(2.58), means that this game is ill-posed or singular. However, this situation can be improved by a proper reformulation of the game (2.57)–(2.58). Like in Sects. 2.2.1, 2.2.2 and 2.2.3, we consider two ways of such a reformulation.

32

2.2.4.1

2 Examples of Singular Extremal Problems and Some Basic Notions

The First Way of the Reformulation of the Game (2.57)–(2.58)

Similar to the previous subsections, in this way we extend in some sense the set U × V of the admissible pairs of controls in the game. As was aforementioned, the supremum value in the equality (2.98) is attained for v(·) ∈ V , while the infimum value does not attained for any u(·) ∈ U . Therefore, we do not change the set V of the admissible controls of the maximizer. However, we extend the set U of the we considerthe admissible controls of the minimizer. Namely, instead of this set,    following set of the admissible minimizer’s controls: U = U c∈R cδ(t) .   × V satisfying Let us show the existence of the pair of controls u ∗ (·), v ∗ (·) ∈ U the condition (2.61). Namely, let us choose u ∗ (t) = −x0 δ(t),

v ∗ (t) = 0,

t ∈ [0, 1].

(2.100)

  First, let us calculate the values Ju u ∗ (·) and Jv v ∗ (·) . We start with the first value. Substituting u ∗ (t) into (2.57) instead of u(t) and solving the resulting equation yield its solution for any v(·) ∈ V  t

0 exp(t − s)v(s)ds, x0 ,

x(t) =

t ∈ (0, 1], t = 0.

Thus, for any v(·) ∈ V , J u (·), v(·) = 

∗



1 0



t

2 exp(t − s)v(s)ds



1

dt − 2

0

v 2 (t)dt. (2.101)

0

Using this equation, as well as Eq. (2.79) and the inequality (2.87), we obtain  J u ∗ (·), v(·) ≤ −γ2



1

v 2 (t)dt ≤ 0

∀v(·) ∈ V ,

(2.102)

0

where the positive number γ2 is given in (2.89). On the other hand, using (2.101), we have  J u ∗ (·), 0 = 0. This equality and the inequality (2.102) yield immediately   Ju u ∗ (·) = sup J u ∗ (·), v(·) = 0, v(·)∈V

(2.103)

and this supremum value is attained for v(t) ≡ 0, t ∈ [0, 1]. Proceed to the value Jv v ∗ (·) . First of all, let us note that, due to (2.58), (2.60) and (2.100),

2.2 Academic Examples

33

  Jv v ∗ (·) = Jv (0) = inf J u(·), 0 ≥ 0. u(·)∈U

(2.104)

Furthermore, the solution of Eq. (2.57), generated by u(·) = u ∗ (·) and v(t) = v ∗ (t) = 0, t ∈ [0, 1], is  x(t) =

0, t ∈ (0, 1], x0 , t = 0.

Therefore,  J u ∗ (·), 0 = 0. This equality, along with the inequality (2.104), yields  Jv (0) = inf J u(·), 0 = 0, u(·)∈U

(2.105)

and the infimum value is attained for u(·) = u ∗ (·).  Equations (2.103) and (2.105) mean that the pair u ∗ (·), v ∗ (·) , given by (2.100), satisfies the condition (2.61). Thus, this pair is the saddle-point solution of the reformulated game (2.57)–(2.58) where the set of the admissible pairs of controls is  × V . The pair of controls, given by (2.100), is convenient rather for the theoU retical analysis and solution of the singular game. Below, we present the second way of the reformulation of the game (2.57)–(2.58), which yields the solution more convenient for a practical implementation.

2.2.4.2

The Second Way of the Reformulation of the Game (2.57)–(2.58)

In this way, we introduce a new definition of saddle-point solution to this game. +∞  Consider a sequence of the pairs of functions u k (·), v ∗ (·) k=1 , satisfying the  conditions u k (·) ∈ U , (k = 1, 2, . . .), v ∗ (·) ∈ V , i.e., u k (·), v ∗ (·) ∈ U × V , (k = 1, 2, . . .). +∞  Definition 2.10 The sequence u k (·), v ∗ (·) k=1 is called a saddle-point sequence (or a saddle-point solution) of the game (2.57)–(2.58) if (a) there exist limk→+∞ Ju u k (·) ; (b) the following equality is valid:   lim Ju u k (·) = Jv v ∗ (·) .

k→+∞

(2.106)

In this case, the value   J ∗ = lim Ju u k (·) = Jv v ∗ (·) k→+∞

(2.107)

34

2 Examples of Singular Extremal Problems and Some Basic Notions

is called a value of the game (2.57)–(2.58). Remark 2.11 Due to Definition 2.10, the solution of the game (2.57)–  saddle-point (2.58) is not a single pair of functions u(·), v(·) ∈ U × V , but it is a sequence of   +∞  pairs of functions from the set U × V , i.e., u ∗ (·), v ∗ (·) = u k (·), v ∗ (·) k=1 . Remark 2.12 It is important to note that the solution of the game (2.57)–(2.58) in the form of a saddle-point sequence of controls’ pairs, belonging to the set U × V , is more convenient for a practical implementation than the solution in the form of the pair, in which one entry is the abstract (generalized) Dirac delta-function δ(t). Similar to Sects. 2.2.1 and 2.2.2, we consider the sequence of positive numbers to the limit equality (2.99), {εk }+∞ k=1 such that lim k→+∞ εk = 0. For this sequence, due  +∞ we have that the sequence of the pairs of the controls u(·, ¯ εk ), 0 k=1 satisfies the conditions of Definition 2.10. Remember that the control u(t, ¯ ε) is given by Eq. +∞  (2.74). Thus, the sequence u(·, ¯ εk ), 0 k=1 is a saddle-point solution (a saddlepoint sequence of controls’ pairs) of the game (2.57)–(2.58). It is important to note that this sequence has been obtained by rather a guess, but not by a rigorous constructive method. Here, we present the rigorous method (the regularization method) allowing to derive a saddle-point sequence of controls’ pairs of the game (2.57)– (2.58). This method consists in approximately replacing this singular game with a new game which has a saddle-point solution in the set U × V , satisfying the firstorder solvability conditions (see, e.g., [2] (Chap. 6) and [4] (Sect. 9.3)). Thus, the new game is well posed (or regular). This new game depends on a small parameter ε > 0, and it becomes the original game (2.57)–(2.58) when ε is replaced with zero. In the new (regularized) game, we keep the differential equation (2.57) and  the set of the admissible pairs of controls U × V , while we replace the functional J u(·), v(·) (see Eq. (2.58)) with the new functional 









1

u 2 (t)dt Jε u(·), v(·) = J u(·), v(·) + ε 0  1  2  = x (t) + ε2 u 2 (t) − 2v 2 (t) dt. 2

(2.108)

0

Thus, the regularized game consists of the differential equation (2.57), the set of the admissible pairs of controls U × V and the functional (2.108) to be minimized by a proper choice of the control u(·) ∈ U (the minimizer’s control) and maximized by a proper choice of the control v(·) ∈ V (the maximizer’s control). Such a definition of the regularized game means that it is a zero-sum differential game. Also, like in the original game (2.57)–(2.58), we assume the following information pattern in the regularized game. The minimizer knows about the set V , but it does not know about the choice of the control v(·) ∈ V by the maximizer. Similarly, the maximizer knows about the set U , but it does not know about the choice of the control u(·) ∈ U by the minimizer. In what follows, for the sake of briefness, we denote the

2.2 Academic Examples

35

regularized game by the numbers of its differential equation and functional, i.e., (2.57), (2.108). Remark 2.13 Note that for any given ε > 0, we define the guaranteed results of the controls, the saddle-point solution and the value of the game (2.57), (2.108) quite similar to Definitions 2.7–2.9. Let us solve the game (2.57), (2.108). By virtue of the first-order solvability condition of a zero-sum differential game [2] (Chap. 6), we have the following assertion. If, for a given ε > 0, the terminal-value problem for the Riccati differential equation dw(t) = −2w(t) + dt



 1 1 − w 2 (t) − 1, w(1) = 0, t ∈ [0, 1] (2.109) ε2 2

has the solution in the entire interval [0, 1], then the game (2.57), (2.108) has the unique saddle-point solution u¯ ∗ (·, ε), v¯ ∗ (·, ε) ∈ U × V . Due to the results of [2] (Chap. 6), the problem (2.109) has the solution in the entire interval [0, 1] if the coefficient for w2 (t) in the right-hand side of the √ Riccati differential equation is positive, i.e., (1/ε2 − 1/2) > 0. Thus, if 0 < ε < 2, then the game (2.57), (2.108) has the unique saddle-point solution. We are going to derive this solution using the results of [4] (Sect. 9.3). By virtue of these results, the entries of the saddle-point solution to the game (2.57), (2.108) have the form u¯ ∗ (t, ε) = −

1 λ(t, ε), 2ε2

v¯ ∗ (t, ε) =

1 λ(t, ε), 4

t ∈ [0, 1],

(2.110)

where the function λ(t, ε) is obtained from the solution of the boundary-value problem   d x(t) 1 1 − = x(t) − λ(t), x(0) = x0 , dt 2ε2 4 dλ(t) = −2x(t) − λ(t), λ(1) = 0. dt Solving this problem and substituting the λ-component of the obtained solution into Eq. (2.110), we have (after a routine algebra) the following expressions for the entries of the saddle-point solution to the game (2.57), (2.108):     t x0 t u¯ (t, μ) = − exp − − x0 exp − μ μ μ +u¯ 1 (t, μ) + c(μ)u¯ 2 (t, μ), ∗

v¯ ∗ (t, μ) = −

μ2 2 u ¯ (t, μ) − c(μ)u¯ 2 (t, μ), 1 2 − μ2 2 − μ2

(2.111)

(2.112)

36

2 Examples of Singular Extremal Problems and Some Basic Notions

where t ∈ [0, 1], √ 2ε μ= √ , 2 + ε2   x0 μ t u¯ 1 (t, μ) = − exp − (1 + μ), 2 − 2μ2 μ

c(μ) =  u¯ 2 (t, μ) =

(2.113)

t ∈ [0, 1],

(2.114)

x0 (1 + μ) exp(−2/μ) 2 − μ2 · , 2 − 2μ2 1 − μ + (1 + μ) exp(−2/μ)

(2.115)

     1 1 t −1 + + 1 exp − , t ∈ [0, 1]. μ μ μ

(2.116)

Equation (2.113) yields that μ > 0 for all ε > 0 and lim μ = +0.

ε→+0

Moreover, using Eqs. (2.115) and (2.116), we obtain (quite similar to (2.34) and (2.35)) the following inequalities for all μ ∈ (0, 0.4] and t ∈ [0, 1]:   2 0 < c(μ) < 2.4x0 exp − , μ  0 < c(μ)u¯ 2 (t, μ) < 2.4x0

    1 1 + exp − . exp − 2μ μ

Furthermore, using Eq. (2.114), we directly have |u¯ 1 (t, μ)| ≤ 0.84x0 μ,

t ∈ [0, 1], μ ∈ [0, 0.4].

Thus, the third and fourth addends on the right-hand side of Eq. (2.111), as well as the entire expression on the right-hand side of Eq. (2.112), tend to zero for μ → +0. This observation means a neglected (for μ → +0) contribution of the aforementioned expressions to the solution of Eq. (2.57) with u(t) = u¯ ∗ (t, μ), v(t) = v¯ ∗ (t, μ). Therefore, we can remove the addends u¯ 1 (t, μ), cu¯ 2 (t, μ) from (2.111), and the entire right-hand side expression from (2.112). Due to this removing, we obtain the controls

2.3 Mathematical Models of Real-Life Problems

u ∗ (t, μ) = −

37

    x0 t t exp − − x0 exp − , t ∈ [0, 1], μ ∈ (0, 0.4], μ μ μ v ∗ (t, μ) = v ∗ (t) = 0, t ∈ [0, 1], μ ∈ (0, 0.4].

Let {μk }+∞ k=1 be a sequence of numbers such that μk ∈ (0, 0.4], (k = 1, 2, . . .), and μ lim k = 0.  Let us show that the sequence of controls’ pairs  k→+∞ +∞ ∗ u (·, μk ), v ∗ (·) k=1 is a solution (a saddle-point sequence of controls’ pairs) to  the game (2.57)–(2.58). It is clear that u ∗ (·, μk ), v ∗ (·) ∈ U × V , (k = 1, 2, . . .). ∗ Let us calculate the limit  ∗ for k → +∞ of the guaranteed result Ju u (·, μk ) and the guaranteed result Jv v (·) . Quite similar to (2.93), we have lim

  sup J u ∗ (·, μk ), v(·) = lim Ju u ∗ (·, μk ) = 0.

k→+∞ v(·)∈V

k→+∞

(2.117)

Moreover, similar to (2.96), we obtain   Jv v ∗ (·) = inf J u(·), v ∗ (·) = −2 u(·)∈U



1

 ∗ 2 v (t) (t)dt = 0.

(2.118)

0

The equalities (2.117) and (2.118) imply immediately the fulfilment of the con +∞ ditions of Definition 2.10, meaning that u ∗ (·, μk ), v ∗ (·) k=1 is the solution (the saddle-point sequence of controls’ pairs) to the game (2.57)–(2.58). Moreover, the value of this game is J ∗ = 0.

2.3 Mathematical Models of Real-Life Problems 2.3.1 Planar Pursuit-Evasion Engagement: Zero-Order Dynamics of Participants Here, we consider a mathematical model of a pursuit-evasion engagement between two flying vehicles. In what follows, we call these vehicles a pursuer and an evader. In a sufficiently small vicinity of the collision course, a nonlinear three-dimensional model of the vehicles’ motion can be linearized and decoupled into two models of planar motions in perpendicular planes (see, e.g., [18, 23]). Due to this observation, in the sequel of this subsection, we consider the linear planar model of the engagement. In this model, the duration of the engagement is td =

R0 , Vc

(2.119)

where R0 is a known initial range from the pursuer to the evader; Vc is the closing speed of the engagement. For instance, in the head-on scenario of the engagement

38

2 Examples of Singular Extremal Problems and Some Basic Notions

Vc = V p + Ve , where V p and Ve are constant magnitudes of the velocity vectors of the pursuer and the evader, respectively, (see, e.g., [26]). In this subsection, we consider the case where both vehicles have zero-order (ideal) linear dynamics. The latter means that the accelerations of the pursuer and the evader, normal to the initial line of sight (the lateral accelerations), a p (t) and ae (t), t ∈ [0, td ], are their controls. Hence, the pursuer’s control is u(t) = a p (t), t ∈ [0, td ], and the evader’s control is v(t) = ae (t), t ∈ [0, td ]. Thus, the mathematical model of the vehicles’ relative motion is (see [24]) d x1 (t) = x2 (t), t ∈ [0, td ], x1 (0) = x10 = 0, dt d x2 (t) = u(t) + v(t), t ∈ [0, td ], x2 (0) = x20 , dt

(2.120)

where x1 (t) is the relative separation of the vehicles, normal to the initial line of sight; x2 (t) is the relative velocity of the vehicles, normal to the initial line of sight. The behaviour of each vehicle in the pursuit-evasion engagement is evaluated by the functional 

td

+ 0

 J u(·), v(·) = f 1 x12 (td )   d1 (t)x12 (t) + d2 (t)x22 (t) − gv (t)v 2 (t) dt,

(2.121)

where f 1 > 0 is a given constant value; d1 (t) ≥ 0, d2 (t) > 0 and gv (t) > 0 are given continuous functions in the interval [0, td ]. The constant f 1 and the functions d1 (t), d2 (t) and gv (t) have proper physical dimensions. For instance, if td is measured in seconds, x1 (t) in meters, x2 (t) in meters per second and u(t) and v(t) in meters per (second)2 , then f 1 is non-dimensional, d1 (t) has the dimension “1/second”, d2 (t) has the dimension “second” and gv (t) has the dimension “(second)3 ”. The objective of the pursuer is to minimize the functional (2.121) along trajectories of the system (2.120) by a proper choice of the continuous control function u(t), t ∈ [0, td ]. The objective of the evader is to maximize the functional (2.121) along trajectories of the system (2.120) by a proper choice of the continuous control function v(t), t ∈ [0, td ]. Similar to Sect. 2.2.4, the problem, consisting of the differential system (2.120), the functional (2.121) and the objectives of the pursuer and the evader, is a zero-sum differential game. In what follows, for the sake of briefness, we denote this game by the numbers of its differential system and functional, i.e., (2.120)–(2.121). Information pattern in this game is the following. The pursuer knows that the evader’s control v(·) is a continuous function in the interval [0, td ], but it does not know about the choice of this control by the evader. Similarly, the evader knows that the pursuer’s control u(·) is a continuous function in the interval [0, td ], but it does not know about the choice of this control by the pursuer.

2.3 Mathematical Models of Real-Life Problems

39

Due to the absence of a cost of the pursuer’s control in the functional (2.121), the is singular. Moreover, it does not have a saddle-point solugame (2.120)–(2.121) tion u(·), v(·) among pairs, in which both controls are regular (non-generalized) functions.

2.3.2 Planar Pursuit-Evasion Engagement: First-Order Dynamics of Participants In this subsection, we present a more general mathematical model of the engagement between two flying vehicles (the pursuer and the evader) than the one studied in the previous subsection. Namely, we consider the case where both vehicles have first-order linear dynamics. In this case, the linearized kinematics of the planar engagement is similar to the kinematics of Sect. 2.3.1. However, in the linear model of the present subsection, the relative velocity of the vehicles is not controlled directly by the lateral accelerations of the pursuer and the evader. The relative velocity is controlled through these accelerations by the so-called acceleration commands a cp (t) and aec (t) of the pursuer and the evader, respectively. These acceleration commands are the controls u(t) and v(t) of the pursuer and the evader, respectively, i.e., u(t) = a cp (t), v(t) = aec (t), t ∈ [0, td ]. Thus, the mathematical model of the vehicles’ relative motion subject to the first-order linear dynamics of the pursuer and the evader has the following form (see, e.g., [14, 22]): d x1 (t) = x2 (t), t ∈ [0, td ], x1 (0) = x10 = 0, dt d x2 (t) = x3 (t) − x4 (t), t ∈ [0, td ], x2 (0) = x20 , dt d x3 (t) v(t) − x3 (t) , t ∈ [0, td ], x3 (0) = x30 , = dt τe u(t) − x4 (t) d x4 (t) , t ∈ [0, td ], x4 (0) = x40 , = dt τp (2.122) where td is determined by (2.119); x1 (t) is the relative separation of the vehicles, normal to the initial line of sight; x2 (t) is the relative velocity of the vehicles, normal to the initial line of sight; x3 (t) and x4 (t) are the lateral accelerations of the evader and the pursuer, respectively, both normal to the initial line of sight; τe and τ p are the time constants of the evader and the pursuer, respectively. In the present subsection, the behaviour of each vehicle in the pursuit-evasion engagement is evaluated by the functional

40

2 Examples of Singular Extremal Problems and Some Basic Notions



td

+ 0

 J u(·), v(·) = f 1 x12 (td ) + f 2 x22 (td )   d1 (t)x12 (t) + d2 (t)x22 (t) + d4 (t)x42 (t) − gv (t)v 2 (t) dt,

(2.123)

where f 1 > 0 and f 2 ≥ 0 are given constant values; d1 (t) ≥ 0, d2 (t) ≥ 0, d4 (t) > 0 and gv (t) > 0 are given continuous functions in the interval [0, td ]. The constants f 1 , f 2 and the functions d1 (t), d2 (t), d4 (t) and gv (t) have proper physical dimensions. Like in the previous subsection, the objective of the pursuer is to minimize the functional (2.123) along trajectories of the system (2.122) by a proper choice of the continuous control function u(t), t ∈ [0, td ]. The objective of the evader is to maximize the functional (2.123) along trajectories of the system (2.122) by a proper choice of the continuous control function v(t), t ∈ [0, td ]. Similar to Sects. 2.2.4 and 2.3.1, the problem, consisting of the differential system (2.122), the functional (2.123), and the objectives of the pursuer and the evader, is a zero-sum differential game. In what follows, we denote this game by the numbers of its differential system and functional, i.e., (2.122)–(2.123). Information pattern in the game (2.122)–(2.123) is similar to the information pattern in the game (2.120)–(2.121). Namely, each of the participants of the pursuitevasion engagement knows that the control of its opponent is a continuous function in the interval [0, td ], but it does not know about the choice of this control by the opponent. Due to the absence of a cost of the pursuer’s control in the functional (2.123), the  game (2.122)–(2.123) is singular, and it does not have a saddle-point solution u(·), v(·) among pairs of the controls, in which both entries are regular (nongeneralized) functions.

2.3.3 Three-Dimensional Pursuit-Evasion Engagement: Zero-Order Dynamics of Participants In this subsection, we present an extension of the model considered in Sect. 2.3.1. Namely, we consider the case of three-dimensional linear pursuit-evasion engagement between two flying vehicles. The kinematics of this engagement is a combination of two planar kinematics in two perpendicular planes (say, horizontal and vertical) in three-dimensional space. Each of these two kinematics is similar to the one considered in Sect. 2.3.1. Thus, the mathematical model of the vehicles’ three-dimensional relative motion subject to their zero-order dynamics is (see [27])

2.3 Mathematical Models of Real-Life Problems

d x1 (t) = x3 (t), t ∈ [0, td ], x1 (0) = x10 = 0, dt d x2 (t) = x4 (t), t ∈ [0, td ], x2 (0) = x20 = 0, dt d x3 (t) = u 1 (t) + v1 (t), t ∈ [0, td ], x3 (0) = x30 , dt d x4 (t) = u 2 (t) + v2 (t), t ∈ [0, td ], x4 (0) = x40 , dt

41

(2.124)

where td is determined by (2.119); x1 (t) is the relative separation of the vehicles in the horizontal plane, normal to the initial line of sight; x2 (t) is the relative separation of the vehicles in the vertical plane, normal to the initial line of sight; x3 (t) is the relative velocity of the vehicles in the horizontal plane, normal to the initial line of sight; x4 (t) is the relative velocity of the vehicles in the vertical plane, normal to the initial line of sight; u 1 (t) and u 2 (t) are the lateral accelerations of the pursuer in the horizontal and vertical planes, respectively; v1 (t) and v2 (t) are the lateral accelerations of the evader in the horizontal and vertical planes, respectively. The behaviour of each vehicle in the pursuit-evasion engagement is evaluated by the functional 

td

+ 0



 J u 1 (·), u 2 (·), v1 (·), v2 (·) = f 1 x12 (td ) + f 2 x22 (td ) d1 (t)x12 (t) + d2 (t)x22 (t) + d3 (t)x32 (t) + d4 (t)x42 (t)  +gu,1 (t)u 21 (t) − gv,1 (t)v12 (t) − gv,2 (t)v22 (t) dt,

(2.125)

where f 1 > 0 and f 2 > 0 are given constant values; d1 (t) ≥ 0, d2 (t) ≥ 0, d3 (t) ≥ 0, d4 (t) > 0 and gu,1 (t) > 0, gv,1 (t) > 0, gv,2 (t) > 0 are given continuous functions in the interval [0, td ]. These constants and functions have proper physical dimensions. The objective of the pursuer is to minimize the functional (2.125) along trajectories of the system (2.124) by a proper choice of the continuous control functions u 1 (t) and u 2 (t), t ∈ [0, td ]. The objective of the evader is to maximize the functional (2.125) along trajectories of the system (2.124) by a proper choice of the continuous control functions v1 (t) and v2 (t), t ∈ [0, td ]. Similar to Sects. 2.2.4 and 2.3.1–2.3.2, the problem, consisting of the differential system (2.124), the functional (2.125), and the objectives of the pursuer and the evader, is a zero-sum differential game. We refer to this game as (2.124)–(2.125). Information pattern in the game (2.124)–(2.125) is similar to the information patterns in the games of Sects. 2.3.1–2.3.2. Namely, each of the participants of the pursuit-evasion engagement knows that the two controls of its opponent are continuous functions in the interval [0, td ], but it does not know about the choice of these controls by the opponent. Due to the absence of a cost of the pursuer’s control u 2 (t) in the functional (2.125), the is singular. solution  game (2.124)–(2.125) It does not have  a saddle-point  u(·), v(·) , where u(·) = col u 1 (·), u 2 (·) and v(·) = col v1 (·), v2 (·) , among pairs

42

2 Examples of Singular Extremal Problems and Some Basic Notions

of the vector-valued controls, in which both entries are regular (non-generalized) vector-valued functions. Remark 2.14 Let us observe that the pursuer’s control u 1 (·) is present in the functional of the game (2.124)–(2.125), and the weight coefficient of this control in the functional is a positive function. This observation, along with the aforementioned absence of the pursuer’s control u 2 (·) in the functional (2.125), means that, among the controls u 1 (·) and u 2 (·), only the latter is “singular” but the former is “regular”. Detailed explanation of such a feature is presented in Chap. 4. Remark 2.15 Instead of the functional (2.125), one of the following functionals can be considered for the evaluation of the behaviour of the pursuer and the evader: 

td

+



0

 J u 1 (·), u 2 (·), v1 (·), v2 (·) = f 1 x12 (td ) + f 2 x22 (td ) d1 (t)x12 (t) + d2 (t)x22 (t) + d3 (t)x32 (t) + d4 (t)x42 (t)  +gu,2 (t)u 22 (t) − gv,1 (t)v12 (t) − gv,2 (t)v22 (t) dt,

(2.126)

where d3 (t) > 0, d4 (t) ≥ 0 and gu,2 (t) > 0 are given continuous functions in the interval [0, td ]; the other coefficients are the same as the corresponding ones in (2.125), or 

td

+ 0



 J u 1 (·), u 2 (·), v1 (·), v2 (·) = f 1 x12 (td ) + f 2 x22 (td ) d1 (t)x12 (t) + d2 (t)x22 (t) + d3 (t)x32 (t) + d4 (t)x42 (t)  −gv,1 (t)v12 (t) − gv,2 (t)v22 (t) dt,

(2.127)

where d3 (t) > 0 is a given continuous function in the interval [0, td ]; the other coefficients are the same as the corresponding ones in (2.125). The games (2.124), (2.126) and (2.124), (2.127) are singular. Moreover, in the first of these games, the pursuer’s control u 1 (·) is “singular”, while the pursuer’s control u 2 (·) is “regular”. In the second of the aforementioned games, both pursuer’s controls, u 1 (·) and u 2 (·), are “singular”.

2.3.4 Infinite-Horizon Robust Vibration Isolation Motion of a body of a mass m in a straight line (say, in the axis X ), generated by a disturbance force v(t) and an actuator force u(t), is modelled by the following differential equation (see [15]):

2.3 Mathematical Models of Real-Life Problems

m

d 2 x(t) d x(t)   = v(t) + u(t), t ≥ 0, x(0) = x ,  = x0 , 0 dt 2 dt t=0

43

(2.128)

where x(t) is a deviation of the body’s position from 0 at the time moment t. The disturbance causes a vibration of the body, while the aim of the actuator is to reject (to isolate) this vibration. The work [15] considers the case where the disturbance force is known. In the present subsection, we consider the case where the disturbance force is unknown. However, it is known that this force belongs to the set V of all functions v(·) given for t ∈ [0, +∞) and such that the Lebesgue +∞ integral 0 v 2 (t)dt exists. Moreover, the disturbance force is subject to the “soft” constraint  +∞ v 2 (t)dt ≤ vg , (2.129) 0

where vg > 0 is a known number. We denote by Vg the set of all v(·) ∈ V , which satisfy the constraint (2.129). We call the set Vg as the set of all admissible disturbance forces. Let U be the set of all functions u(·) given for t ∈ [0, +∞) and such that the  +∞ Lebesgue integral 0 u 2 (t)dt exists. We call the set U as the set of all admissible forces of the actuator. Consider the functional  +∞   2   (2.130) β1 x 2 (t) + β2 d x(t)/dt dt, J u(·), v(·) = 0

where β1 > 0 and β2 > 0 are given constant weight coefficients. Thus, the aforementioned aim of the actuator can be reformulated as follows. The actuator tries to minimize the functional (2.130) by a proper choice of its force u(·) ∈ U . However, since the disturbance force is unknown, this minimization should be robust with respect to v(·) ∈ Vg . Therefore, the minimization of the functional (2.130) by the actuator should be subject to the assumption that the disturbance aims to maximize this functional by a proper choice of its force v(·) ∈ Vg . The problem, consisting of the differential equation (2.128), the sets Vg and U of the admissible forces of the disturbance and the actuator, the functional (2.130) and the aims of the actuator and the disturbance, is a zero-sum differential game. Since the integral in the functional (2.130) is calculated in the infinite time interval [0, +∞), this game is called an infinite-horizon game. We refer to this game as (2.128)–(2.130). Information pattern in this game is the following. The actuator knows the set Vg of the disturbance’s admissible forces, but it does not know about the choice of the disturbance. Similarly, the disturbance knows the set U of the actuator’s admissible forces, but it does not know about the choice of the actuator. Due to the absence of the cost of the actuator’s force u(·) in the functional (2.130), as well as the absence of a constraint imposed on u(·), the game (2.128)–(2.130) is singular. It does not have a

44

2 Examples of Singular Extremal Problems and Some Basic Notions

 saddle-point solution u(·), v(·) , in which both entries are regular (non-generalized) functions. Another mathematical model of the robust vibration isolation is the problem consisting of the differential equation (2.128), the sets V and U of the admissible forces of the disturbance and the actuator and the functional  +∞    2  β1 x 2 (t) + β2 d x(t)/dt − βv v 2 (t) dt, (2.131) J u(·), v(·) = 0

where β1 > 0, β2 > 0 and βv > 0 are given constant weight coefficients. The objective of the actuator is to minimize the functional (2.131) along trajectories of the differential equation (2.128) by a proper choice of its force u(·) ∈ U . This minimization should be carried out by the actuator subject to its assumption on the worse-case behaviour of the disturbance, i.e., that the objective of the disturbance is to maximize the functional (2.131) by a proper choice of the force v(·) ∈ V . Thus, the aforementioned problem is a zero-sum differential game. We refer to this game as (2.128), (2.131). Information pattern in this game is similar to that in the game (2.128)–(2.130) by replacing the set Vg with the set V . The  game (2.128), (2.131) is singular, and it does not have a saddle-point solution u(·), v(·) where both entries are regular (non-generalized) functions.

2.4 Concluding Remarks and Literature Review In this chapter (Sect. 2.2), four academic examples of various singular extremal problems were presented and analysed in detail. Such an analysis clarifies the notion of a singular extremal problem and shows how various singular extremal problems can be treated. Also, in this chapter (Sect. 2.3), four examples of mathematical models of real-life problems were given. These mathematical models represent singular zero-sum differential games. In Sect. 2.2.1, the problem of the minimization of a smooth function in a closed and unbounded set of real numbers Ω was considered. It was shown that this function has an infimum in the set Ω. However, there is not a point in Ω, for which this infimum value is attained. Therefore, the problem of the function’s minimization in the set Ω does not have a solution, i.e., it is ill-posed or singular. Two ways of reformulation of this problem, such that the reformulated problem allows the existence of a solution, were presented. One of these ways consists in a proper extension of the set Ω of possible solutions of the original minimization problem. Namely, the abstract value +∞ is added to this set, and this value is the solution of the reformulated problem. The second way consists in introducing a new definition of solution to the original minimization problem. Due to this definition, the solution of the minimization problem is not a single value (number or ±∞), but it is a sequence of numbers (a minimizing sequence). The deriving of such a sequence by a proper regularization of the original problem was shown. Namely, the original singular

2.4 Concluding Remarks and Literature Review

45

function’s minimization problem is replaced with a new function’s minimization problem. In the latter, the set Ω is unchanged, while the original minimized function is replaced with a new function, depending on a small positive parameter ε. The new function becomes the original one when ε is replaced with zero. The new (regularized) problem has the unique solution in the set Ω, satisfying the first-order extremum condition. Such a condition can be found, for instance, in [5]. Thus, the regularized minimization problem is well posed or regular. The ε-dependent solution of the regularized minimization problem yields the solution (the minimizing sequence) to the original function’s minimization problem. Along this sequence, the original function tends to its infimum in the set Ω. In Sect. 2.2.2, the finite duration optimal control problem with scalar state and control variables was considered. The differential equation in this problem is linear with respect to the state and the control. The initial value of the state variable is not zero. The integral functional contains only the state cost and this cost is positive quadratic. This functional should be minimized by a proper choice of a control variable from the set U of all quadratically integrable functions in the sense of Lebesgue. It was shown that, although there exists an infimum of the functional in the considered optimal control problem, this infimum is not attained for any control from the set U . The latter means that this problem does not have a solution (an optimal control), i.e., this problem is ill-posed or singular. Similar to Sect. 2.2.1, two ways of the reformulation of this problem, allowing the existence of a solution in the reformulated problem, were presented. In the first way, the set U was extended by adding to it the right-hand side δ-function of Dirac with various constant coefficients. Thus, the new set of possible control variables contains not only the regular (non-generalized) functions, but also the generalized functions. Such an extension yields the existence of the solution (the optimal control) of the reformulated problem in the form of this δ-function with a proper coefficient. Analysis and solution of various singular optimal control problems in properly defined sets of generalized functions can be found in [7, 8, 16, 17, 25, 28, 29] and references therein. In the second way, a new definition of solution to the original optimal control problem was introduced. Due to this definition, the solution of the optimal control problem is not a single function (regular or generalized), but it is a sequence of regular functions (a minimizing sequence). The regularization method, allowing to derive such a sequence, was presented. This method consists in approximately replacing the original singular optimal control problem with a new optimal control problem. In this new optimal control problem, the differential equation is the same as in the original problem. The set U also is unchanged, while the functional of the original problem is replaced with a new functional, depending on a small positive parameter ε. The new functional becomes the original one when ε is replaced with zero, meaning that the new (regularized) optimal control problem becomes the original one when this parameter is replaced with zero. The regularized problem has a unique solution (the optimal control) in the set U , satisfying the first-order optimality conditions. Such conditions can be found, for instance, in [4] (Sect. 5.2) and [21] (Sects. 3.2, 5.1, 5.2). Thus, the regularized optimal control problem is well posed or regular. The ε-dependent solution of the regularized optimal control problem yields the solution (the minimizing

46

2 Examples of Singular Extremal Problems and Some Basic Notions

sequence) to the original singular optimal control problem. Along this sequence, the functional in the original problem tends to its infimum in the set U . Analysis and solution of various singular optimal control problems by the regularization method can be found in [3, 9–13, 20, 25] and references therein. In Sect. 2.2.3, the problem of searching a saddle point of a function of two independent scalar variables t and τ was considered. The variable t varies in a closed and unbounded set of real numbers Ωt , while the variable τ varies in a closed and unbounded set of real numbers Ωτ . The considered function is smooth in the set Ωt × Ωτ . It was shown that the values inf t∈Ωt supτ ∈Ωτ and supτ ∈Ωτ inf t∈Ωt of this function are equal to each other. However, there is not any point in the set Ωt × Ωτ for which these values are attained. Therefore, the problem of searching a saddle point of the considered function in the set Ωt × Ωτ , called the saddle-point problem, does not have a solution, i.e., this problem is ill- posed or singular. However, due to the aforementioned equality inf t∈Ωt supτ ∈Ωτ = supτ ∈Ωτ inf t∈Ωt , one can expect to obtain a saddle point of the considered function by a proper reformulation of the original singular saddle-point problem. Similar to Sects. 2.2.1 and 2.2.2, two ways of the reformulation of the original saddle-point problem were presented. Each way leads to a new saddle-point problem, having a proper solution. The first way consists in a proper extension of the set Ωt × Ωτ . Namely, the set Ωτ remains unchanged, while the set Ωt is extended by adding to it the abstract value +∞. Due to such an extension, the reformulated saddle-point problem has a solution. This solution (the saddle point of the considered function) is a pair consisting of the abstract value +∞ and a number from the set Ωτ . The second way consists in introducing a new definition of solution to the original saddle-point problem. Due to this definition, the solution of the saddle-point problem is not a single pair of values (each of which can be a number or ±∞), but it is a sequence of numbers’ pairs (a saddle-point sequence). Obtaining such a sequence by a proper regularization of the original saddle-point problem was shown. Namely, the original singular problem is replaced with a new saddle-point problem. In the latter, the set Ωt × Ωτ remains unchanged, while the original function is replaced with a new function. This new function depends on a small positive parameter ε, and it becomes the original function when ε is replaced with zero. The new (regularized) saddle-point problem has the unique solution in the set Ωt × Ωτ , satisfying the first-order extremum condition with respect to both independent variables. Such a condition can be found, for instance, in [6]. Thus, the regularized saddle-point problem is well posed or regular. The ε-dependent solution of the regularized saddle-point problem yields the solution (the saddle-point sequence) to the original saddle-point problem. Along this sequence, the original function tends to its saddle-point value inf t∈Ωt supτ ∈Ωτ = supτ ∈Ωτ inf t∈Ωt . In Sect. 2.2.4, the finite duration zero-sum differential game with a scalar state variable was considered. Control variables of both players also are scalar. The differential equation in this game is linear with respect to the state and the controls. The initial value of the state variable is not zero. The game’s integral functional contains a positive quadratic state cost and a negative quadratic cost of the second player’s control. However, this functional does not contain the cost of the first player’s control. The aim of the first player is to minimize the game’s functional

2.4 Concluding Remarks and Literature Review

47

by a proper choice of its control variable u(·) from the set U of all quadratically integrable functions in the sense of Lebesgue. The aim of the second player is to maximize this functional by a proper choice of its control variable v(·) from the set V of all quadratically integrable functions in the sense of Lebesgue. Thus, the first player is the minimizing player (the minimizer), while the second player is the maximizing player (the maximizer). It was shown that the values inf u(·)∈U supv(·)∈V and supv(·)∈V inf u(·)∈U of the game’s functional are equal to each other. However, there is not any pair of the controls’ variables in the set U × V for which these values are attained. Therefore, the considered game does not have a saddle-point solution, i.e., this game is ill-posed or singular. However, due to the aforementioned equality inf u(·)∈U supv(·)∈V = supv(·)∈V inf u(·)∈U , one can expect to obtain a saddle-point solution of the considered singular game by its proper reformulation. Similar to Sect. 2.2.3, two ways of the reformulation of the original game were presented. In the first way, the set U was extended by adding to it the right-hand side δ-function of Dirac with various constant coefficients. Thus, the new set of possible pairs of the controls’ variables contains not only the pairs of regular (non-generalized) functions, but also the pairs where the first entry is a generalized function. Such an extension yields the existence of the saddle-point solution to the reformulated game. In this solution, the first entry is the δ-function with a proper coefficient, while the second entry is a regular function from the set V . To the best of our knowledge, the solution of a singular zero-sum differential game by using the Dirac δ-function has been presented before in a single work in the literature (see [1]). In the second way, a new definition of saddle-point solution to the original game was introduced. Due to this definition, the saddle-point solution of the game is not a single pair of functions (each of which can be a regular one or a generalized one), but it is a sequence of pairs of regular functions (a saddle-point sequence). The regularization method, allowing to derive such a sequence, was presented. This method consists in approximately replacing the original singular zero-sum differential game with a new zero-sum differential game. In this new game, the differential equation is the same as in the original game. The sets U and V also are unchanged, while the functional of the original game is replaced with a new functional, depending on a small positive parameter ε. The new functional becomes the original one when ε is replaced with zero, meaning that the new (regularized) game becomes the original one when ε is replaced with zero. The regularized game has the unique saddle-point solution in the set U × V , satisfying the first-order game’s solvability conditions. Such conditions can be found, for instance, in [2] (Chap. 6) and [4] (Sect. 9.3). Thus, the regularized zero-sum differential game is well posed or regular. The ε-dependent saddle-point solution of the regularized game yields the solution (the saddle-point sequence) to the original singular zero-sum differential game. Along this sequence, the functional in the original game tends to the value of the game which is inf u(·)∈U supv(·)∈V = supv(·)∈V inf u(·)∈U of this functional. Detailed analysis and solution of general types’ singular zerosum differential games by the regularization method is presented in the subsequent chapters (see Chaps. 4 and 5). As it was aforementioned, Sect. 2.3 of the present chapter is devoted to several mathematical models of real-life problems.

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2 Examples of Singular Extremal Problems and Some Basic Notions

Thus in Sect. 2.3.1, the mathematical model of a planar engagement between two flying vehicles (the pursuer and the evader) was presented. It is assumed that the engagement takes place in a sufficiently small vicinity of the collision course. This assumption allows linearizing the differential equations, describing the vehicles’ relative motion. The case of zero-order (ideal) linear dynamics for both vehicles was considered, meaning that the accelerations of the pursuer and the evader, normal to the initial line of sight (the lateral accelerations), are their controls. For the evaluation of the behaviour of the pursuer and the evader, a quadratic functional was chosen. This functional is the sum of a terminal part and an integral part. The terminal part is the square of the vehicles’ relative separation with a constant positive coefficient at the end of the engagement. The integrand of the integral part contains the square of the current relative vehicles’ separation with a time-dependent nonnegative coefficient, the square of the current relative vehicles’ velocity with a time-dependent positive coefficient and the square of the current evader’s control with a time-dependent negative coefficient. The objective of the pursuer/evader is to minimize/maximize the functional by a proper choice of its continuous control. Thus, the mathematical model of the engagement between these two flying vehicles is a finite-duration zerosum differential game. Due to the absence of a pursuer’s control cost in the game’s functional, this game is singular. Note that the mathematical model of the engagement between two flying vehicles, presented in this subsection, is an extension of the model of the work [24] where the case of the constant coefficients in the integrand of the functional’s integral part was considered. In Sect. 2.3.2, the extension of the model of Sect. 2.3.1 to the case of the first-order linear dynamics of the flying vehicles was presented. In such a case, the controls of the pursuer and the evader are not their lateral accelerations, but the corresponding acceleration commands. The accelerations themselves are subject to linear differential equations controlled by the acceleration commands. The mathematical model of such an engagement between the flying vehicles with the properly chosen functional for the evaluation of their behaviour is a singular finite-duration zero-sum differential game. The simpler case of this model was considered in the work [24]. In Sect. 2.3.3, another extension of the model of Sect. 2.3.1 was presented. Namely, the case of three-dimensional linear pursuit-evasion engagement between two flying vehicles with the linear zero-order dynamics was presented. In this case, each of the vehicles has two controls, which are its lateral accelerations in two perpendicular planes of the three-dimensional motion. By the properly chosen functional for the evaluation of the vehicles’ behaviour, the mathematical model of this threedimensional linear pursuit-evasion engagement is a singular finite-duration zero-sum differential game. In Sect. 2.3.4, two mathematical models of the infinite-horizon robust rejection (isolation) of a vibration in a straight-line body’s motion were presented. In both models, the body’s motion is described by the second-order differential equation linearly depending on a disturbance force and an actuator force. The disturbance causes a vibration of the body, while the aim of the actuator is to reject (to isolate) this vibration. In this subsection was considered the case of the unknown disturbance force. In the first model, it was assumed that the disturbance force is subject to a “soft”

References

49

(integral) constraint. The functional, evaluating the behaviour of the actuator, is an infinite-horizon integral. The integrand of this integral contains the square of the current body’s deviation from 0 with a constant positive coefficient and the square of the current body’s velocity with a constant positive coefficient. The aim of the actuator is to minimize this functional by a proper choice of its force, subject to the assumption that the disturbance aims to maximize this functional by a proper choice of its force. Thus, this mathematical model of the infinite-horizon robust rejection (isolation) of a vibration in a straight-line body’s motion is an infinite-duration (infinite-horizon) zero-sum differential game. Since the actuator’s force is not present in the game’s functional and is not constrained, this game is singular. In the second model, the disturbance force is unconstrained. However, the integral functional differs from the one in the first model. Namely, the integrand of the functional in the second model contains not only the squares of the current deviation of the body’s position from 0 and body’s velocity with constant positive coefficients, but also the square of the disturbance’s force with a negative coefficient. The objective of the actuator is to minimize this functional by a proper choice of its force, subject to the assumption that the disturbance tries to maximize this functional by a proper choice of its force. The second mathematical model of the robust isolation of a vibration also is a singular infinite-horizon zero-sum differential game. Note that the case of the isolation of a vibration in a straight-line body’s motion, caused by a known disturbance, was considered in the work [15]. This case was not modelled by a zero-sum differential game, but by an optimal control problem.

References 1. Amato, F., Pironti, A.: A note on singular zero-sum linear quadratic differential games. In: Proceedings of the 33rd Conference on Decision and Control, pp. 1533–1535, Lake Buena Vista, FL, USA (1994) 2. Basar, T., Olsder, G.J.: Dynamic Noncooparative Game Theory, 2nd edn. SIAM Books, Philadelphia, PA, USA (1999) 3. Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic Press, New York, NY, USA (1975) 4. Bryson, A.E., Jr., Ho, Y.-C.: Applied Optimal Control: Optimization. Estimation and Control, Taylor & Francis, New York, NY, USA (1975) 5. Courant, R.: Differential and Integral Calculus, vol. I. Wiley, Hoboken, NJ, USA (1988) 6. Courant, R.: Differential and Integral Calculus, vol. II. Wiley, Hoboken, NJ, USA (1988) 7. Geerts, T.: All optimal controls for the singular linear-quadratic problem without stability; a new interpretation of the optimal cost. Linear Algebr. Appl. 116, 135–181 (1989) 8. Geerts, T.: Linear-quadratic control with and without stability subject to general implicit continuous-time systems: coordinate-free interpretations of the optimal costs in terms of dissipation inequality and linear matrix inequality; existence and uniqueness of optimal controls and state trajectories. Linear Algebr. Appl. 203–204, 607–658 (1994) 9. Glizer, V.Y.: Solution of a singular optimal control problem with state delays: a cheap control approach. In: Reich, S., Zaslavski, A.J. (eds.) Optimization Theory and Related Topics, Contemporary Mathematics Series, vol. 568, pp. 77–107. American Mathematical Society, Providence, RI, USA (2012)

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2 Examples of Singular Extremal Problems and Some Basic Notions

10. Glizer, V.Y.: Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence. SIAM J. Control Optim. 50, 2862–2888 (2012) 11. Glizer, V.Y.: Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays. In: Wolansky, G., Zaslavski, A.J. (eds.) Variational and Optimal Control Problems on Unbounded Domains. Contemporary Mathematics Series, vol. 619, pp. 59–98. American Mathematical Society, Providence, RI, USA (2014) 12. Glizer, V.Y., Kelis. O.: Singular infinite horizon quadratic control of linear systems with known disturbances: a regularization approach, p. 36 (2016). arXiv:1603.01839v1 [math.OC]. Last accessed Mar 06 2016 13. Glizer, V.Y., Kelis, O.: Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances. Numer. Algebra Control Optim. 8, 211–235 (2018) 14. Glizer, V.Y., Turetsky, V., Fridman, L., Shinar, J.: History-dependent modified sliding mode interception strategies with maximal capture zone. J. Franklin Inst. 349, 638–657 (2012) 15. Hampton, R.D., Knospe, C.R., Townsend, M.A.: A practical solution to the deterministic nonhomogeneous LQR problem. J. Dyn. Syst. Meas. Control 118, 354–360 (1996) 16. Hautus, M.L.J., Silverman, L.M.: System structure and singular control. Linear Algebr. Appl. 50, 369–402 (1983) 17. Ho, Y.-C.: Linear stochastic singular control problems. J. Optim. Theory Appl. 9, 24–30 (1972) 18. Isaacs, R.: Differential Games. Wiley, New York, NY, USA (1967) 19. Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications, Mineola, NY, USA (2012) 20. Kurina, G.A.: On a degenerate optimal control problem and singular perturbations. Soviet Math. Dokl. 18, 1452–1456 (1977) 21. Lee, B., Markus, L.: Foundations of Optimal Control Theory. Wiley, Hoboken, NJ, USA (1967) 22. Shinar, J.: Solution techniques for realistic pursuit-evasion games. In: Leondes, C. (ed.) Advances in Control and Dynamic Systems, pp. 63–124. Academic Press, New York, NY, USA (1981) 23. Shinar, J., Glizer, V.Y., Turetsky, V.: The effect of pursuer dynamics on the value of linear pursuit-evasion games with bounded controls. In: Krivan, V., Zaccour, G. (eds.) Advances in Dynamic Games-Theory, Applications, and Numerical Methods, Annals of the International Society of Dynamic Games, vol. 13, pp. 313–350. Switzerland, Birkhauser, Basel (2013) 24. Shinar, J., Glizer, V.Y., Turetsky, V.: Solution of a singular zero-sum linear-quadratic differential game by regularization. Int. Game Theory Rev. 16, 1440007-1–1440007-32 (2014) 25. Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Winston, Washington, DC, USA (1977) 26. Turetsky, V., Glizer, V.Y.: Continuous feedback control strategy with maximal capture zone in a class of pursuit games. Int. Game Theory Rev. 7, 1–24 (2005) 27. Turetsky, V., Glizer, V.Y.: Robust controllability of linear systems in non-scalarizable case. In: Proceedings of the 13th UKACC International Conference on Control. Plymouth, UK (2022) 28. Willems, J.C., Kitapci, A., Silverman, L.M.: Singular optimal control: a geometric approach. SIAM J. Control Optim. 24, 323–337 (1986) 29. Zavalishchin, S.T., Sesekin, A.N.: Dynamic Impulse Systems: Theory and Applications. Kluwer Academic Publishers, Dordrecht, Netherlands (1997)

Chapter 3

Preliminaries

3.1 Introduction In this chapter, several problems, allowing first-order solvability conditions, are considered. Namely, we consider zero-sum linear-quadratic differential games in finite-horizon and infinite-horizon settings, as well as H∞ control problems in finitehorizon and infinite-horizon settings. Since these problems allow first-order solvability conditions, they are regular problems. The corresponding solvability conditions are derived. Along with these results, a specific linear state transformation of a linear time-dependent controlled differential system and a quadratic functional is considered. Properties of this transformation are studied. The results of this chapter are used in the subsequent chapters as auxiliary results. The following main notations are applied in the chapter. 1. E n is the n-dimensional real Euclidean space. 2. The superscript ”T ” denotes the transposition of a matrix A, ( A T ) or of a vector x, (x T ). 3. L 2 [a, b; E n ] is the linear space of n-dimensional vector-valued real functions, square-integrable in the finite interval [a, b], and  ·  L 2 [a,b] denotes the norm in this space. 4. L 2 [a, +∞; E n ] is the linear space of n-dimensional vector-valued real functions, square-integrable in the infinite interval [a, +∞), and  ·  L 2 [a,+∞) denotes the norm in this space. 5. On 1 ×n 2 is used for the zero matrix of the dimension n 1 × n 2 , excepting the cases where the dimension of the zero matrix is obvious. In such cases, the notation 0 is used for the zero matrix. 6. In is the n-dimensional identity matrix. 7. Sn is the set of all symmetric matrices of the dimension n × n. 8. Sn+ is the set of all symmetric positive semi-definite matrices of the dimension n × n. 9. Sn++ is the set of all symmetric positive definite matrices of the dimension n × n. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_3

51

52

3 Preliminaries

3.2 Solvability Conditions of Regular Problems 3.2.1 Finite-Horizon Differential Game Consider the following differential equation controlled by two decision makers (players): dz(t) = A(t)z(t) + B(t)u(t) + C(t)v(t), z(0) = z 0 , t ∈ [0, t f ], dt

(3.1)

where t f is a given final time moment; z(t) ∈ E n is a state vector; u(t) ∈ E r and v(t) ∈ E s are controls of the players; A(t), B(t) and C(t) are given matrix-valued functions of corresponding dimensions, continuous in the interval [0, t f ]; z 0 ∈ E n is a given vector. The cost functional, to be minimized by the control u (the minimizer) and maximized by the control v (the maximizer), is J (u, v) = z T (t f )F z(t f ) t f +



 z T (t)D(t)z(t) + u T (t)G u (t)u(t) − v T (t)G v (t)v(t) dt,

(3.2)

0

where a given matrix F ∈ Sn+ ; D(t), G u (t) and G v (t) are given matrix-valued functions of corresponding dimensions, continuous in the interval [0, t f ]; for any t ∈ [0, t f ], D(t) ∈ Sn+ , while G u (t) ∈ Sr++ and G v (t) ∈ Ss++ . Consider the set Ut f of all functions u = u(z, t) : E n × [0, t f ] → E r , which are measurable w.r.t. t ∈ [0, t f ] for any fixed z ∈ E n and satisfy the local Lipschitz condition w.r.t. z ∈ E n uniformly in t ∈ [0, t f ]. Similarly, let Vt f be the set of all functions v = v(z, t) : E n × [0, t f ] → E s , which are measurable w.r.t. t ∈ [0, t f ] for any fixed z ∈ E n and satisfy the local Lipschitz condition w.r.t. z ∈ E n uniformly in t ∈ [0, t f ].   Definition 3.1 By (U V )t f , we denote the set of all pairs u(z, t), v(z, t) , (z, t) ∈ E n × [0, t f ], such that the following conditions are valid: (i) u(z, t) ∈ Ut f , v(z, t) ∈ Vt f ; (ii) the initial-value problem (3.1) for u(t) = u(z, t), v(t) = v(z, t) and any absolutely continuous solution z 0 ∈ E n has the unique    z uv (t; z 0 ) in  the entire interval [0, t f ]; (iii) u z uv (t; z 0 ), t ∈ L 2 [0, t f ; E r ]; (iv) v z uv (t; z 0 ), t ∈ L 2 [0, t f ; E s ]. Such a defined set (U V )t f is called the set of all admissible pairs of the players’ state-feedback controls in the game (3.1)–(3.2).   For a given u(z, t) ∈ Ut f , consider the set Fv,t f u(z, t) = v(z, t) ∈ Vt f :         u(z, t), v(z, t) ∈ (U V )t f . Let Hu,t f = u(z, t) ∈ Ut f : Fv,t f u(z, t)  = ∅ .

3.2 Solvability Conditions of Regular Problems

53

Definition 3.2 For a given u(z, t) ∈ Hu,t f , the value   Ju u(z, t); z 0 =

sup

v(z,t)∈Fv,t f u(z,t)

  J u(z, t), v(z, t) 

(3.3)

is called the guaranteed result of u(z, t) in the game (3.1)–(3.2).   Similarly, for a given v(z, t) ∈ Vt f , consider the set Fu,t f v(z, t) = u(z, t) ∈         Ut f : u(z, t), v(z, t) ∈ (U V )t f . Let Hv,t f = v(z, t) ∈ Vt f : Fu,t f v(z, t)  = ∅ . Definition 3.3 For a given v(z, t) ∈ Hv,t f , the value   Jv v(z, t); z 0 =

inf 

   J u(z, t), v(z, t)

(3.4)

u(z,t)∈Fu,t f v(z,t)

is called the guaranteed result of v(z, t) in the game (3.1)–(3.2).   Consider a pair u ∗ (z, t), v ∗ (z, t) ∈ (U V )t f .   Definition 3.4 The pair u ∗ (z, t), v ∗ (z, t) is called a saddle-point equilibrium solution (or briefly, a saddle-point solution) of the game (3.1)–(3.2) if the guaranteed results of u ∗ (z, t) and v ∗ (z, t) in this game are equal to each other for all z 0 ∈ E n , i.e.,     Ju u ∗ (z, t); z 0 = Jv v ∗ (z, t); z 0

∀z 0 ∈ E n .

(3.5)

If this equality is valid, then the value     J ∗ (z 0 ) = Ju u ∗ (z, t); z 0 = Jv v ∗ (z, t); z 0

(3.6)

is called a value of the game (3.1)–(3.2). Consider the following terminal-value problem for the n × n-matrix-valued function P(t) in the time interval [0, t f ]: d P(t) = −P(t)A(t) − A T (t)P(t) + P(t)[Su (t) − Sv (t)]P(t) − D(t), dt P(t f ) = F, (3.7) where

T Su (t) = B(t)G −1 u (t)B (t),

T Sv (t) = C(t)G −1 v (t)C (t).

(3.8)

Note that the differential equation in the problem (3.7) is a matrix Riccati differential equation. We assume that A1. The problem (3.7) has the solution P(t) = P ∗ (t) in the entire interval [0, t f ].

54

3 Preliminaries

Remark 3.1 Conditions, providing the validity of the assumption A1, can be found in [3] (Chap. 6). General theory of matrix Riccati differential equations can be found in [1]. Also let us note that, due to the uniqueness of P ∗ (t), t ∈ [0, t f ], this matrix belongs to the set Sn . Based on P ∗ (t), we construct the functions T ∗ ∗ −1 T ∗ u ∗ (z, t) = −G −1 u (t)B (t)P (t)z ∈ Ut f , v (z, t) = G v (t)C (t)P (t)z ∈ Vt f . (3.9)

Theorem 3.1 Let the assumption  A1 be valid. Then (a) the pair u ∗ (z, t), v ∗ (z, t) , given by (3.9), is the saddle-point solution of the game (3.1)–(3.2); (b) the value of this game has the form   J ∗ (z 0 ) = J u ∗ (z, t), v ∗ (z, t) = z 0T P ∗ (0)z 0 ;

(3.10)

    (c) for any u(z, t) ∈ Fu,t f v ∗ (z, t) and any v(z, t) ∈ Fv,t f u ∗ (z, t) , the saddle ∗  point solution u (z, t), v ∗ (z, t) of the game (3.1)–(3.2) satisfies the following inequality:       J u ∗ (z, t), v(z, t) ≤ J u ∗ (z, t), v ∗ (z, t) ≤ J u(z, t), v ∗ (z, t) ;

(3.11)

(d) for any t ∈ [0, t f ], P ∗ (t) ∈ Sn+ meaning that J ∗ (z 0 ) ≥ 0. Proof First of all, let us note the following. Since u ∗ (z, t) and v ∗ (z, t) are linear with respect to z with the gain matrices continuous in t ∈ [0, t f ], then  functions u ∗ (z, t), v ∗ (z, t) ∈ (U V )t f . For any t ∈ [0, t f ], consider the Lyapunov-like function V (z, t) = z T P ∗ (t)z,

z ∈ En.

(3.12)

  For any given pair u(z, t), v(z, t) ∈ (U V )t f and any given z 0 ∈ E n , we consider the solution z uv (t; z 0 ), t ∈ [0, t f ], of the initial-value problem (3.1) with u(t) = u(z, t), v(t) = v(z, t). By Vuv (t), let us denote the time realization of the function (3.12) along this solution, i.e.,    Vuv (t) = V z uv (t; z 0 ), t .

(3.13)

Also, by u(z uv , t) and v(z uv , t), we denote the time realizations of the controls u(z, t) and v(z, t), respectively, along the solution z uv (t; z 0 ). Now, calculating the derivative d Vuv (t)/dt, we obtain the following expression:

3.2 Solvability Conditions of Regular Problems

55



d Vuv (t) dz uv (t; z 0 ) T ∗ P (t)z uv (t; z 0 ) =2 dt dt d P ∗ (t) T (t; z 0 ) z uv (t; z 0 ). +z uv dt (3.14) Using (3.1) and (3.7), we can rewrite Eq. (3.14) as d Vuv (t) T (t; z 0 )D(t)z uv (t; z 0 ) = −z uv dt   T +z uv (t; z 0 )P ∗ (t) Su (t) − Sv (t) P ∗ (t)z uv (t; z 0 ) T  +2 B(t)u(z uv , t) + C(t)v(z uv , t) P ∗ (t)z uv (t; z 0 ).

(3.15)

Finally, using (3.9), Eq. (3.15) can be rewritten in the form d Vuv (t) T (t; z 0 )D(t)z uv (t; z 0 ) = −z uv dt  T  T − u(z uv , t) G u (t)u(z uv , t) + v(z uv , t) G v (t)v(z uv , t) T    + u(z uv , t) − u ∗ (z uv , t) G u (t) u(z uv , t) − u ∗ (z uv , t) T    − v(z uv , t) − v ∗ (z uv , t) G v (t) v(z uv , t) − v ∗ (z uv , t) ,

(3.16)

where t ∈ [0, t f ]; u ∗ (z uv , t) and v ∗ (z uv , t) are the time realizations of the controls u ∗ (z, t) and v ∗ (z, t), respectively, along z uv (t; z 0 ). Integrating Eq. (3.16) from t = 0 to t = t f and using Eq. (3.2), the terminal condition in the problem (3.7), as well as Eqs. (3.12)–(3.13), we obtain after a routine rearrangement 

tf

+

  J u(z, t), v(z, t) = z 0T P ∗ (0)z 0  T   u(z uv , t) − u ∗ (z uv , t) G u (t) u(z uv , t) − u ∗ (z uv , t)

0

 T   − v(z uv , t) − v ∗ (z uv , t) G v (t) v(z uv , t) − v ∗ (z uv , t) dt.

(3.17)

This equation, along with Definitions 3.2–3.4, yields immediately the statements (a), (b) and (c) of the theorem. Proceed to statement (d). Consider the pair of the players’ feedback controls, consisting of u = u ∗ (z, t) and v ≡ 0. This pair is admissible in the game (3.1)–(3.2). ∗ (t; z 0 ), t ∈ [0, t f ], be the solution of the problem Let, for any z 0 ∈ E n , the function z u0 ∗ (3.1) with u(t) = u (z, t) and v(t) ≡ 0. By Vu0 (t), we denote the time realization of the function (3.12) along this solution, i.e.,  ∗   (t; z 0 ), t . Vu0 (t) = V z u0

(3.18)

56

3 Preliminaries

Based on these notations and using Eq. (3.16), we obtain the equation T  ∗ d Vu0 (t) ∗ (t; z 0 ) D(t)z u0 (t; z 0 ) = − z u0 dt T  T  ∗ ∗ ∗ ∗ , t) G u (t)u ∗ (z u0 , t) − v ∗ (z u0 , t) G v (t)v ∗ (z u0 , t), − u ∗ (z u0

(3.19)

 ∗ ∗ where t ∈ [0, t f ]; u ∗ (z u0 , t) and v ∗ (z u0 , t are the time realizations of the controls ∗ (t; z 0 ). u ∗ (z, t) and v ∗ (z, t), respectively, along the solution z u0 n r Since for all t ∈ [0, t f ], D(t) ∈ S+ , G u (t) ∈ S++ and G v (t) ∈ Ss++ , then Eq. (3.19) yields the inequality d Vu0 (t)/dt ≤ 0, t ∈ [0, t f ]. Integration of this inequality from any given t ∈ [0, t f ] to t = t f , as well as use of Eqs. (3.12), (3.18), the terminal condition in the problem (3.7) and the inclusion F ∈ Sn+ directly yield the chain of the inequalities  Thus,

T

∗ (t; z 0 ) z u0



 ∗ T ∗ ∗ P ∗ (t)z u0 (t; z 0 ) ≥ z u0 (t f ; z 0 ) F z u0 (t f ; z 0 ) ≥ 0. T

∗ (t; z 0 ) z u0

∗ P ∗ (t)z u0 (t; z 0 ) ≥ 0,

t ∈ [0, t f ].

(3.20)

Since, for a given t ∈ [0, t f ] and a properly pre-chosen z 0 , we can obtain any value of ∗ (t; z 0 ) ∈ E n , then the inequality (3.20) implies the inclusion P ∗ (t) ∈ the vector z u0 n ⊔ ⊓ S+ for any t ∈ [0, t f ]. This completes the proof of the theorem.

3.2.2 Infinite-Horizon Differential Game Here, we consider the following differential equation controlled by two decision makers (players): dz(t) = Az(t) + Bu(t) + Cv(t), z(0) = z 0 , t ≥ 0, dt

(3.21)

where z(t) ∈ E n is a state vector; u(t) ∈ E r and v(t) ∈ E s are controls of the players; A, B and C are given constant matrices of corresponding dimensions; z 0 ∈ E n is a given vector. The cost functional in the infinite-horizon game differs from (3.2). Namely, it has the form J (u, v) =

+∞ 

 z T (t)Dz(t) + u T (t)G u u(t) − v T (t)G v v(t) dt,

(3.22)

0

where D, G u and G v are given constant matrices; D ∈ Sn+ , while G u ∈ Sr++ and G v ∈ Ss++ .

3.2 Solvability Conditions of Regular Problems

57

The functional (3.22) is minimized by the control u (the minimizer) and maximized by the control v (the maximizer). Consider the set U of all functions u = u(z) : E n → E r satisfying the local Lipschitz condition. Similarly, let V be the set of all functions v = v(z) : E n → E s satisfying the local Lipschitz condition.   Definition 3.5 By (U V ), we denote the set of all pairs u(z), v(z) , z ∈ E n , such that the following conditions are valid: (i) u(z) ∈ U, v(z) ∈ V; (ii) the initial-value problem (3.21) for u(t) = u(z), v(t) = v(z) and any z 0 ∈ E n has the unique soluentire interval [0, +∞); (iii) z uv (t; z 0 ) ∈ L 2 [0, +∞; E n ]; (iv) tion  z uv (t; z0 ) in the 2 u z uv(t; z 0 ) ∈ L [0, +∞; E r ]; (v) v z uv (t; z 0 ) ∈ L 2 [0, +∞; E s ]. Such a defined set (U V ) is called the set of all admissible pairs of the players’ state-feedback controls in the game (3.21)–(3.22).     For a given u(z) ∈ U, consider the set Fv u(z) = v(z) ∈ V : u(z), v(z) ∈       (U V ) . Let Hu = u(z) ∈ U : Fv u(z) = ∅ . Definition 3.6 For a given function u(z) ∈ Hu , the value   Ju u(z); z 0 =

   J u(z), v(z)

sup

(3.23)

v(z)∈Fv u(z)

is called the guaranteed result of u(z) in the game (3.21)–(3.22).   Similarly, for a given v(z) ∈ V, consider the set Fu v(z) = u(z) ∈ U :         u(z), v(z) ∈ (U V ) . Let Hv = v(z) ∈ V : Fu v(z)  = ∅ . Definition 3.7 For a given function v(z) ∈ Hv , the value   Jv v(z); z 0 =

inf

   J u(z), v(z)

(3.24)

u(z)∈Fu v(z)

is called the guaranteed result of v(z) in the game (3.21)–(3.22).   Consider a pair u ∗ (z), v ∗ (z) ∈ (U V ).   Definition 3.8 The pair u ∗ (z), v ∗ (z) is called a saddle-point equilibrium solution (or briefly, a saddle-point solution) of the game (3.21)–(3.22) if the guaranteed results of u ∗ (z) and v ∗ (z) in this game are equal to each other for all z 0 ∈ E n , i.e.,     Ju u ∗ (z); z 0 = Jv v ∗ (z); z 0

∀z 0 ∈ E n .

(3.25)

If this equality is valid, then the value     J ∗ (z 0 ) = Ju u ∗ (z); z 0 = Jv v ∗ (z); z 0 is called a value of the game (3.21)–(3.22).

(3.26)

58

3 Preliminaries

Consider the following matrix Riccati algebraic equation with respect to the n × nmatrix P: (3.27) P A + A T P − P(Su − Sv )P + D = 0, where

T Su = BG −1 u B ,

T Sv = C G −1 v C .

(3.28)

We assume A2. Equation (3.27) has a solution P = P ∗ ∈ Sn such that the trivial solution of each of the following systems is asymptotically stable: dz(t) = (A − Su P ∗ + Sv P ∗ )z(t), dt dz(t) = (A − Su P ∗ )z(t), dt

t ≥ 0,

t ≥ 0.

(3.29)

(3.30)

Remark 3.2 Conditions, providing the validity of the assumption A2, can be found in [3] (Chap. 6). General theory of matrix Riccati algebraic equations can be found in [1]. Using the matrix P ∗ , mentioned in the assumption A2, we construct the functions T ∗ ∗ −1 T ∗ u ∗ (z) = −G −1 u B P z ∈ U , v (z) = G v C P z ∈ V.

(3.31)

Theorem 3.2 A2 be valid. Then   Let the assumption (a) the pair u ∗ (z), v ∗ (z) , given by (3.31), is the saddle-point solution of the game (3.21)–(3.22); (b) the value of this game has the form   J ∗ (z 0 ) = J u ∗ (z), v ∗ (z) = z 0T P ∗ z 0 ;

(3.32)

    u(z) ∈ Fu v ∗ (z) and any v(z) ∈ Fv u ∗ (z) , the saddle-point solution  (c)∗ for any u (z), v ∗ (z) of the game (3.21)–(3.22) satisfies the following inequality:       J u ∗ (z), v(z) ≤ J u ∗ (z), v ∗ (z) ≤ J u(z), v ∗ (z) ;

(3.33)

(d) P ∗ ∈ Sn+ , meaning that J ∗ (z 0 ) ≥ 0. Proof Since u ∗ (z) and v ∗ (z) are linear functions of z, and the trivial solution of the  system (3.29) is asymptotically stable, then u ∗ (z), v ∗ (z) ∈ (U V ). Consider the Lyapunov-like function

3.2 Solvability Conditions of Regular Problems

V (z) = z T P ∗ z,

59

z ∈ En.

(3.34)

  For any given pair u(z), v(z) ∈ (U V ) and any given z 0 ∈ E n , we consider the solution z uv (t; z 0 ), t ∈ [0, +∞), of the initial-value problem (3.21) generated by u(t) = u(z), v(t) = v(z). By virtue of Definition 3.5, we obtain lim z uv (t; z 0 ) = 0.

(3.35)

t→+∞

By Vuv (t), let us denote the time realization of the function (3.34) along this solution. Thus,    (3.36) Vuv (t) = V z uv (t; z 0 ) , t ∈ [0, +∞). Furthermore, by u(z uv ) and v(z uv ), we denote the time realizations of the controls u(z) and v(z), respectively, along the solution z uv (t; z 0 ). Now, calculating the derivative d Vuv (t)/dt, we obtain the following expression:

d Vuv (t) dz uv (t; z 0 ) T ∗ P z uv (t; z 0 ), =2 dt dt

t ∈ [0, +∞).

(3.37)

Using (3.21) and (3.27), we can rewrite Eq. (3.37) in the form d Vuv (t) T (t; z 0 )Dz uv (t; z 0 ) = −z uv dt T +z uv (t; z 0 )P ∗ (Su − Sv )P ∗ z uv (t; z 0 )  T +2 Bu(z uv ) + Cv(z uv ) P ∗ z uv (t; z 0 ), t ∈ [0, +∞).

(3.38)

Finally, using (3.31), Eq. (3.38) can be rewritten as follows: d Vuv (t) T (t; z 0 )Dz uv (t; z 0 ) = −z uv dt  T  T − u(z uv ) G u u(z uv ) + v(z uv ) G v v(z uv ) T    + u(z uv ) − u ∗ (z uv ) G u u(z uv ) − u ∗ (z uv ) T    − v(z uv ) − v ∗ (z uv ) G v v(z uv ) − v ∗ (z uv ) , t ∈ [0, +∞),

(3.39)

where u ∗ (z uv ) and v ∗ (z uv ) are the time realizations of the controls u ∗ (z) and v ∗ (z), respectively, along z uv (t; z 0 ). The following should be noted. Since z uv (t; z 0 ) ∈ L 2 [0, +∞; E n ], then due to (3.31), u ∗ (z uv ) ∈ L 2 [0, +∞; E r ] and v ∗ (z uv ) ∈ L 2 [0, +∞; E s ]. Furthermore, since  u(z uv ) ∈ L 2 [0, +∞; E r ] and v(z uv ) ∈ L 2 [0, +∞; E s ], then (u(z uv ) − u ∗ (z uv ) ∈ L 2 [0, +∞; E r ] and (v(z uv ) − v ∗ (z uv ) ∈ L 2 [0, +∞; E s ]. Based on this observation, let us integrate Eq. (3.39) from t = 0 to +∞. This integration and use of Eqs. (3.22), (3.34), (3.35) and (3.36) yield after a routine rearrangement

60

3 Preliminaries

  J u(z), v(z) = z 0T P ∗ z 0  +∞  T   u(z uv ) − u ∗ (z uv ) G u u(z uv ) − u ∗ (z uv ) + 0

 T   − v(z uv ) − v ∗ (z uv ) G v v(z uv ) − v ∗ (z uv ) dt.

This equation, along with Definitions 3.6–3.8, yields immediately the statements (a), (b) and (c) of the theorem. Proceed to statement (d). Consider the pair of the players’ feedback controls, consisting of u = u ∗ (z) and v ≡ 0. Since the trivial solution of the system (3.30) is asymptotically stable, this pair is admissible in the game (3.21)–(3.22). Let, for any ∗ (t; z 0 ), t ∈ [0, +∞), be the solution of the problem (3.21) z 0 ∈ E n , the function z u0 ∗ generated by u(t) = u (z) and v(t) ≡ 0. Thus, lim z ∗ (t; z 0 ) t→+∞ u0

= 0.

(3.40)

By Vu0 (t), we denote the time realization of the function (3.34) along this solution, i.e.,  ∗   (t; z 0 ) . (3.41) Vu0 (t) = V z u0 Using these notations and Eq. (3.39), we obtain the equation T ∗  ∗ d Vu0 (t) (t; z 0 ) Dz u0 (t; z 0 ) = − z u0 dt  T  T ∗ ∗ ∗ ∗ − u ∗ (z u0 ) G u u ∗ (z u0 ) − v ∗ (z u0 ) G v v ∗ (z u0 ), t ∈ [0, +∞),

(3.42)

 ∗ ∗ where u ∗ (z u0 are the time realizations of the controls u ∗ (z) and v ∗ (z), ) and v ∗ (z u0 ∗ (t; z 0 ). respectively, along the solution z u0 n r Since D ∈ S+ , G u ∈ S++ and G v ∈ Ss++ , then Eq. (3.42) yields the inequality d Vu0 (t)/dt ≤ 0, t ∈ [0, +∞). Integration of this inequality from t = 0 to +∞ and use of Eqs. (3.34), (3.40) and (3.41) directly yield the inequality z 0T P ∗ z 0 ≥ 0

∀z 0 ∈ E n .

The latter implies immediately the inclusion P ∗ ∈ Sn+ . This completes the proof of the item (d) and the proof of the theorem. ⊔ ⊓

3.2.3 Finite-Horizon H∞ Problem The following controlled system is under the consideration:

3.2 Solvability Conditions of Regular Problems

dz(t) = A(t)z(t) + B(t)u(t) + C(t)w(t), dt

61

z(0) = 0, t ∈ [0, t f ],

(3.43)

where t f is a given final time moment; z(t) ∈ E n is a state vector; u(t) ∈ E r is a control; w(t) ∈ E s is an unknown disturbance; A(t), B(t) and C(t) are given matrixvalued functions of corresponding dimensions, continuous in the interval [0, t f ]. Along with the system (3.43), we consider the output equation   W (t) = col L(t)z(t), M(t)u(t) ,

t ∈ [0, t f ],

(3.44)

where W (t) ∈ E p is an output; L(t) and M(t) are given matrix-valued functions, continuous in the interval [0, t f ]; the dimensions of the matrices L(t) and M(t) are p1 × n and p2 × r ( p1 + p2 = p, r ≤ p2 ), respectively. Assuming that w(t) ∈ L 2 [0, t f ; E s ], we consider the functional  2  2 J (u, w) = z T (t f )F z(t f ) + W (t) L 2 [0,t f ] − γ 2 w(t) L 2 [0,t f ] ,

(3.45)

where a given matrix F ∈ Sn+ ; γ > 0 is a given constant called the performance level. Definition 3.9 By U H,t f , we denote the set of all functions u = u(z, t) : E n × [0, t f ] → E r such that the following conditions are valid: (i) u(z, t) is measurable w.r.t. t ∈ [0, t f ] for any fixed z ∈ E n and satisfies the local Lipschitz condition w.r.t. z ∈ E n uniformly in t ∈ [0, t f ]; (ii) the initial-value problem (3.43) for u(t) = u(z, t) continuous solution z uw (t) and any w(t) ∈ L 2 [0, t f ; E s ] has the unique absolutely  in the entire interval t ∈ [0, t f ]; (iii) u z uw (t), t ∈ L 2 [0, t f ; E r ]. Such a defined set U H,t f is called the set of all admissible controllers for the system (3.43)–(3.44) and the functional (3.45). Definition 3.10 The H∞ control problem for the system (3.43)–(3.44) with the functional (3.45) is to find a controller u ∗ (z, t) ∈ U H,t f that ensures the inequality J (u ∗ , w) ≤ 0

(3.46)

along trajectories of (3.43) with u(t) = u ∗ (z, t) for all w(t) ∈ L 2 [0, t f ; E s ]. Controller u ∗ (z, t) is called the solution of the H∞ control problem (3.43)–(3.45). Consider the matrices D(t) = L T (t)L(t), G(t) = M T (t)M(t), t ∈ [0, t f ].

(3.47)

Let us assume that A3. For all t ∈ [0, t f ], rank M(t) = r . The assumption A3 implies the invertibility of the matrix G(t) for all t ∈ [0, t f ]. Using the above-introduced matrices D(t) and G(t) in (3.47), as well as the assumption A3, we consider the following terminal-value problem for the n × nmatrix-valued function K (t) in the time interval [0, t f ]:

62

3 Preliminaries

d K (t) = −K (t)A(t) − A T (t)K (t) + K (t)[Su (t) − Sw (t)]K (t) − D(t), dt K (t f ) = F, (3.48) where

Su (t) = B(t)G −1 (t)B T (t),

Sw (t) = γ −2 C(t)C T (t).

(3.49)

We assume that A4. The problem (3.48) has the solution K (t) = K ∗ (t) in the entire interval [0, t f ]. By the same arguments as in Sect. 3.2.1, the matrix K ∗ (t) ∈ Sn for all t ∈ [0, t f ]. Theorem 3.3 Let the assumptions A3 and A4 be valid. Then (a) the controller u ∗ (z, t) = −G −1 (t)B T (t)K ∗ (t)z

(3.50)

solves the H∞ control problem for the system (3.43)–(3.44) with the functional (3.45); (b) for any t ∈ [0, t f ], K ∗ (t) ∈ Sn+ . Proof Let us start with the item (a). Since u ∗ (z, t) is a linear function with respect to z with the gain matrix continuous in t ∈ [0, t f ], then u ∗ (z, t) ∈ U H,t f . Let u(z, t) be any given function from U H,t f , w(t) be any given function from L 2 [0, t f ; E s ] and z uw (t), t ∈ [0, t f ], be the solution of the initial-value problem (3.43) with this w(t) and u(t) = u(z, t). Now, quite similar to the proof of Theorem 3.1 (see Eq. (3.17)) we obtain  tf   T    u(z uw , t) − u ∗ (z uw , t) G(t) u(z uw , t) − u ∗ (z uw , t) J u(z, t), w(t) = 0

 T   −γ 2 w(t) − w ∗ (t) w(t) − w ∗ (t) dt, (3.51)

    where u(z uw , t) = u z uw (t), t , u ∗ (z uw , t) = u ∗ z uw (t), t , w∗ (t) = γ −2 C T (t)K ∗ (t)z uw (t). Equation (3.51) directly yields the inequality (3.46), which proves the item (a). ⊔ ⊓ The item (b) is proven similar to the item (d) of Theorem 3.1.

3.2.4 Infinite-Horizon H∞ Problem Let us consider the system consisting of the controlled differential equation dz(t) = Az(t) + Bu(t) + Cw(t), dt and the algebraic output equation

z(0) = 0, t ≥ 0,

(3.52)

3.2 Solvability Conditions of Regular Problems

63

  W (t) = col Lz(t), Mu(t) ,

t ≥ 0,

(3.53)

where z(t) ∈ E n is a state vector; u(t) ∈ E r is a control; w(t) ∈ E s is an unknown disturbance; W (t) ∈ E p is an output; A, B and C are given constant matrices of corresponding dimensions; L and M also are given constant matrices having the dimensions p1 × n and p2 × r ( p1 + p2 = p, r ≤ p2 ), respectively. Subject to the assumption that w(t) ∈ L 2 [0, +∞; E s ], we consider the functional  2  2 J (u, w) = W (t) L 2 [0,+∞) − γ 2 w(t) L 2 [0,+∞) ,

(3.54)

where γ > 0 is a given constant called the performance level. Definition 3.11 By U H , we denote the set of all functions u = u(z) : E n → E r such that the following conditions are valid: (i) u(z) satisfies the local Lipschitz condition; (ii) the initial-value problem (3.52) for u(t) = u(z) and any w(t) ∈ L 2 [0, +∞; E s ] has the unique solution  z uw (t)  in the entire interval t ∈ [0, +∞); (iii) z uw (t) ∈ L 2 [0, +∞; E n ]; (iv) u z uw (t) ∈ L 2 [0, +∞; E r ]. Such a defined set U H is called the set of all admissible controllers for the system (3.52)–(3.53) and the functional (3.54). Definition 3.12 The H∞ control problem for the system (3.52)–(3.53) with the functional (3.54) is to find a controller u ∗ (z) ∈ U H that ensures the inequality J (u ∗ , w) ≤ 0

(3.55)

along trajectories of (3.52) with u(t) = u ∗ (z) for all w(t) ∈ L 2 [0, +∞; E s ]. Consider the matrices D = L T L , G = M T M.

(3.56)

Thus, the matrix D belongs at least to the set Sn+ , and the matrix G belongs at least to the set Sr+ . We assume that A5. rank M = r . The assumption A5 yields the invertibility of the matrix G, i.e., G ∈ Sr++ . Based on the above-introduced matrices D and G (see Eq. (3.56)), as well as the assumption A5, we consider the following Riccati algebraic equation with respect to n × n-matrix K : K A + A T K − K (Su − Sw )K + D = 0, where

Su = BG −1 B T ,

Sw = γ −2 CC T .

We assume that A6. Equation (3.57) has a solution K = K ∗ ∈ Sn such that

(3.57)

(3.58)

64

3 Preliminaries 

(I) the matrix Q(K ∗ ) = D + K ∗ Su K ∗ ∈ Sn++ ; (II) the trivial solution of the following system is asymptotically stable: dz(t) = (A − Su K ∗ )z(t), t ≥ 0. dt Theorem 3.4 Let the assumptions A5 and A6 be valid. Then (a) K ∗ ∈ Sn++ ; (b) the controller u ∗ (z) = −G −1 B T K ∗ z

(3.59)

(3.60)

solves the H∞ control problem for the system (3.52)–(3.53) with the functional (3.54). Proof Let us start with the item (a). Using the expression for the matrix Q(K ∗ ) and the fact that K ∗ is a solution of Eq. (3.57), we have the following equality:   K ∗ (A − Su K ∗ ) + (A − Su K ∗ )T K ∗ = − Q(K ∗ ) + K ∗ Sw K ∗ .

(3.61)

We can consider this equality as the algebraic Lyapunov equation (see, e.g., [8]) with ∗ T ∗ matrices ( A − respect to K ∗ with the coefficients’  Su K ) and ( A − Su K ), and the  ∗ ∗ ∗ non-homogeneity matrix − Q(K ) + K Sw K . Therefore, due to the item (II) of the assumption A6, the equality (3.61) yields K∗ =



+∞

     exp (A − Su K ∗ )T χ Q(K ∗ ) + K ∗ Sw K ∗ exp (A − Su K ∗ )χ dχ .

0

(3.62) A6), the matrix Since the matrix Q(K ∗ ) ∈ Sn++ (see the item (I) of the assumption   K ∗ Sw K ∗ belongs at least to the set Sn+ and the matrix exp (A − Su K ∗ )χ is nonsingular for all χ ∈ [0, +∞), then the matrix on the right-hand side of (3.62) belongs to the set Sn++ . Hence, K ∗ ∈ Sn++ . Proceed to item (b). Consider the Lyapunov-like function V (z) = z T K ∗ z,

z ∈ En.

(3.63)

∗ By z uw (t), t ≥ 0, we denote the solution of the initial-value problem (3.52) with  ∗   ∗ u(t) = u ∗ (z) and any given w(t) ∈ L 2 [0, +∞; E s ]. Let Vuw (t) = V z uw (t) . Then, ∗ (t) and using routing algebraic transformations (similar to those differentiating Vuw in the proofs of Theorems 3.5, 3.2), we obtain ∗ T ∗  T  ∗ d Vuw (t) ∗ ∗ (t) Dz uw (t) − u ∗ (z uw ) Gu ∗ (z uw ) + γ 2 w T (t)w(t) = − z uw dt  T   −γ 2 w(t) − w ∗ (t) w(t) − w ∗ (t) , t ≥ 0, (3.64) ∗ ∗ where u ∗ (z uw ) is the time realizations of the controller u ∗ (z) along z uw (t); w ∗ (t) = −2 T ∗ ∗ γ C K z uw (t).

3.2 Solvability Conditions of Regular Problems

65

From Eq. (3.64), we have the inequality for all t ≥ 0 

∗ (t) z uw

T

 T d V ∗ (t) ∗ ∗ ∗ Dz uw (t) + u ∗ (z uw ) Gu ∗ (z uw ) − γ 2 w T (t)w(t) ≤ − uw . dt

Integrating the latter from t = 0 to any t ≥ 0, as well as taking into account the initial condition in (3.52), Eq. (3.63) and the inclusion K ∗ ∈ Sn++ , yields the inequality  t  0

 T ∗  T ∗ ∗ ∗ (σ ) Dz uw (σ ) + u ∗ (z uw ) Gu ∗ (z uw ) − γ 2 w T (σ )w(σ ) dσ ≤ 0. z uw (3.65)

∗ Note that in the integrand in (3.65), u ∗ (z uw ) depends on σ ∈ [0, t]. 2 s Since w(t) ∈ L [0, +∞; E ], the inequality (3.65) yields

 t  0

 T ∗  T ∗ ∗ ∗ (σ ) Dz uw (σ ) + u ∗ (z uw ) Gu ∗ (z uw ) dσ ≤ z uw  2 γ 2 w(t) L 2 [0,+∞) ,

t ≥ 0.

(3.66)

Using Eqs. (3.58), (3.60) and the expression for the matrix Q(K ∗ ) (see the item (I) of the assumption A6), we can rewrite (3.66) as 

t 0



 2 T ∗ ∗ (σ ) Q(K ∗ )z uw (σ )dσ ≤ γ 2 w(t) L 2 [0,+∞) , z uw

t ≥ 0.

(3.67)

Since Q(K ∗ ) ∈ Sn++ , the integral on the left-hand side of the inequality (3.67) is  +∞  ∗ T a non-decreasing function of t ∈ [0, +∞). Therefore, the integral 0 z uw (σ ) ∗ ∗ 2 n Q(K ∗ )z uw (σ )dσ converges, meaning   that 2z uw (t) ∈ L r[0, +∞; E ]. The latter, ∗ ∗ along with Eq. (3.60), yields u z uw (t) ∈ L [0, +∞; E ]. Thus, due to Definition 3.11, u ∗ (z) ∈ U H . Now, the inequality (3.66) yields 

+∞ 0



 T ∗  T ∗ ∗ ∗ (σ ) Dz uw (σ ) + u ∗ (z uw ) Gu ∗ (z uw ) − γ 2 w T (σ )w(σ ) dσ ≤ 0, z uw

which implies the fulfilment of the inequality (3.55). This completes the proof of the item (b). ⊔ ⊓

66

3 Preliminaries

3.3 State Transformation in Linear System and Quadratic Functional Consider the following system: dζ (t) = A(t)ζ (t) + B(t)u(t) + C(t)ω(t), dt

ζ (0) = ζ0 , t ∈ [0, t f ],

(3.68)

where t f is a given final time moment; ζ (t) ∈ E n is a state vector; u(t) ∈ E r , (r ≤ n) and ω(t) ∈ E s are inputs; A(t), B(t) and C(t) are given matrix-valued functions of corresponding dimensions in the interval [0, t f ]; ζ0 ∈ E n is a given vector. Along with the system (3.68), we consider the following functional calculated on solutions of this system:  J (u, ω) = ζ T (t f )F ζ (t f ) +

tf

ζ T (t)D(t)ζ (t)  +u T (t)G(t)u(t) − ω T (t)Ω(t)ω(t) dt, 0

(3.69)

where a given matrix F ∈ Sn+ ; D(t), G(t) and Ω(t) are given matrix-valued functions of corresponding dimensions in the interval [0, t f ]; for any t ∈ [0, t f ], D(t) ∈ Sn+ and G(t) ∈ Sr+ , while Ω(t) ∈ Ss++ . Remark 3.3 In (3.68)–(3.69), u(t) and ω(t) represent either the players’ controls (the case of the zero-sum differential game), or the control and the disturbance, respectively (the case of the H∞ problem). In the latter case, ζ0 = 0; Ω(t) = γ 2 Is , where γ is a given positive number. In what follows of this section, we assume that the matrix G(t) has the form G(t) =

¯ G(t) 0

0 , 0

t ∈ [0, t f ],

¯ ∈ S++ (0 < q < r ). where for all t ∈ [0, t f ], the matrix G(t) Let us partition the matrix B(t) into blocks as q

  B(t) = B1 (t), B2 (t) , t ∈ [0, t f ], where the matrices B1 (t) and B2 (t) are of the dimensions n × q and n × (r − q). We assume that B(t) has full AI. The matrix  column rank r for all t ∈ [0, t f ].  AII. det B2T (t)D(t)B2 (t) = 0, t ∈ [0, t f ]. AIII. FB2 (t f ) = 0. AIV. The matrix-valued functions A(t), C(t), G(t) and Ω(t) are continuously differentiable in the interval [0, t f ].

3.3 State Transformation in Linear System and Quadratic Functional

67

AV. The matrix-valued functions B(t) and D(t) are twice continuously differentiable in the interval [0, t f ]. Remark 3.4 Let us clarify the aforementioned assumptions. Thus, the assumption AI means that all coordinates of the input u(t) influence the system (3.68). The assumptions AII–AIII are technical assumptions, allowing to transform the system (3.68) and the functional (3.69) to a much simpler form presented in the forthcoming Theorem 3.5. The assumptions AIV–AV provide a proper smoothness of corresponding matrix-valued time-dependent coefficients in the simplified form of the system (3.68) and the functional (3.69). This simplified form and the smoothness of its coefficients are used in the analysis and solution of the singular zero-sum differential games and H∞ control problems studied in Chaps. 4–7. In the further analysis, we use the notion of a complement matrix-valued function to B(t) and some properties of this complement matrix-valued function. Definition 3.13 For r < n and all t ∈ [0, t f ], the matrix-valued function Bc (t) is called a complement  to B(t) if the dimension of Bc (t) is n × (n − r ), and the block matrix Bc (t), B(t) is nonsingular for all t ∈ [0, t f ]. Remark 3.5 If r = n, B(t) is a square matrix and it is nonsingular for all t ∈ [0, t f ] (see the assumption AI). Therefore, in this case, there is not a necessity to complete B(t) in the sense of Definition 3.13. Due to this definition, if r < n, then the block   c (t) = Bc (t), B1 (t) is the complement to B2 (t) for form matrix-valued function B c (t) = B1 (t). all t ∈ [0, t f ]. If r = n, then the complement to B2 (t) is B Lemma 3.1 Let r < n. Let the assumption AI and the assumption AV with respect to B(t) be valid. Then, there exists a complement matrix-valued function Bc (t) to B(t), which is twice continuously differentiable for t ∈ [0, t f ]. Proof Let us construct an n-dimensional vector-valued function bc,1 (t)  = 0, t ∈ [0, t f ], which is orthogonal to each column of the matrix-valued function B(t) for all t ∈ [0, t f ]. Necessary and sufficient condition for the column-vector bc,1 (t) to be orthogonal to each column of B(t) is the fulfilment of the equality B T (t)bc,1 (t) = 0,

t ∈ [0, t f ].

(3.70)

This equality can be considered as a system of r linear algebraic equations with n unknowns (the entries of bc,1 (t)). In this system, the number of the equations is smaller than the number of the unknowns (r < n), the coefficients (the entries of B T (t)) are twice continuously differentiable functions of t ∈ [0, t f ] and rankB T (t) = r for all t ∈ [0, t f ]. Therefore, by virtue of the results of [7], there exists a solution bc,1 (t) of the system (3.70) such that bc,1 (t)  = 0 for all t ∈ [0, t f ] and bc,1 (t) is a twice continuously differentiable function of t ∈ [0, t f ]. Since this vector bc,1 (t) is orthogonal to each column of B(t) for all t ∈ [0, t f ], then the block    matrix Bc,1 (t) = bc,1 (t), B(t) has the column rank r + 1 for all t ∈ [0, t f ]. If

68

3 Preliminaries

r + 1 = n, then bc,1 (t) is the complement to B(t). If r + 1 < n, then one can construct, similar to the construction of bc,1 (t), an n-dimensional vector-valued function bc,2 (t), which is non-zero, orthogonal to each column of Bc,1 (t) and twice 

for all t ∈ [0, t f ]. Hence, the block matrix Bc,2 (t) =  continuously differentiable bc,2 (t), bc,1 (t), B(t) has the column rank r + 2 for all t ∈ [0, t f ]. Repeating the above-described procedure n − r times, we   construct the matrix-valued function Bc (t) = bc,n−r (t), bc,n−r −1 (t), . . . , bc,1 (t) , which is the twice continuously differentiable complement to B(t) for all t ∈ [0, t f ]. This completes the proof of the lemma. ⊔ ⊓ For all t ∈ [0, t f ], we consider the following matrices: −1   c (t), H(t) = B2T (t)D(t)B2 (t) B2T (t)D(t)B

 c (t) − B2 (t)H(t). L(t) = B (3.71) Using the matrix L(t), we transform the state in the system (3.68) and the functional (3.69) as    (3.72) ζ (t) = R(t)z(t), R(t) = L(t), B2 (t) , t ∈ [0, t f ],

where z(t) ∈ E n is a new state. Lemma 3.2 Let the assumptions AI, AII and AV be valid. Then, for any t ∈ [0, t f ], the matrix R(t) is invertible. Moreover, the matrix-valued function R(t) is twice continuously differentiable in the interval [0, t f ]. point inthe Proof We start with the first statement of the lemma. Let t be any given  c (t), B2 (t)  = interval [0, t f ]. Due to Definition 3.13 and Remark 3.5, we have det B 0. Using this inequality, the form of the matrices L(t) and R(t), the assumption AII and the invariance property of the determinant of a square matrix, we obtain     c (t), B2 (t)  = 0, c (t) − B2 (t)H(t), B2 (t) = det B det R(t) = det B which means the invertibility of R(t) for any t ∈ [0, t f ]. The second statement of the lemma directly follows from the form of R(t), the ⊔ ⊓ assumptions AI, AII, AV and Lemma 3.1. Let us partition the matrix H(t) into blocks as   H(t) = H1 (t), H2 (t) ,

(3.73)

where the blocks H1 (t) and H2 (t) are of the dimensions (r − q) × (n − r ) and (r − q) × q. Theorem 3.5 Let the assumptions AI–AV be valid. Then, the transformation (3.72) converts the system (3.68) and the functional (3.69) to the new system dz(t) = A(t)z(t) + B(t)u(t) + C(t)ω(t), dt

t ∈ [0, t f ], z(0) = z 0 ,

(3.74)

3.3 State Transformation in Linear System and Quadratic Functional

69

and the new functional  J (u, ω) = z (t f )F z(t f ) +

tf



z T (t)D(t)z(t)  +u T (t)G(t)u(t) − ω (t)Ω(t)ω(t) dt, T

0 T

(3.75)

where   A(t) = R−1 (t) A(t)R(t) − dR(t)/dt , C(t) = R−1 (t)C(t), z 0 = R−1 (0)ζ0 , (3.76) ⎛ ⎞ O(n−r )×q O(n−r )×(r −q) ⎠, Oq×(r −q) (3.77) B(t) = R−1 (t)B(t) = ⎝ Iq H2 (t) Ir −q F = RT (t f )FR(t f ) =

F1 O(n−r +q)×(r −q) O(r −q)×(n−r +q) O(r −q)×(r −q)

D(t) = RT (t)D(t)R(t) =

D1 (t) O(r −q)×(n−r +q)



O(n−r +q)×(r −q) D2 (t)

,

(3.78)

,

(3.79)

F1 = LT (t f )FL(t f ), D1 (t) = LT (t)D(t)L(t), D2 (t) = B2T (t)D(t)B2 (t). (3.80) n−r +q

r −q

, while the matrix D2 (t) ∈ S++ . The For all t ∈ [0, t f ], the matrix D1 (t) ∈ S+ n−r+q matrix F1 ∈ S+ . Moreover, the matrix-valued functions A(t), B(t), C(t) and D(t) are continuously differentiable in the interval [0, t f ]. Proof We prove the theorem for the case r < n. The case r = n is treated similarly. We start with the proof of the form (3.76)–(3.77) of the coefficients and the initial state value in the system (3.74). The representations for the coefficients A(t), C(t) and the initial state value z 0 are obtained straightforward using the smoothness and the invertibility of R(t) in the interval [0, t f ]. Proceed to the coefficient B(t). Since the matrix R(t) is invertible, then in order to prove the expression for B(t), it is necessary and sufficient to show that R(t)B(t) = B(t),

t ∈ [0, t f ].

(3.81)

c (t) (see Using Eqs. (3.71)–(3.72), (3.73) and the block form of the matrix B Remark 3.5) yields   R(t) = Bc (t) − B2 (t)H1 (t), B1 (t) − B2 (t)H2 (t), B2 (t) , t ∈ [0, t f ]. Substitution of this block matrix and the block matrix in the expression for B(t) (see (3.77)) into the left-hand side of (3.81) yields after a routine algebra the validity of

70

3 Preliminaries

this equation. Thus, the expression for B in (3.77) is proven. Now, let us prove the expressions (3.78) and (3.79) for the coefficients F and D(t) in the new functional (3.75). We start with (3.78). Using the expression for R(t) (see Eq. (3.72)), we obtain  T   F = RT (t f )FR(t f ) = L(t f ), B2 (t f ) F L(t f ), B2 (t f )



T T   L (t f )FL(t f ) LT (t f )FB2 (t f ) L (t f ) F L(t f ), B2 (t f ) = . (3.82) = B2T (t f ) B2T (t f )FL(t f ) B2T (t f )FB2 (t f ) This equation, along with the assumption AIII and the expression for F1 in (3.80), directly yields Eq. (3.78). Proceed to the proof of (3.79). Similar to (3.82), we have D(t) = RT (t)D(t)R(t) =

LT (t)D(t)L(t) LT (t)D(t)B2 (t) . B2T (t)D(t)L(t) B2T (t)D(t)B2 (t)

(3.83)

Using (3.71) yields

c (t) B2T (t)D(t)L(t) = B2T (t)D(t) B   −1 c (t) = 0, t ∈ [0, t f ]. −B2 (t) B2T (t)D(t)B2 (t) B2T (t)D(t)B The latter, along with (3.83) and (3.80), proves (3.79). The positive semi-definiteness of D1 (t) directly follows from the expression for this matrix in (3.80) and such a property of the matrix D(t). In the similar way, we obtain the positive semidefiniteness of F1 . The positive definiteness of the matrix D2 (t) directly follows from the expression for this matrix in (3.80) and the assumption AII. The smoothness of the matrix-valued functions A(t), B(t), C(t) and D(t) follows immediately from the expressions for these functions (3.76), (3.77), (3.79), the assumptions AIV-AV and Lemma 3.2. ⊔ ⊓

3.4 Concluding Remarks and Literature Review In this chapter (Sects. 3.2.1 and 3.2.2), the regular finite- and infinite-horizon linearquadratic zero-sum differential games were considered. The definitions of the admissible state-feedback players’ controls in these games, as well as the definitions of their saddle-point solutions, were presented. The solvability conditions and the solutions of these games were derived. The regular linear-quadratic zero-sum differential game in both, finite-horizon and infinite-horizon, settings were extensively studied in the literature. Various approaches to this study, including definitions of admissible players’ controls, definitions of saddle-point solutions and methods of deriving such solutions, can be found for instance in [2–4, 16–18] and references therein. In the present chapter, the study

References

71

of the regular linear-quadratic zero-sum differential games is based on the results of the works [9, 10, 13]. However, the definitions of the saddle-point solution and the game value proposed in the present chapter differ from the definitions of such notions in these works. Moreover, the material of Sects. 3.2.1 and 3.2.2 is presented in a much more detailed form than in [9, 10, 13]. In these subsections, the derivation of the saddle-point solutions to the considered games is carried out by the analysis of properly chosen Lyapunov-like functions. In Sects. 3.2.3 and 3.2.4 of the present chapter, the regular finite-horizon and infinite-horizon linear-quadratic H∞ control problems were considered. The definitions of the admissible state-feedback controls in these problems, as well as the definitions of their solutions, were presented. The solvability conditions and the solutions of these problems were derived. The regular H∞ control problems were studied extensively in the literature (see, e.g., [2, 5, 6, 19] and references therein). In the present chapter, the controller solving the H∞ problem is derived by treatment of some Lyapunov-like function. In Sect. 3.2.4, where the infinite-horizon problem is analysed, the novel assumption is proposed (see the item (I) of the assumption A6). This assumption guarantees that the solutions of the differential system in the H∞ problem subject to the controller, solving this problem, are quadratically integrable in the infinite interval for all disturbances with the same feature. In Sect. 3.3 of the present chapter, the linear state transformation in the linear timedependent controlled differential system and the quadratic functional was considered. This transformation allows converting the original system to a new system consisting of three modes. One of these modes does not contain the control. The second mode contains only a part of the control’s coordinates, while the third mode contains all the coordinates of the control. Moreover, this transformation decomposes the quadratic state cost in the functional into two parts. The first part contains the state’s coordinates of the first two modes, while the second part contains the state’s coordinates of the third mode. This transformation will be used in the next chapters for analysis and solution of singular games and H∞ problems. Transformations, similar to the one studied in this chapter, were considered in the literature (see [11, 12, 14, 15, 20]). In all these works, excepting [15], the transformations were applied to differential systems and functionals with constant coefficients. In [15], the transformation was applied to time-dependent system and functional. Moreover, in [11, 20] a particular case was considered. In this case, the transformation converts the system to a new system consisting of two modes. In the present chapter, the above-mentioned transformation is studied in much more detail. This study includes detailed proofs of the invertibility and smoothness of the transformation’s time-dependent gain matrix.

References 1. Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkh¨auser, Basel, Switzerland (2003) 2. Basar, T., Bernhard, P.: H∞ -Optimal Control and Related Minimax Design Problems: A Dynamic Games Approach, 2nd edn. Birkhauser, Boston, MA, USA (1995)

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3. Basar, T., Olsder, G.J.: Dynamic Noncooparative Game Theory, 2nd edn. SIAM Books, Philadelphia, PA, USA (1999) 4. Bryson, A.E., Ho, Y.C.: Applied Optimal Control: Optimization. Estimation and Control, Hemisphere, New York, NY, USA (1975) 5. Chang, X.H.: Robust Output Feedback H∞ Control and Filtering for Uncertain Linear Systems. Springer, Berlin, Germany (2014) 6. Doyle, J.C., Glover, K., Khargonekar, P.P., Francis, B.: State-space solution to standard H2 and H∞ control problems. IEEE Trans. Automat. Control 34, 831–847 (1989) 7. Fefferman, C., Kollar, ´ J.: Continuous solutions of linear equations. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds.) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol. 28, 233–282. Springer, New York, NY, USA (2013) 8. Gajic, Z., Qureshi, M.T.J.: Lyapunov Matrix Equation in System Stability and Control. Dover Publications, Mineola, NY, USA (2008) 9. Glizer, V.Y.: Saddle-point equilibrium sequence in one class of singular infinite horizon zerosum linear-quadratic differential games with state delays. Optimization 68, 349–384 (2019) 10. Glizer, V.Y.: Saddle-point equilibrium sequence in a singular finite horizon zero-sum linearquadratic differential game with delayed dynamics. Pure Appl. Funct. Anal. 6, 1227–1260 (2021) 11. Glizer, V.Y., Fridman, L., Turetsky, V.: Cheap suboptimal control of an integral sliding mode for uncertain systems with state delays. IEEE Trans. Automat. Control 52, 1892–1898 (2007) 12. Glizer, V.Y., Kelis, O.: Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser. J. Control Decis. 2, 155–184 (2015) 13. Glizer, V.Y., Kelis, O.: Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence. Numer. Algebra Control Optim. 7, 1–20 (2017) 14. Glizer, V.Y., Kelis, O.: Solution of a singular H∞ control problem: a regularization approach. In: Proceedings of the 14th International Conference on Informatics in Control. Automation and Robotics, pp. 25–36, Madrid, Spain (2017) 15. Glizer, V.Y., Kelis, O.: Finite-horizon H∞ control problem with singular control cost. In: Gusikhin, O., Madani, K. (eds.) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol. 495, pp. 23–46. Springer Nature, Switzerland (2020) 16. Jacobson, D.H.: On values and strategies for infinite-time linear quadratic games. IEEE Trans. Automat. Control 22, 490–491 (1977) 17. Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer, New York, NY, USA (1988) 18. Mageirou, E.F.: Values and strategies for infinite time linear quadratic games. IEEE Trans. Automat. Control 21, 547–550 (1976) 19. Petersen, I., Ugrinovski, V., Savkin, A.V.: Robust Control Design using H∞ Methods. Springer, London, UK (2000) 20. Shinar, J., Glizer, V.Y., Turetsky, V.: Solution of a singular zero-sum linear-quadratic differential game by regularization. Int. Game Theory Rev. 16, 1440007-1–1440007-32 (2014)

Chapter 4

Singular Finite-Horizon Zero-Sum Differential Game

4.1 Introduction In this chapter, a finite-horizon zero-sum linear-quadratic differential game, which cannot be solved by application of the first-order solvability conditions, presented in Sect. 3.2.1, is considered. Thus, this game is singular. Its singularity is due to the singularity of the weight matrix in the control cost of a minimizing player in the game’s functional. For this game, novel definitions of a saddle-point equilibrium (a saddle-point equilibrium sequence) and a game value are introduced. Regularization method is proposed for obtaining these saddle-point equilibrium sequence and game value. This method consists in an approximate replacement of the original singular game with an auxiliary regular finite-horizon zero-sum linear-quadratic differential game depending on a small positive parameter. Thus, the first-order solvability conditions are applicable for this new game. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix differential equation, arising in these conditions, yields the solution (the saddle-point equilibrium sequence and the game value) to the original singular game. The following main notations are applied in the chapter. 1. E n is the n-dimensional real Euclidean space. 2.  ·  denotes the Euclidean norm either of a vector or of a matrix. 3. The superscript “T ” denotes the transposition of a matrix A, (A T ) or of a vector x, (x T ). 4. L 2 [a, b; E n ] is the linear space of n-dimensional vector-valued real functions, square-integrable in the finite interval [a, b], and  ·  L 2 [a,b] denotes the norm in this space. 5. On 1 ×n 2 is used for the zero matrix of the dimension n 1 × n 2 , excepting the cases where the dimension of the zero matrix is obvious. In such cases, the notation 0 is used for the zero matrix. 6. In is the n-dimensional identity matrix. 7. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block-vector of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_4

73

74

4 Singular Finite-Horizon Zero-Sum Differential Game

dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 8. diag(a1 , a2 , ..., an ), where ai , (i = 1, 2, ..., n) are real numbers, is the diagonal matrix with these numbers on the main diagonal. 9. Sn is the set of all symmetric matrices of the dimension n × n. 10. Sn+ is the set of all symmetric positive semi-definite matrices of the dimension n × n. 11. Sn++ is the set of all symmetric positive definite matrices of the dimension n × n.

4.2 Initial Game Formulation The dynamics of the game is described by the following differential equation: dζ (t) = A(t)ζ (t) + B(t)u(t) + C(t)v(t), ζ (0) = ζ0 , t ∈ [0, t f ], dt

(4.1)

where t f is a given final time moment; ζ (t) ∈ E n is a state vector; u(t) ∈ E r , (r ≤ n) and v(t) ∈ E s are controls of the game’s players; A(t), B(t) and C(t), t ∈ [0, t f ] are given matrix-valued functions of corresponding dimensions; ζ0 ∈ E n is a given vector. The cost functional of the game, to be minimized by the control u (the minimizer) and maximized by the control v (the maximizer), is J (u, v) = ζ T (t f )F ζ (t f ) t f +



 ζ T (t)D(t)ζ (t) + u T (t)G u (t)u(t) − v T (t)G v (t)v(t) dt,

(4.2)

0

where a given matrix F ∈ Sn+ ; D(t), G u (t), G v (t), t ∈ [0, t f ] are given matrix-valued functions of corresponding dimensions; for any t ∈ [0, t f ], D(t) ∈ Sn+ , G v (t) ∈ Ss++ , while G u (t) has the form  G u (t) = where

 G¯ u (t) 0 , 0 0

q G¯ u (t) ∈ S++ ,

0 < q < r,

t ∈ [0, t f ],

(4.3)

t ∈ [0, t f ].

Remark 4.1 Due to (4.3), the matrix G u (t) is not invertible, meaning that the results of Sect. 3.2.1 are not applicable to the game (4.1)–(4.2). Thus, this game is singular which requires another approach to its solution. t f of all functions u =  u (ζ, t) : E n × [0, t f ] → E r , which are Consider the set U measurable w.r.t. t ∈ [0, t f ] for any fixed ζ ∈ E n and satisfy the local Lipschitz t f be the set of all condition w.r.t. ζ ∈ E n uniformly in t ∈ [0, t f ]. Similarly, let V v (ζ, t) : E n × [0, t f ] → E s , which are measurable w.r.t. t ∈ [0, t f ] functions v = 

4.2 Initial Game Formulation

75

for any fixed ζ ∈ E n and satisfy the local Lipschitz condition w.r.t. ζ ∈ E n uniformly in t ∈ [0, t f ].   V )t f , we denote the set of all pairs  Definition 4.1 By (U v (ζ, t) , (ζ, t) ∈ u (ζ, t), t f ,  u (ζ, t) ∈ U v (ζ, t) ∈ E n × [0, t f ], such that the following conditions are valid: (i)  t f ; (ii) the initial-value problem (4.1) for u(t) =  V u (ζ, t), v(t) =  v (ζ, t) and any absolutely continuous solution ζ0 ∈ E n has the unique   ζuv (t; ζ0 ) in the entire interu ζuv (t; ζ0 ), t ∈ L 2 [0, t f ; E r ]; (iv)  v ζuv (t; ζ0 ), t ∈ L 2 [0, t f ; E s ]. val [0, t f ]; (iii)   Such a defined set (U V )t f is called the set of all admissible pairs of the players’ state-feedback controls in the game (4.1)–(4.2).  

t f : t f , consider the set F v,t f  u (ζ, t) =  v (ζ, t) ∈ V For a given  u (ζ, t) ∈ U   

   u,t f = t f : F v,t f  u (ζ, t)  = ∅ . Sim u (ζ, t) ∈ U  u (ζ, t), v (ζ, t) ∈ (U V )t f . Let H  

t f , consider the set F t f : u,t f  v (ζ, t) =  u (ζ, t) ∈ U ilarly, for a given  v (ζ, t) ∈ V     

 v,t f =  t f : F u,t f   u (ζ, t), v (ζ, t) ∈ (U V )t . Let H v (ζ, t)  = ∅ . v (ζ, t) ∈ V f

u,t f , the value u (ζ, t) ∈ H Definition 4.2 For a given   u (ζ, t); ζ0 = Ju 

 u (ζ, t), v (ζ, t) J 

sup

(4.4)

v,t   v (ζ,t)∈F u (ζ,t) f

is called the guaranteed result of  u (ζ, t) in the game (4.1)–(4.2). v,t f , the value Definition 4.3 For a given  v (ζ, t) ∈ H  v (ζ, t); ζ0 = Jv 

inf 

 u (ζ, t), v (ζ, t) J 

u,t   u (ζ,t)∈F v (ζ,t) f

(4.5)

is called the guaranteed result of  v (ζ, t) in the game (4.1)–(4.2).  ∗  u k (ζ, t), Consider a sequence of the pairs  v ∗ (ζ, t) ∈ (U V )t f , (k = 1, 2, ...).

 ∗  +∞ u k (ζ, t), v ∗ (ζ, t) k=1 is called a saddle-point equiDefinition 4.4 The sequence  librium sequence (or briefly, a saddle-point sequence) of the game (4.1)–(4.2) if for any ζ0 ∈ E n :  ∗ u k (ζ, t); ζ0 ; (a) there exist limk→+∞ Ju  (b) the following equality is valid:  ∗  ∗ u k (ζ, t); ζ0 = Jv  v (ζ, t); ζ0 . lim Ju 

k→+∞

(4.6)

In this case, the value  ∗  ∗ J ∗ (ζ0 ) = lim Ju  u k (ζ, t); ζ0 = Jv  v (ζ, t); ζ0 k→+∞

is called a value of the game (4.1)–(4.2).

(4.7)

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4 Singular Finite-Horizon Zero-Sum Differential Game

4.3 Transformation of the Initially Formulated Game  Let us partition the matrix B(t) into blocks as: B(t) = B1 (t), B2 (t) , t ∈ [0, t f ], where the matrices B1 (t) and B2 (t) are of the dimensions n × q and n × (r − q). We assume that: B(t) has full A4.I. The matrix column rank r for all t ∈ [0, t f ].  A4.II. det B2T (t)D(t)B2 (t) = 0, t ∈ [0, t f ]. A4.III. FB2 (t f ) = 0. A4.IV. The matrix-valued functions A(t), C(t), G¯ u (t) and G v (t) are continuously differentiable in the interval [0, t f ]. A4.V. The matrix-valued functions B(t) and D(t) are twice continuously differentiable in the interval [0, t f ]. Remark 4.2 Note that the assumptions A4.I–A4.V allow to construct and carry out the state transformation (3.72) in the system (4.1) and the functional (4.2) as it is presented below. This transformation converts the system (4.1) and the functional (4.2) to a much simpler form for the analysis and solution of the corresponding singular differential game. Let Bc (t) be a complement matrix-valued function to B(t) in the interval [0, t f ]   c (t) = Bc (t), B1 (t) is a complement matrix-valued (see Definition 3.13). Then, B function to B2 (t) in the interval [0, t f ]. Due to the assumptions A4.I and A4.V, as c (t) to be twice continuously well as Remark 3.5 and Lemma 3.1, we can choose B c (t) differentiable in the interval [0, t f ]. Using such a chosen matrix-valued function B and the above introduced matrix-valued function B2 (t), we construct the matrixvalued functions H(t) and L(t), t ∈ [0, t f ] (see the Eq. (3.71)). Then, based on these matrices, we construct and carry out the state transformation (3.72) in the system (4.1) and the functional (4.2). As a direct consequence of Lemma 3.2 and Theorem 3.5, we have the following assertion. Proposition 4.1 Let the assumptions A4.I–A4.V be valid. Then, the transformation (3.72) converts the system (4.1) and the functional (4.2) to the following system and functional: dz(t) = A(t)z(t) + B(t)u(t) + C(t)v(t), dt 

t ∈ [0, t f ], z(0) = z 0 ,

tf

J (u, v) = z T (t f )F z(t f ) +



z T (t)D(t)z(t)  +u T (t)G u (t)u(t) − v (t)G v (t)v(t) dt, 0 T

(4.8)

(4.9)

where the matrix-valued coefficients A(t), B(t), C(t), F, D(t) and the vector z 0 are given by (3.76)–(3.80). The coefficients A(t), B(t), C(t), D(t), G u (t), G v (t)

4.3 Transformation of the Initially Formulated Game

77

are continuously differentiable in the interval [0, t f ]. Moreover, in (3.80), D1 (t) ∈ n−r+q r −q n−r +q S+ , D2 (t) ∈ S++ for all t ∈ [0, t f ], F1 ∈ S+ . Like in the game (4.1)–(4.2), in the new game with the dynamics (4.8) and the functional (4.9) the objective of the control u is to minimize the functional, while the objective of the control v is to maximize the functional. Moreover, like the game (4.1)–(4.2), the new game (4.8)–(4.9) is singular. Remark 4.3 We choose the set (U V )t f (see Definition 3.1) to be the set of all  admissible pairs u(z, t), v(z, t) , (z, t) ∈ E n × [0, t f ] of the players’ state-feedback controls in the game (4.8)–(4.9). Moreover, we keep Definitions 3.2 and 3.3 (see Sect. 3.2.1) of the guaranteed results of the controls u(z, t) and v(z, t) to be valid in the game (4.8)–(4.9). Let ζ0 ∈ E n and z 0 ∈ E n be any given vectors such that ζ0 = R(0)z 0 and, therefore, z 0 = R−1 (0)ζ0 (the n × n-matrix R(t), t ∈ [0, t f ] is given by (3.71)–(3.72)). As a direct consequence of Definition 4.1, Remark 4.3, Lemma 3.2 and Proposition 4.1, we have the following assertion.   u (ζ, t), v (ζ, t) ∈ (U V)f Corollary 4.1 Let the assumptions A4.I–A4.V be valid. If  and ζuv (t; ζ0 ), t ∈ [0, t f ] is the solution of the initial-value problem (4.1) gener   u R(t)z, t , v R(t)z, t ∈ ated by this pair of the players’ controls, then the pair  (U V ) f and ζuv (t; ζ0 ) = R(t)z uv (t; z 0 ), t ∈ [0, t f ], where z uv (t; z 0 ), t ∈ [0, t f ] is the unique solution  of the initial-value  problem (4.8) generated  by the players’ v (ζ, t) = u(t) =  u R(t)z, t , v(t) =  v R(t)z, t . Moreover, J  u (ζ, t), controls 

  J  u R(t)z, t , v R(t)z, t . Vice versa: if u(z, t), v(z, t) ∈ (U V ) f and z uv (t; z 0 ), generated by this pair of the t ∈ [0, t f ] is the solution of the initial-value problem (4.8)  −1  −1  players’ controls, then u R (t)ζ, t , v R (t)ζ, t ∈ (U V ) f and z uv (t; z 0 ) = R−1 (t)ζuv (t; ζ0 ), t ∈ [0, t f ], where ζuv (t; ζ0 ), t ∈ [0, t f ] is the unique solution  −1 of the u(t) = u R (t)ζ, t , initial-value problem (4.1) generated by the players’ controls     v(t) = v R−1 (t)ζ, t . Moreover, J u(z, t), v(z, t) = J u R−1 (t)ζ, t , v R−1 (t) ζ, t .  Consider a sequence of the pairs u ∗k (z, t), v ∗ (z, t) ∈ (U V )t f , (k = 1, 2, ...).

 +∞ Definition 4.5 The sequence u ∗k (z, t), v ∗ (z, t) k=1 is called a saddle-point equilibrium sequence (or briefly, a saddle-point sequence) of the game (4.8)–(4.9) if for any z 0 ∈ E n :  (a) there exist limk→+∞ Ju u ∗k (z, t); z 0 ; (b) the following equality is valid:   lim Ju u ∗k (z, t); z 0 = Jv v ∗ (z, t); z 0 ,

k→+∞

(4.10)

  where Ju u ∗k (z, t); z 0 and Jv v ∗ (z, t); z 0 are the guaranteed results of the controls u ∗k (z, t) and v ∗ (z, t), respectively, in this game.

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4 Singular Finite-Horizon Zero-Sum Differential Game

The value   J ∗ (z 0 ) = lim Ju u ∗k (z, t); z 0 = Jv v ∗ (z, t); z 0 k→+∞

(4.11)

is called a value of the game (4.8)–(4.9). Lemma 4.1 Let the assumptions +∞ A4.I–A4.V be valid. If the sequence v ∗ (ζ, t) k=1 is a saddle-point sequence of the game (4.1)– of the pairs  u ∗k (ζ, t),   ∗ +∞ (4.2), then the sequence of the pairs  u ∗k R(t)z, t , v R(t)z, t is a saddlek=1 point sequence of  the game (4.8)–(4.9). Vice versa: if the sequence of the pairs

 +∞ u ∗k (z, t), v ∗ (z, t) k=1 is a saddle-point sequence of the game (4.8)–(4.9), then    +∞ the sequence of the pairs u ∗k R−1 (t)ζ, t , v ∗ R−1 (t)ζ, t is a saddle-point k=1 sequence of the game (4.1)–(4.2). Proof We start with the first statement of the lemma. Since the sequence

 ∗ +∞  u k (ζ, t), v ∗ (ζ, t) k=1 is a saddle-point sequence of the game (4.1)–(4.2), then  ∗ for any k ∈ {1, 2, ...} the pair of the players’ controls  v ∗ (ζ, t) is admisu k (ζ, t), sible in this game. by virtue of Corollary 4.1, the pair of the play  Therefore, ∗ ∗ u k R(t)z, t , v R(t)z, t is admissible in the game (4.8)–(4.9) ers’ controls   ∗ u k (ζ, t), v ∗ (ζ, t) = for (k = 1, 2, ...), and the following equality is valid: J  any  ∗ v R(t)z, t , (k = 1, 2, ...). Moreover, due to Definitions 4.2–4.3, J  u ∗k R(t)z, t , Remark 4.3 and Corollary 4.1, we have   ∗ u k (ζ, t); ζ0 = Ju  u ∗k R(t)z, t ; z 0 , k = 1, 2, ..., Ju    ∗ v (ζ, t); ζ0 = Jv  v ∗ R(t)z, t ; z 0 . Jv  These equalities, along with Definitions 4.4 and 4.5, prove the first statement of the lemma. The second statement is proven similarly. ⊔ ⊓ Remark 4.4 Due to Lemma 4.1, the initially formulated game (4.1)–(4.2) is equivalent to the new game (4.8)–(4.9). From the other hand, due to Proposition 4.1, the latter game is simpler than the former one. Therefore, in what follows of this chapter, we deal with the game (4.8)–(4.9). We consider this game as an original one and call it the Singular Finite-Horizon Game (SFHG).

4.4 Regularization of the Singular Finite-Horizon Game

79

4.4 Regularization of the Singular Finite-Horizon Game 4.4.1 Partial Cheap Control Finite-Horizon Game The first step in the solution of the SFHG is its regularization. This means that the SFHG is replaced with a regular differential game, which is close in some sense to the SFHG. This new game has the same dynamics (4.8) as the SFHG. However, in contrast with the SFHG, the functional in the new game has the “regular” form, i.e., it contains a quadratic control cost of the minimizer with a “regular” (positive definite) weight matrix:  tf  T z (t)D(t)z(t) Jε (u, v) = z T (t f )F z(t f ) + 0   +u T (t) G u (t) + Λ(ε) u(t) − v T (t)G v (t)v(t) dt, where

 Λ(ε) = diag 0, ..., 0, ε2 , ..., ε 2 ,       q

(4.12)

(4.13)

r −q

ε > 0 is a small parameter. Thus, taking into account the Eqs.  (4.3), (4.13) and the q inclusion G¯ u (t) ∈ S++ , we have immediately the inclusion G u (t) + Λ(ε) ∈ Sr++ for all t ∈ [0, t f ] and ε > 0. Remark 4.5 In the game (4.8), (4.12), the functional is minimized by the control u and maximized by the control v. Since the parameter ε > 0 is small, this game is a partial cheap control differential game, i.e., it is a differential game where a cost of some control coordinates of at least one player is much smaller than the costs of the other control coordinates and a state cost. Since for any ε > 0 the weight matrix for the minimizer’s control cost in the functional (4.12) is positive definite, game. the game (4.8), (4.12) is a regular differential The set of all admissible pairs  of players’ state-feedback controls u(z, t), v(z, t) in this game coincides with such a set in the SFHG, i.e., it is (U V )t f . Also, we keep Definitions 3.2 and 3.3 of the guaranteed results of the controls u(z, t) and v(z, t), as well as Definition 3.4 of the saddle-point solution, to be valid in the game (4.8), (4.12) for any given ε > 0. The above mentioned definitions can be found in Sect. 3.2.1.

4.4.2 Saddle-Point Solution of the Game (4.8), (4.12) Let us consider the following terminal-value problem for the n × n-matrix-valued function P(t) in the time interval [0, t f ]:

80

4 Singular Finite-Horizon Zero-Sum Differential Game

d P(t) = −P(t)A(t) − A T (t)P(t) + P(t)[Su (t, ε) − Sv (t)]P(t) − D(t), dt P(t f ) = F, (4.14) where  −1 T Su (t, ε) = B(t) G u (t) + Λ(ε) B T (t), Sv (t) = C(t)G −1 v (t)C (t).

(4.15)

We assume that: A4.VI. For a given ε > 0, the problem (4.14) has the solution P(t) = P ∗ (t, ε) in the entire interval [0, t f ]. Note that, due to the uniqueness of the solution to the problem (4.14), P ∗ (t, ε) ∈ n S for all [0, t f ]. Using this solution, we construct the functions u ∗ε (z, t) = −[G u (t) + Λ(ε)]−1 B T (t)P ∗ (t, ε)z, T ∗ vε∗ (z, t) = G −1 v (t)C (t)P (t, ε)z.

(4.16)

The following assertion is a direct consequence of Theorem 3.1. Proposition 4.2 Let the assumptions A4.I–A4.VI be valid. Then: (a) the pair u ∗ε (z, t), vε∗ (z, t) , given by (4.16), is the saddle-point solution of the game (4.8), (4.12); (b) the value of this game has the following form:  Jε∗ (z 0 ) = Jε u ∗ε (z, t), vε∗ (z, t) = z 0T P ∗ (0, ε)z 0 ;

(4.17)

  (c) for any u(z, t) ∈ Fu,t f vε∗ (z, t) and any v(z, t) ∈ Fv,t f u ∗ε (z, t) , the saddle ∗ point solution u ε (z, t), vε∗ (z, t) of the game (4.8), (4.12) satisfies the following inequality:    Jε u ∗ε (z, t), v(z, t) ≤ J u ∗ε (z, t), vε∗ (z, t) ≤ J u(z, t), vε∗ (z, t) ;

(4.18)

(d) for any t ∈ [0, t f ], P ∗ (t, ε) ∈ Sn+ meaning that Jε∗ (z 0 ) ≥ 0. Remember that the sets Fu,t f (·) and Fv,t f (·) are introduced in Sect. 3.2.1, and these sets are used in Definitions 3.2 and 3.3. Remark 4.6 In the next section, an asymptotic analysis (for ε → +0) of the game (4.8), (4.12) is carried out. Using this analysis, ε-free conditions for the existence of the solution P ∗ (t, ε) to the problem (4.14), mentioned in the assumption A4.VI, are established. All these results, will be used to derive the saddle-point sequence and the value of the SFHG.

4.5 Asymptotic Analysis of the Game (4.8), (4.12)

81

4.5 Asymptotic Analysis of the Game (4.8), (4.12) We start this analysis with an asymptotic solution with respect to the small parameter ε > 0 of the terminal value problem (4.14).

4.5.1 Transformation of the Problem (4.14) Using the block form of the matrix B(t) and the block-diagonal form of the matrix G u (t) + Λ(ε) (see the Eqs. (3.77) and (4.3), (4.13)), we can represent the matrix Su (t, ε) (see the Eq. (4.15)) in the block form as: ⎛ Su (t, ε) = ⎝

Su 1 (t)

Su 2 (t)

SuT2 (t)

(1/ε )Su 3 (t, ε) 2

⎞ ⎠,

(4.19)

where the (n − r + q) × (n − r + q)-matrix Su 1 (t), the (n − r + q) × (r − q)matrix Su 2 (t) and (r − q) × (r − q)-matrix Su 3 (t, ε) have the form ⎛ Su 1 (t) = ⎝

0 0 0 G¯ −1 u (t)

⎛

⎞ ⎠,

Su 2 (t) = ⎝

⎞

0 T G¯ −1 u (t)H2 (t)

⎠, (4.20)

T Su 3 (t, ε) = ε2 H2 (t)G¯ −1 u (t)H2 (t) + Ir −q .

Due to the form of the matrices Su (t, ε) and Su 3 (t, ε), the right-hand side of the differential equation in (4.14) has the singularity at ε = 0. To remove this singularity and to transform this equation to a differential equation of an explicit singular perturbation form, we look for the solution of the terminal-value problem (4.14) in the block form ⎛ ⎞ P1 (t, ε) ε P2 (t, ε) ⎠, (4.21) P(t, ε) = ⎝ ε P2T (t, ε) ε P3 (t, ε) where the matrices P1 (t, ε), P2 (t, ε) and P3 (t, ε) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively, and P1T (t, ε) = P1 (t, ε), P3T (t, ε) = P3 (t, ε). We also partition the matrices A(t) and Sv (t) into blocks as: ⎛ A(t) = ⎝

A1 (t) A2 (t) A3 (t) A4 (t)

⎞ ⎠,

⎛ Sv (t) = ⎝

Sv1 (t) Sv2 (t) SvT2 (t)

Sv3 (t)

⎞ ⎠,

(4.22)

82

4 Singular Finite-Horizon Zero-Sum Differential Game

where the blocks A1 (t), A2 (t), A3 (t) and A4 (t) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively; the blocks Sv1 (t), Sv2 (t) and Sv3 (t) have the form Sv1 (t)=C1 (t)G −1 v (t) T T −1 (t)C (t), S (t) = C (t)G (t)C (t), and C (t) and C1T (t), Sv2 (t) = C1 (t)G −1 v3 2 1 v v 2 2 C2 (t) are the upper and lower blocks of the matrix C(t) of the dimensions (n − r + q) × s and (r − q) × s, respectively. Substitution of (3.78), (3.79), (4.19), (4.21), (4.22) into the problem (4.14) converts the latter after a routine algebra into the following equivalent problem: d P1 (t, ε) = −P1 (t, ε)A1 (t) − ε P2 (t, ε) A3 (t) − A1T (t)P1 (t, ε) dt  −ε A3T (t)P2T (t, ε) + P1 (t, ε) Su 1 (t) − Sv1 (t) P1 (t, ε)   +ε P2 (t, ε) SuT2 (t) − SvT2 (t) P1 (t, ε) + ε P1 (t, ε) Su 2 (t) − Sv2 (t) P2T (t, ε)  +P2 (t, ε) Su 3 (t, ε) − ε2 Sv3 (t) P2T (t, ε) − D1 (t), P1 (t f , ε) = F1 , (4.23)

d P2 (t, ε) = −P1 (t, ε)A2 (t) − ε P2 (t, ε) A4 (t) − ε A1T (t)P2 (t, ε) dt  −ε A3T (t)P3 (t, ε) + ε P1 (t, ε) Su 1 (t) − Sv1 (t) P2 (t, ε)   +ε2 P2 (t, ε) SuT2 (t) − SvT2 (t) P2 (t, ε) + ε P1 (t, ε) Su 2 (t) − Sv2 (t) P3 (t, ε)  +P2 (t, ε) Su 3 (t, ε) − ε2 Sv3 (t) P3 (t, ε), P2 (t f , ε) = 0, ε

(4.24)

d P3 (t, ε) = −ε P2T (t, ε)A2 (t) − ε P3 (t, ε)A4 (t) − ε A2T (t)P2 (t, ε) dt  −ε A4T (t)P3 (t, ε) + ε2 P2T (t, ε) Su 1 (t) − Sv1 (t) P2 (t, ε)   +ε2 P3 (t, ε) SuT2 (t) − SvT2 (t) P2 (t, ε) + ε2 P2T (t, ε) Su 2 (t) − Sv2 (t) P3 (t, ε)  +P3 (t, ε) Su 3 (t, ε) − ε2 Sv3 (t) P3 (t, ε) − D2 (t), P3 (t f , ε) = 0. ε

(4.25) The problem (4.23)–(4.25) is a singularly perturbed terminal-value problem for a set of Riccati-type matrix differential equations. In the next subsection, based on the Boundary Function Method (see e.g. [16]), an asymptotic solution of this problem is constructed and justified.

4.5 Asymptotic Analysis of the Game (4.8), (4.12)

83

4.5.2 Asymptotic Solution of the Terminal-Value Problem (4.23)–(4.25) Due to the Boundary Function Method [16], we look for the zero-order asymptotic solution to the problem (4.23)–(4.25) in the form o b (t) + Pi,0 (τ ), i = 1, 2, 3, τ = (t − t f )/ε, Pi,0 (t, ε) = Pi,0

(4.26)

where the terms with the upper index “o” constitute the so called outer solution, while the terms with the upper index “b” are the boundary correction terms in a left-hand neighbourhood of t = t f ; τ ≤ 0 is a new independent variable, called the stretched time. For any t ∈ [0, t f ), τ → −∞ as ε → +0. Equations and conditions for these terms are obtained by substitution of (4.26) into the problem (4.23)–(4.25) instead of Pi (t, ε), (i = 1, 2, 3), and equating coefficients for the same power of ε on both sides of the resulting equations, separately the coefficients depending on t and on τ . b (τ ), we have the equation For the boundary correction term P1,0 b d P1,0 (τ )

dτ

= 0, τ ≤ 0.

(4.27)

b (τ ) → 0 for τ → −∞. Due to the Boundary Function Method, we require that P1,0 Subject to this requirement, the Eq. (4.27) yields the solution b (τ ) = 0, P1,0

τ ≤ 0.

(4.28)

Using the equality Su 3 (t, 0) = Ir−q , t ∈ [0, t f ], we obtain the equations for the o (t), (i = 1, 2, 3) in the interval [0, t f ]: outer solution terms Pi,0 o d P1,0 (t) o o (t)A1 (t) − A1T (t)P1,0 (t) = −P1,0 dt  o o (t) Su 1 (t) − Sv1 (t) P1,0 (t) +P1,0  T o o o +P2,0 (t) P2,0 (t) − D1 (t), P1,0 (t f ) = F1 ,

(4.29)

o o o (t)A2 (t) + P2,0 (t)P3,0 (t), 0 = −P1,0

(4.30)

2  o (t) − D2 (t). 0 = P3,0

(4.31)

o Solving the algebraic equations (4.31) and (4.30) with respect to P3,0 (t) and we have

o (t), P2,0

84

4 Singular Finite-Horizon Zero-Sum Differential Game

 1/2  −1/2 o o o P3,0 (t) = D2 (t) , P2,0 (t) = P1,0 (t)A2 (t) D2 (t) , t ∈ [0, t f ],

(4.32)

where the superscript “1/2” denotes the unique symmetric positive definite square root of corresponding symmetric positive definite matrix, the superscript “−1/2” denotes the inverse matrix of this square root. Remark 4.7 Since the matrix-valued function D2 (t) is continuously differentiable 1/2 −1/2   in the interval [0, t f ], then the matrix-valued functions D2 (t) and D2 (t) are continuously differentiable for t ∈ [0, t f ]. o Substitution of the expression for P2,0 (t) from (4.32) into (4.29) yields o d P1,0 (t) o o (t)A1 (t) − A1T (t)P1,0 (t) = −P1,0 dt o o o +P1,0 (t)S1,0 (t)P1,0 (t) − D1 (t), P1,0 (t f ) = F1 , t ∈ [0, t f ],

where

S1,0 (t) = Su 1 (t) − Sv1 (t) + A2 (t)D2−1 (t)A2T (t).

(4.33)

(4.34)

In what follows of this chapter, we assume: o (t) in the entire interval [0, t f ]. A4.VII. The problem (4.33) has the solution P1,0 Remark 4.8 Note that the assumption A4.VII is an ε-free assumption, which guarantees the existence of the solution to the problem (4.23)–(4.25) in the entire interval [0, t f ] for all sufficiently small ε > 0 (see Lemma 4.2 and its proof below). The latter provides the existence of the saddle-point solution to the game (4.8), (4.12) for all such ε > 0 (see Corollary 4.2 below). Moreover, the assumption A4.VII guarantees the existence of a saddle-point solution to the Reduced Finite-Horizon Game (see Sect. 4.6 below). n−r +q

Remark 4.9 By virtue of the results of [2] (Chap. 6), if S1,0 (t) ∈ S+ t ∈ [0, t f ], then the assumption A4.VII is valid.

for all

b Now, let us proceed to obtaining the boundary correction terms P2,0 (τ ) and For these terms we have the terminal-value problem

b P3,0 (τ ).

b d P2,0 (τ )

o b b o b b (t f )P3,0 (τ ) + P2,0 (τ )P3,0 (t f ) + P2,0 (τ )P3,0 (τ ), = P2,0 dτ b  b d P3,0 (τ ) o b b o (t f )P3,0 (τ ) + P3,0 (τ )P3,0 (t f ) + P3,0 (τ ))2 , = P3,0 dτ b o b o P2,0 (0) = −P2,0 (t f ), P3,0 (0) = −P3,0 (t f ),

(4.35)

where τ ≤ 0. o (t) in (4.33), we can rewrite the Using (4.32) and the terminal condition for P1,0 problem (4.35) as:

4.5 Asymptotic Analysis of the Game (4.8), (4.12)

85

 1/2  b b (τ ) D2 (t f ) + P3,0 (τ ) = P2,0 dτ  −1/2 b  −1/2 b +F1 A2 (t f ) D2 (t f ) P3,0 (τ ), P2,0 (0) = −F1 A2 (t f ) D2 (t f ) , b d P2,0 (τ )

b d P3,0 (τ )

dτ

1/2 b  1/2  b P3,0 (τ ) + P3,0 (τ ) D2 (t f ) = D2 (t f )  b 2  1/2 b + P3,0 (τ ) , P3,0 (0) = − D2 (t f ) , (4.36)

where τ ≤ 0. This problem consists of two subproblems, which can be solved consecutively: b (τ ) is solved, then the subproblem with first the subproblem with respect to P3,0 b b respect to P2,0 (τ ) is solved. The subproblem with respect to P3,0 (τ ) is a terminalvalue problem for a Bernoulli-type matrix differential equation, [4]. This subproblem has the following unique solution for all τ ≤ 0:   1/2 1/2  b (τ ) = −2 D2 (t f ) exp 2 D2 (t f ) τ Ir −q P3,0  1/2 −1 + exp 2 D2 (t f ) τ .

(4.37)

Substituting (4.37) into the first subproblem of (4.36) and solving the obtained b (τ ) yield for all τ ≤ 0: terminal-value problem with respect to P2,0   −1/2 1/2  b P2,0 (τ ) = −2F1 A2 (t f ) D2 (t f ) exp 2 D2 (t f ) τ Ir −q  1/2 −1 + exp 2 D2 (t f ) τ .

(4.38)

1/2  r−q ∈ S++ , there exist constants c > 0 and β > 0 such that the soluSince D2 (t f ) tion (4.37)–(4.38) of the problem (4.36) satisfies the inequality 

b b (τ ), P3,0 (τ ) ≤ c exp(βτ ), max P2,0

τ ≤ 0.

(4.39)

Lemma 4.2 Let the assumptions A4.I–A4.V, A4.VII be valid. Then, there exists a positive number ε0 such that, for all ε ∈ (0, ε0 ], the problem (4.23)–(4.25) has the unique solution Pi (t, ε) = Pi∗ (t,ε), (i = 1, 2, 3), in the entire interval [0, t f ]. This solution satisfies the inequalities  Pi∗ (t, ε) − Pi,0 (t, ε) ≤ aε, t ∈ [0, t f ], where Pi,0 (t, ε), (i = 1, 2, 3), are given by (4.26); a > 0 is some constant independent of ε. Proof We make the transformation of variables in the problem (4.23)–(4.25) Pi (t, ε) = Pi,0 (t, ε) + δi (t, ε),

i = 1, 2, 3,

where δi (t, ε), (i = 1, 2, 3) are new unknown matrix-valued functions.

(4.40)

86

4 Singular Finite-Horizon Zero-Sum Differential Game

Consider the block-form matrix-valued function    δ1 (t, ε) εδ2 (t, ε) . δ(t, ε) = εδ2T (t, ε) εδ3 (t, ε)

(4.41)

Substitution of (4.40) into the problem (4.23)–(4.25) and use of the Eqs. (4.28), (4.29)–(4.31), (4.35) and the block representations of the matrices Su (t, ε), P(t, ε), A(t), Sv (t) (see the Eqs. (4.19), (4.21), (4.22)) yield after a routine algebra the terminal-value problem for δ(t, ε) dδ(t, ε) = −δ(t, ε)α(t, ε) − α T (t, ε)δ(t, ε) dt +δ(t, ε)[Su (t, ε) − Sv (t)]δ(t, ε) − γ (t, ε), δ(t f , ε) = 0, t ∈ [0, t f ], (4.42) where α(t, ε) = A(t) − [Su (t, ε) − Sv (t)]P0 (t, ε),  P0 (t, ε) =

 P1,0 (t, ε) ε P2,0 (t, ε) ; T (t, ε) ε P3,0 (t, ε) ε P2,0

the matrix-valued function γ (t, ε) is expressed in a known form by the matrix-valued functions P0 (t, ε), Su (t, ε) and Sv (t); for any ε > 0, γ (t, ε) is a continuous function of t ∈ [0, t f ]. Let us represent the matrix γ (t, ε) in the block form as:  γ (t, ε) =

 γ1 (t, ε) γ2 (t, ε) , γ2T (t, ε) γ3 (t, ε)

where the dimensions of the blocks are the same as in the matrix P0 (t, ε). Using the inequality (4.39), the following estimates can be established for the blocks of γ (t, ε): γ1 (t, ε) ≤ b1 [ε + exp(βτ )], γl (t, ε) ≤ b1 ε, l = 2, 3, τ = (t − t f )/ε, t ∈ [0, t f ], ε ∈ (0, ε1 ], (4.43) where b1 is some positive number independent of ε; β is the positive number introduced in (4.39); ε1 > 0 is some sufficiently small number. Using the results of [1], we rewrite the problem (4.42) in the equivalent integral form  t   Ω T (σ, t, ε) δ(σ, ε) Su (σ, ε) − Sv (σ ) δ(σ, ε) δ(t, ε) = tf

 −γ (σ, ε) Ω(σ, t, ε)dσ, t ∈ [0, t f ], ε ∈ (0, ε1 ],

(4.44)

4.5 Asymptotic Analysis of the Game (4.8), (4.12)

87

where, for any given t ∈ [0, t f ] and ε ∈ (0, ε1 ], the n × n-matrix-valued function Ω(σ, t, ε) is the unique solution of the problem dΩ(σ, t, ε) = α(σ, ε)Ω(σ, t, ε), Ω(t, t, ε) = In , σ ∈ [t, t f ]. dσ Let Ω1 (σ, t, ε), Ω2 (σ, t, ε), Ω3 (σ, t, ε) and Ω4 (σ, t, ε) be the upper left-hand, upper right-hand, lower left-hand and lower right-hand blocks of the matrix Ω(σ, t, ε) of the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively. By virtue of the results of [5], we have the following estimates of these blocks for all 0 ≤ t ≤ σ ≤ t f :     Ωk (σ, t, ε) ≤ b2 , k = 1, 3, Ω2 (σ, t, ε) ≤ b2 ε,      Ω4 (σ, t, ε) ≤ b2 ε + exp − 0.5β(σ − t)/ε , ε ∈ (0, ε2 ],

(4.45)

where b2 is some positive number independent of ε; ε2 > 0 is some sufficiently small number. Applying the method of successive approximations to the Eq. (4.44), let us con

+∞ sider the sequence of the matrix-valued functions δ j (t, ε) j=0 given as:  δ j+1 (t, ε) =

t

  Ω T (σ, t, ε) δ j (σ, ε) Su (σ, ε) − Sv (σ ) δ j (σ, ε)

tf

 −γ (σ, ε) Ω(σ, t, ε)dσ, j = 0, 1, ..., t ∈ [0, t f ], ε ∈ (0, ε1 ],

(4.46)

where δ0 (t, ε) =0, t ∈ [0, t f ], ε ∈ (0, ε1]; the matrices δ j (σ, ε) have the block εδ j,2 (t, ε) δ j,1 (t, ε) , ( j = 1, 2, ...), and the dimensions of form δ j (σ, ε) = εδ Tj,2 (t, ε) εδ j,3 (t, ε) the blocks in each of these matrices are the same as the dimensions of the corresponding blocks in (4.41). Using the block form of all the matrices appearing in the Eq. (4.46), as well as using the inequalities (4.43) and (4.45), we obtain the existence number ε0 ≤ min{ε1 , ε2 } such that for any ε ∈ (0, ε0 ] the sequence of

a positive +∞ δ j (t, ε) j=0 converges in the linear space of all n × n-matrix-valued functions continuous in the interval [0, t f ]. Moreover, the following inequalities are fulfilled: δ j,i (t, ε) ≤ aε,

i = 1, 2, 3,

j = 1, 2, ...,

t ∈ [0, t f ],

where a > 0 is some number independent of ε, i and j. Thus, for any ε ∈ (0, ε0 ], 

δ ∗ (t, ε) = lim δ j (t, ε) j→+∞

(4.47)

88

4 Singular Finite-Horizon Zero-Sum Differential Game

is a solution of the Eq. (4.44) and, therefore, of the problem (4.42) in the entire interval [0, t f ]. Moreover, this solution has the block form similar to (4.41) and satisfies the inequalities δi∗ (t, ε) ≤ aε, i = 1, 2, 3, t ∈ [0, t f ].

(4.48)

Since the right-hand side of the differential equation in the problem (4.42) satisfies the local Lipschitz condition w.r.t. δ uniformly in t ∈ [0, t f ], this problem cannot have more than one solution. Therefore, δ ∗ (t, ε) defined by (4.47) is the unique solution of the problem (4.42). This observation, along with the Eq. (4.40) and the ⊔ ⊓ inequalities in (4.48), proves the lemma. As a direct consequence of Lemma 4.2 and the equivalence of the problems (4.14) and (4.23)–(4.25), we have the following assertion. Corollary 4.2 Let the assumptions A4.I–A4.V, A4.VII be valid. Then, for any ε ∈ (0, ε0 ], the assumption A4.VI is fulfilled and ∗



P (t, ε) =

 P1∗ (t, ε) ε P2∗ (t, ε)  ∗ T . ε P2 (t, ε) ε P3∗ (t, ε)

Moreover, all the statements of Proposition 4.2 are valid.

4.5.3 Asymptotic Representation of the Value of the Game (4.8), (4.12) Let us consider the following value: 

o (0)x0 , J o (x0 ) = x0T P1,0

(4.49)

o where P1,0 (t) is the solution of the terminal-value problem (4.33); x0 ∈ E n−r +q is the upper block of the vector z 0 .

Lemma 4.3 Let the assumptions A4.I–A4.V, A4.VII be valid. Then, for all ε ∈ (0, ε0 ], the following inequality is satisfied:  ∗   J (z 0 ) − J o (x0 ) ≤ c(z 0 )ε, ε where Jε∗ (z 0 ) is the value of the game (4.8), (4.12); c(z 0 ) > 0 is some constant independent of ε, while depending on z 0 . Proof The statement of the lemma directly follows from Proposition 4.2 (item (b)), ⊔ ⊓ Lemma 4.2 and Corollary 4.2.

4.6 Reduced Finite-Horizon Differential Game

89

4.6 Reduced Finite-Horizon Differential Game Let us consider the following block-form matrices 

B, A2 (t) ,  B= B1,0 (t) = 



O(n−r )×q Iq



 , Θ(t) =

G¯ u (t) Oq×(r −q) O(r −q)×q D2 (t)

 . (4.50)

Proposition 4.3 The matrix S1,0 (t), defined by (4.34), can be represented as: T S1,0 (t) = B1,0 (t)Θ −1 (t)B1,0 (t) − Sv1 (t).

Proof Using the block form of the matrices B1,0 (t) and Θ(t), we obtain





 B, A2 (t)



T B1,0 (t)Θ −1 (t)B1,0 (t) =    −1 T  G¯ u (t) Oq×(r −q) B = A2T (t) O(r −q)×q D2−1 (t) −1 T T  B G¯ −1 u (t) B + A2 (t)D2 (t)A2 (t).

(4.51)

T B and the Eq. (4.20), we have:  B G¯ −1 Due to the block form of the matrix  u (t) B = Su 1 (t). This equality and the Eqs. (4.34), (4.51) directly yield the statement of the proposition. ⊔ ⊓ Consider the finite-horizon zero-sum linear-quadratic differential game, the dynamics of which is d xr (t) = A1 (t)xr (t) + B1,0 (t)u r (t) + C1 (t)vr (t), xr (0) = x0 , dt

(4.52)

where t ∈ [0, t f ]; xr (t) ∈ E n−r+q is a state variable; u r (t) ∈ E r and vr (t) ∈ E s are controls of the game’s players. The functional, to be minimized by u r (t) and maximized by vr (t), has the form 

tf

+ 0

Jr (u r , vr ) = xrT (t f )F1 xr (t f )  T  xr (t)D1 (t)xr (t) + u rT (t)Θ(t)u r (t) − vrT (t)G v (t)vr (t) dt.

(4.53)

We call the game (4.52)–(4.53) the Reduced Finite-Horizon Game (RFHG). Since Θ(t) ∈ Sr++ and G v (t) ∈ Ss++ for all t ∈ [0, t f ], the  RFHG is regular. The set of all the admissible pairs of the state-feedback controls u r (xr , t), vr (xr , t) , the guaranteed results of u r (xr , t) and vr (xr , t), the saddle-point solution and the game value in the RFHG are defined similarly to those notions in Sect. 3.2.1 (see Definitions 3.1–3.4). Based on Proposition 4.3, we have (quite similarly to Theorem 3.1) the following assertion.

90

4 Singular Finite-Horizon Zero-Sum Differential Game

Proposition4.4 Let the assumptions A4.I–A4.V, A4.VII be valid. Then: (a) the pair u ∗r (xr , t), vr∗ (xr , t) , where T o T o u ∗r (xr , t) = −Θ −1 (t)B1,0 (t)P1,0 (t)xr , vr∗ (xr , t) = G −1 v (t)C 1 (t)P1,0 (t)x r , (4.54)

is the saddle-point solution in the RFHG; (b) the game value in the RFHG has the form Jr∗ = J o (x0 ), where J o (x0 ) is defined by (4.49). Remark 4.10 Using the expressions for B1,0 (t) and Θ(t) (see the Eq. (4.50)), we can represent the minimizer’s control u ∗r (xr , t) in the saddle point solution of the RFHG as:   ∗ u r,1 (xr , t) , u ∗r (xr , t) = u ∗r,2 (xr , t) −1 ∗ T o T o u ∗r,1 (xr , t) = −G¯ −1 u (t) B P1,0 (t)x r , u r,2 (x r , t) = −D2 (t)A2 (t)P1,0 (t)x r . (4.55)

4.7 Saddle-Point Sequence of the SFHG 4.7.1 Main Assertions For any given ε ∈ (0, ε0 ], consider the following function of (z, t) ∈ E n × [0, t f ]:  u o∗ ε,0 (z, t)

=

 1

−ε

 u ∗r,1 (x, t)  , T o o (t) x + P3,0 (t)y P2,0

(4.56)

o o (t), P3,0 (t) are given in (4.32) and where z = col(x, y), x ∈ E n−r+q , y ∈ E r −q ; P2,0 o P1,0 (t) is the solution of the problem (4.33).

A4.VII be valid. Then, for any given Lemma 4.4 Let theassumptions A4.I–A4.V, ∗ (z, t), v (x, t) is an admissible pair of the players’ stateε ∈ (0, ε0 ], the pair u o∗ r ε,0  (z, t), vr∗ (x, t) ∈ (U V )t f . feedback controls in the SFHG, i.e., u o∗ ε,0 Proof The statement of the lemma directly follows from the linear dependence n ∗ n−r+q and the continuity with respect to of u o∗ ε,0 (z, t) on z ∈ E , vr (x, t) on x ∈ E o∗ ⊔ ⊓ t ∈ [0, t f ] of the gain matrices in u ε,0 (z, t) and vr∗ (x, t). Lemma 4.5 Let the assumptions A4.I–A4.V, A4.VII be valid. Then, there exists a o o o positive  o∗ number εu , (εu ≤ ε0 ) such that, for allo∗ε ∈ (0, εu ], the guaranteed result Ju u ε,0 (z, t), z 0 of the minimizer’s control u ε,0 (z, t) in the SFHG satisfies the inequality    o∗  Ju u (z, t); z 0 − J o (x0 ) ≤ aε, ε,0

4.7 Saddle-Point Sequence of the SFHG

91

where a > 0 is some constant independent of ε; J o (x0 ) is defined by (4.49). Proof of the lemma is presented in Sect. 4.7.2. Lemma 4.6 Let A4.I–A4.V, A4.VII be valid. Then, the guaran ∗the assumptions (x, t); z of the maximizer’s control vr∗ (x, t) in the SFHG is v teed result J v r 0  ∗ o Jv vr (x, t); z 0 = J (x0 ). Proof of the lemma is presented in Sect. 4.7.3. Let {εk }+∞ k=1 be a sequence of numbers, satisfying the following conditions: (i) εk ∈ (0, εuo ], (k = 1, 2, ...), where εuo > 0 is defined in Lemma 4.5; (ii) εk → +0 for k → +∞. The following assertion is an immediate consequence of Definition 4.5 and Lemmas 4.4–4.6. Theorem 4.1 Let the assumptions +∞ A4.I–A4.V, A4.VII be valid. Then the sequence of ∗ the pairs u o∗ εk ,0 (z, t), vr (x, t) k=1 is the saddle-point sequence of the SFHG. Moreover, J o (x0 ) is the value of this game. the upper block of the minimizer’s control sequence Remark 4.11

o∗ +∞Note that u εk ,0 (z, t) k=1 is independent of k, and it coincides with the upper block of the minimizer’s control in the saddle-point solution of the RFHG. Similarly, the maximizer’s control in the saddle-point sequence of the SFHG and the value of this game coincide with the saddle-point maximizer’s control and the game value, respectively, in the RFHG. The latter game is regular, and it is of a smaller dimension than the SFHG. Thus, in order to solve the SFHG, one has to solve the smaller dimension rego o (t) and P2,0 (t) using the Eq. (4.32). ular RFHG and construct two gain matrices P3,0 Remark 4.12 If q = 0, i.e., all the coordinates of the control of the minimizing player in the SFHG (4.8)–(4.9) are singular, then Su 1 (t) ≡ O(n−r )×(n−r ) , Su 2 (t) ≡ O(n−r )×r , Su 3 (t) ≡ Ir and S1,0 (t) = A2 (t)D2−1 (t)A2T (t) − Sv1 (t). The matrix-valued o (t) is the solution of the terminal-value problem (4.33) with this S1,0 (t). function P1,0 The upper block of the control u o∗ ε,0 (z, t) (see the Eq. (4.56)) vanishes, while the lower  o T 1 block remains unchanged. Thus, in this case we have u o∗ ε,0 (z, t) = − ε P2,0 (t) x +  o (t)y , z = col(x, y), x ∈ E n−r , y ∈ E r . Theorem 4.1 is valid with this particular P3,0 o form of u o∗ ε,0 (z, t) and the aforementioned P1,0 (t). More details on the case q = 0 can be found in the work [12].

4.7.2 Proof of Lemma 4.5 Using the expression for u ∗r,1 (x, t) in (4.55) and the matrix M(t, ε), given as:  M(t, ε) = −

o G¯ −1 (t)  B T P1,0 (t) Oq×(r−q) u o T 1 1 o P (t) P (t) 2,0 ε ε 3,0

 , t ∈ [0, t f ], ε ∈ (0, ε0 ], (4.57)

92

4 Singular Finite-Horizon Zero-Sum Differential Game

we can rewrite the control (4.56) in the form u o∗ ε,0 (z, t) = M(t, ε)z,

(z, t) ∈ E n × [0, t f ].

Substitution of u(t) = u o∗ ε,0 (z, t) into the game (4.8)–(4.9), and use of Definition 3.2  (z, t); z yield that Ju u o∗ 0 coincides with the optimal value of the functional in the ε,0 following optimal control problem: dz(t)  = A(t) + B(t)M(t, ε) z(t) + C(t)v(t), t ∈ [0, t f ], z(0) = z 0 , (4.58) dt  T J u o∗ ε,0 (z, t), v = z (t f )F z(t f )  tf  T  z (t) D(t) + M T (t, ε)G u (t)M(t, ε) z(t) + 0  −v T (t)G v (t)v(t) dt → sup  .

(4.59)

v(t)=v(z,t)∈Fv,t f u o∗ ε,0 (z,t)

For any given ε ∈ (0, ε0 ], consider the following terminal-valued problem with respect to the unknown n × n-matrix-valued function K (t, ε):  d K (t, ε) = −K (t, ε) A(t) + B(t)M(t, ε) dt  T − A(t) + B(t)M(t, ε) K (t, ε) − K (t, ε)Sv (t)K (t, ε)  − D(t) + M T (t, ε)G u (t)M(t, ε) , K (t f , ε) = F, t ∈ [0, t f ],

(4.60)

where Sv (t) is given in (4.15). By virtue of the results of [10], we have the following. If the problem (4.60) has the solution K (t, ε) in the entire interval [0, t f ], then the optimal control problem (4.58)–(4.59) is solvable and the optimal value of the functional in this problem is z 0T K (0, ε)z 0 . Therefore,  T Ju u o∗ ε,0 (z, t); z 0 = z 0 K (0, ε)z 0 .

(4.61)

Now, we are going to prove the existence of the solution of (4.60) in the entire interval [0, t f ]. To do this, we will construct and justify an asymptotic solution of this problem for all sufficiently small ε > 0. Similarly to (4.21), we look for the solution K (t, ε) of the problem (4.60) in the block form  K (t, ε) =

 K 1 (t, ε) εK 2 (t, ε) , K 1T (t, ε) = K 1 (t, ε), K 3T (t, ε) = K 3 (t, ε), εK 2T (t, ε) εK 3 (t, ε) (4.62)

4.7 Saddle-Point Sequence of the SFHG

93

where the matrices K 1 (t, ε), K 2 (t, ε) and K 3 (t, ε) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively. Substitution of (3.77), (3.78), (3.79), (4.3), (4.22), (4.57) and (4.62) into (4.60) converts the latter after a routine algebra into the following equivalent problem for t ∈ [0, t f ]: d K 1 (t, ε) = −K 1 (t, ε)A1 (t) − εK 2 (t, ε)A3 (t) − A1T (t)K 1 (t, ε) dt o o −ε A3T (t)K 2T (t, ε) + K 1 (t, ε)Su 1 (t)P1,0 (t) + εK 2 (t, ε)SuT2 (t)P1,0 (t)  o T o o T +K 2 (t, ε) P2,0 (t) + P1,0 (t)Su 1 (t)K 1 (t, ε) + ε P1,0 (t)Su 2 (t)K 2 (t, ε) o +P2,0 (t)K 2T (t, ε) − K 1 (t, ε)Sv1 (t)K 1 (t, ε) − εK 2 (t, ε)SvT2 (t)K 1 (t, ε)

−εK 1 (t, ε)Sv2 (t)K 2T (t, ε) − ε2 K 2 (t, ε)Sv3 (t)K 2T (t, ε) o o −D1 (t) − P1,0 (t)Su 1 (t)P1,0 (t), K 1 (t f , ε) = F1 , (4.63)

ε

d K 2 (t, ε) = −K 1 (t, ε)A2 (t) − εK 2 (t, ε)A4 (t) − ε A1T (t)K 2 (t, ε) dt o o −ε A3T (t)K 3 (t, ε) + K 2 (t, ε)P3,0 (t) + ε P1,0 (t)Su 1 (t)K 2 (t, ε)

o o +ε P1,0 (t)Su 2 (t)K 3 (t, ε) + P2,0 (t)K 3 (t, ε) − εK 1 (t, ε)Sv1 (t)K 2 (t, ε)

−ε2 K 2 (t, ε)SvT2 (t)K 2 (t, ε) − εK 1 (t, ε)Sv2 (t)K 3 (t, ε) −ε2 K 2 (t, ε)Sv3 (t)K 3 (t, ε), K 2 (t f , ε) = 0,

ε

(4.64)

d K 3 (t, ε) = −εK 2T (t, ε) A2 (t) − εK 3 (t, ε)A4 (t) − ε A2T (t)K 2 (t, ε) dt o o −ε A4T (t)K 3 (t, ε) + K 3 (t, ε)P3,0 (t) + P3,0 (t)K 3 (t, ε) −ε2 K 2T (t, ε)Sv1 (t)K 2 (t, ε) − ε2 K 3 (t, ε)SvT2 (t)K 2 (t, ε) −ε2 K 2T (t, ε)Sv2 (t)K 3 (t, ε) − ε2 K 3 (t, ε)Sv3 (t)K 3 (t, ε) −D2 (t), K 3 (t f , ε) = 0, (4.65)

where Su 1 (t) and Su 2 (t) are given in (4.20). Similarly to Sect. 4.5.2, we look for the zero-order asymptotic solution of the problem (4.63)–(4.65) in the form o b (t) + K i,0 (τ ), i = 1, 2, 3, τ = (t − t f )/ε. K i,0 (t, ε) = K i,0

(4.66)

94

4 Singular Finite-Horizon Zero-Sum Differential Game

By the same arguments as in Sect. 4.5.2 (see the Eqs. (4.27), (4.28)), we obtain b K 1,0 (τ ) = 0,

τ ≤ 0.

(4.67)

o Furthermore, the outer solution terms K i,0 (t), (i = 1, 2, 3) satisfy the following set of equations in the interval [0, t f ]: o

T d K 1,0 (t) o o o o (t) A1 (t) − A1T (t)K 1,0 (t) + K 2,0 (t) P2,0 (t) = −K 1,0 dt

T o o o o o o +K 1,0 (t)Su 1 (t)P1,0 (t) + P2,0 (t) K 2,0 (t) + P1,0 (t)Su 1 (t)K 1,0 (t) o o (t)Sv1 (t)K 1,0 (t) − D1 (t) −K 1,0 o o o −P1,0 (t)Su 1 (t)P1,0 (t), K 1,0 (t f ) = F1 ,

(4.68)

o o o o o (t)A2 (t) + K 2,0 (t)P3,0 (t) + P2,0 (t)K 3,0 (t) = 0, − K 1,0

(4.69)

o o o o (t)P3,0 (t) + P3,0 (t)K 3,0 (t) − D2 (t) = 0. K 3,0

(4.70)

o Solving the algebraic equations (4.70) and (4.69) with respect to K 3,0 (t) and and using the Eq. (4.32), we obtain

o (t), K 2,0

o K 3,0 (t) =

1/2 1 D2 (t) , t ∈ [0, t f ], 2

(4.71)

 −1/2 1 o  −1/2 o o K 2,0 (t) = K 1,0 (t)A2 (t) D2 (t) − P1,0 (t)A2 (t) D2 (t) , t ∈ [0, t f ]. 2 (4.72) o o (t) and K 2,0 (t) from (4.32) and (4.72), respectively, into (4.68) Substitution of P2,0 yields after a routine algebra the following terminal-value problem in the time interval [0, t f ]: o d K 1,0 (t)

o o (t)A1 (t) − A1T (t)K 1,0 (t) = −K 1,0 dt  o o o o +K 1,0 (t) S1,0 (t) + Sv1 (t) P1,0 (t) + P1,0 (t) S1,0 (t) + Sv1 (t) K 1,0 (t)  o o o o (t)Sv1 (t)K 10 (t) − D1 (t) − P1,0 (t) S1,0 (t) + Sv1 (t) P1,0 (t), −K 1,0 o K 10 (t f ) = F1 ,



(4.73)

where S1,0 (t) is given by (4.34). o o (t) into (4.73) instead of K 1,0 (t) and It is verified directly by substitution of P1,0 using (4.33), that the terminal-value problem (4.73) has a solution in the entire interval o (t), i.e., [0, t f ] and this solution equals to P1,0

4.7 Saddle-Point Sequence of the SFHG o o K 1,0 (t) = P1,0 (t),

95

t ∈ [0, t f ].

(4.74)

Moreover, due to the linear and quadratic dependence of the right-hand side of the o (t), this solution is unique. differential equation in (4.73) on K 1,0 b b Now, let us obtain K 2,0 (τ ) and K 3,0 (τ ). Similarly to Sect. 4.5.2 (see the Eqs. (4.35), (4.36)), we derive the following equations and the conditions for finding these matrixvalued functions: b  1/2  −1/2 b d K 2,0 (τ ) b (τ ) D2 (t f ) + F1 A2 (t f ) D2 (t f ) K 3,0 (τ ), = K 2,0 dτ

b  1/2  1/2 b d K 3,0 (τ ) b (τ ) D2 (t f ) + D2 (t f ) K 3,0 (τ ), = K 3,0 dτ

 −1/2 1 b K 2,0 (0) = − F1 A2 (t f ) D2 (t f ) , 2

τ ≤ 0,

τ ≤ 0, (4.75) (4.76)

1/2 1 D2 (t f ) . 2

(4.77)



−1/2  1 b exp 2(D2 (t f ))1/2 τ , τ ≤ 0, K 2,0 (τ ) = − F1 A2 (t f ) D2 (t f ) 2

(4.78)

b K 3,0 (0) = −

The problem (4.75)–(4.77) yields the unique solution

b K 3,0 (τ ) = −

1/2  1 exp 2(D2 (t f ))1/2 τ , τ ≤ 0, D2 (t f ) 2

(4.79)

and this solution satisfies the inequality  b   K (τ ) ≤ c exp(βτ ), τ ≤ 0, j,0

j = 2, 3,

(4.80)

where c > 0 and β > 0 are some constants. Now, based on the Eqs. (4.71), (4.72), (4.74) and the inequality (4.80), it is shown (quite similarly to Lemma 4.2) the existence of a positive number εuo , (εuo ≤ ε0 ) such problem (4.63)–(4.65) has the unique that, for all ε ∈ (0, εuo ], the terminal-value  solution K 1 (t, ε), K 2 (t, ε), K 3 (t, ε) in the entire interval [0, t f ]. Consequently, the equivalent problem (4.60) has the unique solution (4.62) in the entire interval [0, t f ] for all ε ∈ (0, εuo ]. Moreover, the solution of (4.63)–(4.65) satisfies the inequalities    K i (t, ε) − K i,0 (t, ε) ≤ aε, t ∈ [0, t f ], ε ∈ (0, εo ], u

(4.81)

where K i0 (t, ε), (i = 1, 2, 3) are given in (4.66); a > 0 is some constant independent of ε. Let us represent the vector z 0 in the block form as: z 0 = col(x0 , y0 ), where x0 ∈ E n−r+q , y0 ∈ E r−q . Substitution of this representation and (4.62) into (4.61) yields for all ε ∈ (0, εuo ]:

96

4 Singular Finite-Horizon Zero-Sum Differential Game



 T T T Ju u o∗ ε,0 (z, t); z 0 = x 0 K 1 (0, ε)x 0 + ε 2x 0 K 2 (0, ε)y0 + y0 K 3 (0, ε)y0 . (4.82) Subtracting (4.49) from (4.82), estimating the resulting expression and taking into account (4.74), we obtain the following inequality for all ε ∈ (0, εuo ]:   o∗   Ju u (z, t); z 0 − J o (x0 ) ≤ K 1 (0, ε) − K o (0)x0 2 ε,0 1,0  +ε 2K 2 (0, ε)x0 y0  + K 3 (0, ε)y0 2 .

(4.83)

Due to the Eq. (4.66) for i = 2, 3, (4.71)–(4.72) and the inequalities (4.80) and (4.81) for i = 2, 3, the matrices K 2 (0, ε) and K 3 (0, ε) are bounded with respect to ε ∈ (0, εuo ]. Moreover, due to the Eq. (4.66) for i = 1, (4.67) and the inequality (4.81) for i = 1, o (0) ≤ aε, ε ∈ (0, εuo ], (4.84) K 1 (0, ε) − K 10 where a > 0 is some constant independent of ε. Now, the inequalities (4.83) and (4.84), along with the above mentioned boundedness of the matrices K 2 (0, ε) and K 3 (0, ε), directly yield the statement of the lemma.

4.7.3 Proof of Lemma 4.6 Substitution of v(t) = vr∗ (x, t)into the system (4.8) and the functional (4.9), and use of Definition 3.3 yield that Jv vr∗ (x, t); z 0 coincides with the optimal value of the functional in the following optimal control problem: dz(t)  = A(t)z(t) + B(t)u(t), t ∈ [0, t f ], z(0) = z 0 , dt  tf  T   ∗ T   z (t) D(t)z(t) J (u) = J u, vr (x, t) = z (t f )F z(t f ) + 0  +u T (t)G u (t)u(t) dt → inf  ,

(4.85)

(4.86)

u(t)=u(z,t)∈Fu,t f vr∗ (x,t)

where  = A(t)  = D(t) Thus,





1 (t) A 2 (t) A 4 (t) 3 (t) A A

1 (t) 0 D 2 (t) 0 D







=

 o A1 (t) + Sv1 (t)P1,0 (t) A2 (t) , o (t) A4 (t) A3 (t) + SvT2 (t)P1,0

(4.87)

 o o D1 (t) − P1,0 (t)Sv1 (t)P10 (t) 0 . 0 D2 (t)

(4.88)



= 



4.7 Saddle-Point Sequence of the SFHG

 Jv vr∗ (x, t); z 0 =

97

J(u)

inf 

(4.89)

u(t)=u(z,t)∈Fu,t f vr∗ (x,t)

in the optimal control problem (4.85)–(4.86), which is singular (see e.g. [7, 8] and references therein). The value in the right-hand side of (4.89) can be calculated in the way, similar to that proposed in [8] for solution of a stochastic singular linearquadratic optimal control problem with state delays. The further proof consists of three stages.

4.7.3.1

Stage 1: Regularization of (4.85)–(4.86)

Let us replace the singular optimal control problem (4.85)–(4.86) with a regular one, consisting of the equation of dynamics (4.85) and the functional Jε (u) = z T (t f )F z T (t f ) +   + + u T (t) G u (t) + Λ(ε) u(t) dt →



tf



 z T (t) D(t)z(t)

0

inf 

,

(4.90)

u(t)=u(z,t)∈Fu,t f vr∗ (x,t)

where ε > 0 is a small parameter; Λ(ε) is given by (4.13). For a given ε > 0, consider the following terminal-value problem for the n × n(t, ε) in the time interval [0, t f ]: matrix-valued function K (t, ε) dK (t, ε) (t, ε) A(t)  −A T (t) K = −K dt (t, ε) − D(t),  (t, ε)Su (t, ε) K K˜ (t f , ε) = F, +K

(4.91)

where Su (t, ε) is given in (4.15). By virtue of the results of [10], if the terminal-value problem (4.91) has a (t, ε) in the entire interval [0, t f ], then the problem (4.85), (4.90) is solution K    u ∗ε z, t = − G u (t) + solvable, i.e., it has the optimal control of the form u =  −1 (t, ε)z, and the optimal value of its functional has the form Λ(ε) B T (t) K (0, ε)z 0 . Jε∗ (z 0 ) = z 0T K 4.7.3.2

(4.92)

Stage 2: Asymptotic Analysis of the Problem (4.85), (4.90)

Let us start with an asymptotic solution of the problem (4.91). Similarly to (4.21) (t, ε) of the problem (4.91) in the block form and (4.62), we seek the solution K

98

4 Singular Finite-Horizon Zero-Sum Differential Game

 1 (t, ε) ε K 2 (t, ε) K T  T  3 (t, ε) , K 1 (t, ε) = K 1 (t, ε), K 3 (t, ε) = K 3 (t, ε), 2T (t, ε) ε K εK (4.93) 2 (t, ε) and K 3 (t, ε) have the dimensions (n − r + 1 (t, ε), K where the matrices K q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively. Using the block representations (3.78), (4.19), (4.87), (4.88) and (4.93), one can rewrite the problem (4.91) in the following equivalent form: (t, ε) = K



1 (t, ε) dK 1 (t) − ε K 2 (t, ε) A 3 (t) − A 1T (t) K 1 (t, ε) 1 (t, ε) A = −K dt 3T (t) K 2T (t, ε) + K 1 (t, ε)Su 1 (t) K 1 (t, ε) + ε K 2 (t, ε)SuT (t) K 1 (t, ε) −ε A 2 T 2 (t, ε) + K 2 (t, ε)Su 3 (t, ε) K 2T (t, ε) 1 (t, ε)Su 2 (t) K +ε K 1 (t), −D

ε

1 (t f , ε) = F1 , K

(4.94)

2 (t, ε) dK 2 (t) − ε K 2 (t, ε) A 4 (t) − ε A 1T (t) K 2 (t, ε) 1 (t, ε) A = −K dt 3T (t) K 3 (t, ε) + ε K 1 (t, ε)Su 1 (t) K 2 (t, ε) −ε A 2 T 2 (t, ε) + ε K 1 (t, ε)Su 2 (t) K 3 (t, ε) +ε K 2 (t, ε)Su (t) K 2

ε

2 (t, ε)Su 3 (t, ε) K 3 (t, ε), K 2 (t f , ε) = 0, +K

(4.95)

3 (t, ε) dK 2 (t) − ε K 3 (t, ε) A 4 (t) − ε A 2T (t) K 2 (t, ε) 2T (t, ε) A = −ε K dt 4T (t) K 3 (t, ε) + ε2 K 2 (t, ε) 2T (t, ε)Su 1 (t) K −ε A 2 T 2 T 2 (t, ε) + ε K 3 (t, ε) 2 (t, ε)Su 2 (t) K +ε K 3 (t, ε)Su 2 (t) K 3 (t, ε)Su 3 (t, ε) K 3 (t, ε) − D 2 (t), K 3 (t f , ε) = 0. +K

(4.96)

Similarly to Sects. 4.5.2 and 4.7.2, we seek the zero-order asymptotic solution of the problem (4.94)–(4.96) in the form o b i,0 i,0 i,0 (t, ε) = K (t) + K (τ ), i = 1, 2, 3, τ = (t − t f )/ε. K

(4.97)

Like in the above mentioned subsections (see the Eqs. (4.28) and (4.67)), we have b 1,0 (τ ) = 0, K

τ ≤ 0.

(4.98)

o i,0 (t), (i = 1, 2, 3) satisfy the following set Furthermore, the outer solution terms K of equations for t ∈ [0, t f ]:

4.7 Saddle-Point Sequence of the SFHG

99

o 1,0 dK (t) o o o o 1 (t) − A 1T (t) K 1,0 1,0 1,0 1,0 (t) A (t) + K (t)Su 1 (t) K (t) = −K dt  T o o o 2,0 20 1 (t), K 1,0 +K (t) K (t) − D (t f ) = F1 ,

o o o 2 (t) + K 2,0 3,0 1,0 (t) A (t) K (t) = 0, −K 2  o 2 (t) = 0. 3,0 (t) − D K

(4.99)

(4.100) (4.101)

o 3,0 (t) and Solving the algebraic equations (4.101) and (4.100) with respect to K and using the Eqs. (4.88), (4.87), we obtain

o 2,0 (t), K

 1/2  −1/2 o o o 2,0 1,0 3,0 (t) = D2 (t) , K (t) = K (t)A2 (t) D2 (t) , K

t ∈ [0, t f ]. (4.102)

o 1 (t) and K 2,0 1 (t), D (t) (see the Eqs. (4.87), Now, substitution of the expressions for A (4.88) and (4.102)) into (4.99) yields after a routine algebra the following terminalo 1,0 (t) in the time interval [0, t f ]: value problem for K o 1,0 dK (t)

 o o 1,0 (t) A1 (t) + Sv1 (t)P1,0 (t) = −K dt T o  o  o o 1,0 (t) 1,0 1,0 (t) + K (t) K (t) S1,0 (t) + Sv1 (t) K − A1 (t) + Sv1 (t)P1,0  o o o 1,0 (t f ) = F1 , (4.103) − D1 (t) − P1,0 (t)Sv1 (t)P10 (t) , K where S1,0 (t) is given by (4.34). o o 1,0 (t) into (4.103) instead of K (t) and use of (4.33), one By substitution of P1,0 verifies immediately that the terminal-value problem (4.103) has a solution in the o (t), i.e., entire interval [0, t f ] and this solution equals to P1,0 o o 1,0 (t) = P1,0 (t), K

t ∈ [0, t f ].

(4.104)

Moreover, due to the linear and quadratic dependence of the right-hand side of the o 1,0 (t), this solution is unique. differential equation in (4.103) on K b b   (τ ). These matrix-valued functions satisfy Proceed to obtaining K 2,0 (τ ) and K 30 the same terminal-value problem as the problem (4.36). Therefore, b b 2,0 (τ ) = P2,0 (τ ), K

b b 3,0 K (τ ) = P3,0 (τ ), τ ≤ 0,

(4.105)

b b (τ ) and P2,0 (τ ) are given by (4.37) and (4.38), respectively. where P3,0 Now, based on the Eqs. (4.102), (4.104) and (4.98), (4.105), it is shown (similarly ε such that, for all ε ∈ (0, ε], to Lemma 4.2) the existence of a positive number  the problem (4.94)–(4.96) (and, consequently, the problem (4.91), (4.93)) has the 2 (t, ε), K 3 (t, ε)} in the entire interval [0, t f ]. Moreover, 1 (t, ε), K unique solution { K ε]: the following inequalities are satisfied for all t ∈ [0, t f ] and ε ∈ (0,

100

4 Singular Finite-Horizon Zero-Sum Differential Game

  K i,0 (t, ε) ≤ aε, i = 1, 2, 3, i (t, ε) − K

(4.106)

i,0 (t, ε), (i = 1, 2, 3) are given by (4.97); a > 0 is some constant indepenwhere K dent of ε. Further, by using the Eqs. (4.92), (4.93), (4.98), (4.102), (4.104) and (4.105), as well as the inequalities (4.39) and (4.106), we prove (quite similarly to the proof of Lemma 4.5) the validity of the inequality   ∗  J (z 0 ) − J o (x0 ) ≤ aε, ε

ε ∈ (0, ε],

(4.107)

where J o (x0 ) is defined by (4.49); a > 0 is some constant independent of ε.

4.7.3.3

Stage 3: Deriving the Expression for the Optimal Value of the Functional in the Problem (4.85)–(4.86)

First, let us observe that the inequality (4.107) is equivalent to J o (x0 ) − aε ≤ Jε∗ (z 0 ) ≤ J o (x0 ) + aε, ε ∈ (0, ε].

(4.108)

Using this inequality, we have for any ε ∈ (0, ε]: inf 

 ∗  ∗ u ε (z, t) u ε (z, t) ≤ Jε  J(u) ≤ J 

u(t)=u(z,t)∈Fu,t f vr∗ (x,t)

= Jε∗ (z 0 ) ≤ J o (x0 ) + aε,

(4.109)

yielding o J(u) ≤ J (x0 ).

inf 

u(t)=u(z,t)∈Fu,t f

vr∗ (x,t)

Let us show that inf 

o J(u) = J (x0 ).

(4.110)

u(t)=u(z,t)∈Fu,t f vr∗ (x,t)

For this purpose, let us assume the opposite, i.e., inf 

u(t)=u(z,t)∈Fu,t f

o J(u) < J (x0 ).

vr∗ (x,t)

 The latter means the existence of  u (z, t) ∈ Fu,t f vr∗ (x, t) , such that

(4.111)

4.8 Examples

101

inf 

 u (z, t) < J o (x0 ). J(u) < J 

(4.112)

u(t)=u(z,t)∈Fu,t f vr∗ (x,t)

Since  u ∗ε (z, t) is an optimal control in the problem (4.85), (4.90), and (4.108) ε]: holds, we obtain for any ε ∈ (0,  ∗   u ε (z, t) ≤ Jε  u (z, t) = J  u (z, t) + bε 2 , (4.113) J o (x0 ) − aε ≤ Jε∗ (z 0 ) = Jε  t T  u  z(t), t ∈ [0, t f ] is the solution of (4.85) z(t), t Λ(1) u z(t), t dt; where b = 0 f  u (z, t); Λ(ε) is given by (4.13). generated by the control u =   u (z, t) , The chain of the inequalities and the equalities (4.113) implies J o (x0 )≤ J  which contradicts the right-hand inequality in (4.112). Therefore, the inequality (4.111) is wrong, meaning the validity of (4.110). The latter, along with (4.89), directly yields the statement of the lemma.

4.8 Examples 4.8.1 Example 1 Consider the particular case of the SFHG (4.8)–(4.9) with the following data: n = 2, r = 2, s = 2, q = 1, t f = 2, z 0 = col(3, 4),     1 0 −3 2(t + 1) , , B(t) ≡ A(t) = 2 1 −1 4 √      5 0 2 0 t +2 1 C(t) = , , D(t) ≡ , F= 0 2 0 0 2 −1     1 0 1 0 , G v (t) ≡ . G u (t) ≡ 0 0 0 1

(4.114)

Due to these data, as well as due to the expression for Su 1 (t), the expression for Sv (t), the block form of this matrix and the expression for S1,0 (t) (see the Eqs. (4.20), (4.15), (4.22) and (4.34)), we obtain Su 1 (t) ≡ 1,

Sv1 (t) = t + 3,

S1,0 (t) ≡ −1,

t ∈ [0, 2].

Using these functions and the data (4.114), the terminal-value problem (4.33) becomes as: o  o 2 d P1,0 (t) o o (t) − P1,0 (t) − 5, P1,0 (2) = 2, t ∈ [0, 2]. = 6P1,0 dt

The solution of this problem has the form

102

4 Singular Finite-Horizon Zero-Sum Differential Game

  −1 o P1,0 (t) = 1 + 4 3 exp − 4(t − 2) + 1 , t ∈ [0, 2].

(4.115)

o o Using the Eqs. (4.32), (4.114), (4.115), we calculate the gains P3,0 (t) and P2,0 (t), appearing in the minimizer’s control (4.56), as: o P3,0 (t) ≡

 √ √   −1  o 2, P2,0 (t) = 2(t + 1) 1 + 4 3 exp − 4(t − 2) + 1 , (4.116)

where t ∈ [0, 2]. Now, by virtue of Theorem 4.1, the Eqs. (4.50), (4.54), (4.55), (4.56) and (4.115)– (4.116), we directly obtain the entries of the saddle-point sequence in the SFHG with the data (4.114):  u o∗ εk ,0 (z, t)

=−

o P1,0 (t)x √   2 o (t + 1)P1,0 (t)x + y εk

 , vr∗ (x, t) =

√

o t + 2P1,0 (t)x o P1,0 (t)x

 ,

where z = col(x, y), x and y are scalar variables, t ∈ [0, 2]. Moreover, by virtue of Theorem 4.1 and the Eqs. (4.49), (4.115), the value of the SFHG with the data (4.114) is   J o (x0 ) = J o (3) = 9 1 + 4[3 exp(8) + 1]−1 ≈ 9.004.

4.8.2 Example 2: Solution of Planar Pursuit-Evasion Game with Zero-Order Dynamics of Players In this example, we solve the game (2.120)–(2.121), which is the model of the planar pursuit-evasion engagement between two flying vehicles with zero-order dynamics (for details, see Sect. 2.3.1). For the sake of the book’s reading convenience, we rewrite this game here as: d x1 (t) = x2 (t), t ∈ [0, td ], x1 (0) = x10 = 0, dt d x2 (t) = u(t) + v(t), t ∈ [0, td ], x2 (0) = x20 , dt



td

+ 0

 J u(·), v(·) = f 1 x12 (td )   d1 (t)x12 (t) + d2 (t)x22 (t) − gv (t)v 2 (t) dt,

(4.117)

(4.118)

where td > 0 is the duration of the game; x1 (t) and x2 (t), t ∈ [0, td ] are the scalar state variables of the game; u(t) and v(t), t ∈ [0, td ] are the scalar controls of the

4.8 Examples

103

pursuer and the evader, respectively; f 1 > 0 is a given constant value; d1 (t) ≥ 0, d2 (t) > 0, gv (t) > 0, t ∈ [0, td ] are given functions. Here, we assume that these functions are continuously differentiable in the interval [0, td ]. The objective of the pursuer is to minimize the functional (4.118) by a proper choice of its control, while the objective of the evader is to maximize this functional by a proper choice of its control. The game (4.117)–(4.118) can be considered as a particular case of the SFHG (4.8)–(4.9) with the following data: n = 2, r = 1, s = 1, q = 0, z 0 = col(0, x20 ),         f1 0 0 0 0 1 , , F= , C(t) ≡ , B(t) ≡ A(t) ≡ 0 0 1 1 0 0   d1 (t) 0 , G u (t) ≡ 0, G v (t) = gv (t). D(t) = 0 d2 (t)

(4.119)

Using these data, as well as the expression for Sv (t), the block form of this matrix and the expression for S1,0 (t) (see the Eqs. (4.15), (4.22) and Remark 4.12), we obtain Sv1 (t) ≡ 0,

S1,0 (t) =

1 , d2 (t)

t ∈ [0, td ].

For these functions and the data (4.119), the terminal-value problem (4.33) becomes as: o d P1,0 (t)

dt

=

2 1  o P1,0 (t) − d1 (t), d2 (t)

o P1,0 (td ) = f 1 , t ∈ [0, td ]. (4.120)

The problem (4.120) is a terminal-value problem for a scalar Riccati differential equation. In general, this problem cannot be solved in a closed analytical form. Here, we present two cases where it is possible to obtain the exact analytical solution of the problem (4.120). Case I: d1 (t) = c/d2 (t), c > 0 is a given number, t ∈ [0, td ]. √ √ can be distinguished in this case. Namely: (I.1) c = f 1 ; (I.2) √ Three subcases c < f 1 ; (I.3) c > f 1 . Let us consider each of these cases. Subcase I.1 o In this subcase, the solution of the problem (4.120) is P1,0 (t) ≡ f 1 , t ∈ [0, td ]. Subcase I.2 In this subcase, the solution of the problem (4.120) is o P1,0 (t) =

where

√ 1 + R(t) c , t ∈ [0, td ], 1 − R(t)

(4.121)

104

4 Singular Finite-Horizon Zero-Sum Differential Game

R(t) =

√  t  √ dσ f1 − c ≤ 0. √ exp 2 cQ(t) , Q(t) = d f1 + c 2 (σ ) td

(4.122)

It is seen that √ in this subcase, 0 < R(t) < 1, t ∈ [0, td ]. Therefore, in this subcase, o (t) > c, t ∈ [0, td ]. P1,0 Subcase I.3 In this subcase, the solution of the problem (4.120) has the same form as (4.121)– (4.122). However, in this √ subcase, −1 < R(t) < 0, t ∈ [0, td ]. Therefore, in this o (t) < c, t ∈ [0, td ]. subcase, 0 < P1,0 Now, proceed to the second case where it is possible to obtain the exact analytical solution of the problem (4.120). Case II: d1 (t) ≡ 0, t ∈ [0, td ]. In this case, the solution of the problem (4.120) is  o (t) = P1,0

−1

1 − Q(t) f1

,

t ∈ [0, td ],

where Q(t) is given in (4.122). o (t), t ∈ [0, td ] of the problem (4.120) is obtained, the scalar Once the solution P1,0 o o (t) are calculated using the Eq. (4.32) and the data (4.119). functions P3,0 (t) and P2,0 Thus, o P3,0 (t) =

d2 (t),

o (t) P1,0 o P2,0 (t) = √ , d2 (t)

t ∈ [0, td ].

Using these functions, as well as the Eq. (4.54), Remark 4.12 and Theorem 4.1, we directly obtain the entries of the saddle-point sequence in the singular finite-horizon zero-sum game (4.117)–(4.118) u o∗ εk ,0 (z, t)

1 =− εk

!

o P1,0 (t) x1 + √ d2 (t)

"

d2 (t)x2 , vr∗ (x1 , t) ≡ 0,

where z = col(x1 , x2 ), x1 and x2 are scalar variables, t ∈ [0, td ]. Moreover, by virtue of Remark 4.12, Theorem 4.1 and the Eq. (4.49), the value o 2 (0)x10 = 0. of the game (4.117)–(4.118) is J o (x10 ) = P1,0

4.8.3 Example 3: Solution of Three-Dimensional Pursuit-Evasion Game with Zero-Order Dynamics of Players In this example, we solve two games, modeling the three-dimensional pursuit-evasion engagement between two flying vehicles with zero-order dynamics. These games

4.8 Examples

105

have been formulated in Sect. 2.3.3 (see the games (2.124)–(2.125) and (2.124), (2.127)). Remember that in the game (2.124)–(2.125), one coordinate of the pursuer’s two-dimensional control vector is “regular”, while the other coordinate is “singular”. In the game (2.124), (2.127), both coordinates of the pursuer’s control vector are “singular”.

4.8.3.1

Pursuit-Evasion Game with One “Regular” and One “Singular” Coordinates of the Pursuer’s Control

Here, we consider the following particular case of the game (2.124)–(2.125): d x1 (t) = x3 (t), t ∈ [0, td ], x1 (0) = x10 = 0, dt d x2 (t) = x4 (t), t ∈ [0, td ], x2 (0) = x20 = 0, dt d x3 (t) = u 1 (t) + v1 (t), t ∈ [0, td ], x3 (0) = x30 , dt d x4 (t) = u 2 (t) + v2 (t), t ∈ [0, td ], x4 (0) = x40 , dt

 + 0

td

(4.123)

 J u 1 (·), u 2 (·), v1 (·), v2 (·) = f 1 x12 (td ) + f 2 x22 (td )   d4 (t)x42 (t) + gu,1 (t)u 21 (t) − gv,1 (t)v12 (t) − gv,2 (t)v22 (t) dt, (4.124)

 where td > 0 is the duration of the  game; col x1 (t), x2(t), x3 (t), x4 (t) , t ∈ [0, td ] is the state vector of the game; col u 1 (t), u 2 (t) and col v1 (t), v2 (t) , t ∈ [0, td ] are the control vectors of the pursuer and the evader, respectively; f 1 > 0 and f 2 > 0 are given constant values; d4 (t) > 0, gu,1 (t) > 0, gv,1 (t) > 0, gv,2 (t) > 0, t ∈ [0, td ] are given functions. Here, we assume that these functions are continuously differentiable in the interval [0, td ]. The objective of the pursuer is to minimize the functional (4.124) by a proper choice of its control vector, while the objective of the evader is to maximize this functional by a proper choice of its control vector. The game (4.123)–(4.124) can be considered as a particular case of the SFHG (4.8)–(4.9) with the following data:

106

4 Singular Finite-Horizon Zero-Sum Differential Game

n = 4,

r = 2, ⎛ 0 0 1 ⎜0 0 0 A(t) ≡ ⎜ ⎝0 0 0 0 0 0

s = 2, q = 1, ⎞ ⎛ 0 0 ⎜0 1⎟ ⎟ , B(t) ≡ ⎜ ⎝1 0⎠ 0 0

z 0 = col(0, 0, x30 , x40 ), ⎞ ⎛ ⎞ 0 0 0 ⎜ ⎟ 0⎟ ⎟ , C(t) ≡ ⎜ 0 0 ⎟ , ⎠ ⎝ 0 1 0⎠ 1 0 1  F = diag( f 1 , f 2 , 0, 0), D(t) = diag 0, 0, 0, d4 (t) ,   G u (t) = diag gu,1 (t), 0 , G v (t) = diag gv,1 (t), gv,2 (t) .

(4.125)

Due to these data, as well as due to the expression for Su 1 (t), the expression for Sv (t), the block form of this matrix and the expression for S1,0 (t) (see the Eqs. (4.20), (4.15), (4.22) and (4.34)), we obtain 

   1 1 Su 1 (t) = diag 0, 0, Sv1 (t) = diag 0, 0, , gu,1 (t) gv,1 (t)   1 1 1 , − , t ∈ [0, td ]. S1,0 (t) = diag 0, d4 (t) gu,1 (t) gv,1 (t) Using these matrix-valued functions and the data (4.125), the terminal-value problem (4.33) becomes the following terminal-value problem for the 3 × 3-matrix-valued o (t): function P1,0 o d P1,0 (t)

o o (t) A1 (t) − A1T (t)P1,0 (t) = −P1,0 dt o o o +P1,0 (t)S1,0 (t)P1,0 (t), P1,0 (td ) = F1 , t ∈ [0, td ],

(4.126)

⎛

⎞ 0 0 1 A1 (t) ≡ A1 = ⎝ 0 0 0 ⎠ , 0 0 0

where

F1 = diag( f 1 , f 2 , 0).

This problem has the unique solution in the entire interval [0, td ] if and only if the following inequality is satisfied: 



td

φ1 (t) = 1 + f 1 t



 1 1 − (σ − td )2 dσ > 0 ∀t ∈ [0, td ]. (4.127) gu,1 (σ ) gv,1 (σ )

Subject to this condition, the solution of the problem (4.126) has the form −1 1/2   1/2  o (t) = I3 − A1T (t − td ) F1 Φ(t) F1 I3 − A1 (t − td ) , P1,0 where t ∈ [0, td ],

(4.128)

4.8 Examples

107



= diag f1 , f2 , 0 ,  Φ(t) = diag φ1 (t), φ2 (t), 1 ,  td 1 φ2 (t) = 1 + f 2 dσ. d4 (σ ) t 1/2

F1

o o Using the Eqs. (4.32), (4.125), (4.128), we calculate the gains P3,0 (t) and P2,0 (t), appearing in the minimizer’s control (4.56), as:

o P3,0 (t) =

d4 (t),

  1 f2 o P2,0 (t) = √ , 0 , t ∈ [0, td ]. (4.129) col 0, φ2 (t) d4 (t)

Now, by virtue of Theorem 4.1, the Eqs. (4.50), (4.54), (4.55), (4.56) and (4.128)– (4.129), we directly obtain the entries of the saddle-point sequence in the SFHG with the data (4.125), i.e, in the game (4.123)–(4.124): ⎛



⎞ 1 o 0, 0, (t)x P 1,0 g (t) u,1 ⎝  ⎠,

u o∗ √ εk ,0 (z, t) = − f2 1 √1 d (t)y , 0 x + 0, 4 εk φ2 (t) d4 (t)  

o 0, 0, gv,11(t) P1,0 (t)x ∗ vr (x, t) = , 0 where z = col(x, y), x = col(x1 , x2 , x3 ), y = x4 ; t ∈ [0, td ]. Furthermore, by virtue of Theorem 4.1 and the Eqs. (4.49), (4.128), the value of the game (4.123)–(4.124) is  J o (x0 ) = J o col(0, 0, x30 ) o = (0, 0, x30 )P1,0 (0)col(0, 0, x30 ) =

4.8.3.2

2 2 td f 1 x30 . φ1 (0)

Pursuit-Evasion Game with Both “Singular” Coordinates of the Pursuer’s Control

Here we consider a particular case of the game (2.124), (2.127). In this case, the dynamics of the game is described by the system of the differential equations (4.123). The functional in this case has the form  J u 1 (·), u 2 (·), v1 (·), v2 (·) = f 1 x12 (td ) + f 2 x22 (td )  td  + d3 (t)x32 (t) + d4 (t)x42 (t) 0  −gv,1 (t)v12 (t) − gv,2 (t)v22 (t) dt,

(4.130)

108

4 Singular Finite-Horizon Zero-Sum Differential Game

where d3 (t) > 0 is a given function, continuously differentiable in the interval [0, td ]; the other coefficients are the same as the corresponding ones in (4.124). Like in the game (4.123)–(4.124), the objective of the pursuer in the game (4.123), (4.130) is to minimize the functional by a proper choice of its control vector  col u 1 (·), u 2 (·) , while the objective of the evader is to maximize this functional by a proper choice of its control vector col v1 (·), v2 (·) . The game (4.123), (4.130) can be considered as a particular case of the SFHG (4.8)–(4.9) with the following data: z 0 = col(0, 0, x30 , x40 ), ⎛ ⎞ ⎞ 0 0 0 ⎜ ⎟ 0⎟ ⎟ , C(t) ≡ ⎜ 0 0 ⎟ , ⎝ ⎠ 0 1 0⎠ 1 0 1  F = diag( f 1 , f 2 , 0, 0), D(t) = diag 0, 0, d3 (t), d4 (t) ,   G u (t) = diag 0, 0 , G v (t) = diag gv,1 (t), gv,2 (t) .

n = 4,

r = 2, ⎛ 0 0 1 ⎜0 0 0 A(t) ≡ ⎜ ⎝0 0 0 0 0 0

s = 2, q = 0, ⎛ ⎞ 0 0 ⎜0 1⎟ ⎟ , B(t) ≡ ⎜ ⎝1 0⎠ 0 0

(4.131)

Using these data, as well as the expression for Sv (t), the block form of this matrix and the expression for S1,0 (t) (see the Eqs. (4.15), (4.22) and Remark 4.12), we obtain   1 1 Sv1 (t) ≡ O2×2 , S1,0 (t) = diag t ∈ [0, td ]. , d3 (t) d4 (t) For these functions and the data (4.131), the terminal-value problem (4.33) becomes as:   o d P1,0 (t) 1 1 o o (t)diag (t), P1,0 = P1,0 , dt d3 (t) d4 (t) o P1,0 (td ) = diag( f 1 , f 2 ), t ∈ [0, td ]. The solution of this problem exists in the entire interval [0, td ] and has the form  !   "−1   td 1 1 1 1 o , diag . P1,0 (t) = diag dσ , + f1 f2 d3 (σ ) d4 (σ ) t

(4.132)

o (t) and Using the Eqs. (4.32), (4.131), (4.132), we calculate the gain matrices P3,0 appearing in the minimizer’s control (4.56), as:

o (t), P2,0



o P3,0 (t) = diag d3 (t), d4 (t) ,   1 1 o o ,√ , t ∈ [0, td ]. (t) = P1,0 (t)diag √ P2,0 d3 (t) d4 (t)

(4.133)

4.9 Concluding Remarks and Literature Review

109

Now, using the Eqs. (4.132)–(4.133), as well as the Eq. (4.54), Remark 4.12 and Theorem 4.1, we directly obtain the entries of the saddle-point sequence in the singular finite-horizon zero-sum game (4.123), (4.130) u o∗ εk ,0 (z, t)

 !  1 1 1 o (t)x =− ,√ P1,0 diag √ εk d3 (t) d4 (t)

" +diag d3 (t), d4 (t) y , vr∗ (x, t) ≡ col(0, 0),

where z = col(x, y), x = col(x1 , x2 ), y = col(x3 , x4 ), t ∈ [0, td ]. Moreover, by virtue of Remark 4.12, Theorem 4.1 and the Eq. (4.49), the value of the game (4.123), (4.130) is  o (0)col(x10 , x20 ) = 0. J o col(x10 , x20 ) = (x10 , x20 )P1,0

4.9 Concluding Remarks and Literature Review In this chapter, the singular finite-horizon zero-sum linear-quadratic differential game was considered. The singularity of this game is due to the singularity of the weight matrix in the control cost of the minimizing player (the minimizer) in the game’s functional. If this weight matrix is non-zero, only a part of the coordinates of the minimizer’s control is singular, while the others are regular. For this game, the novel definitions of the saddle-point equilibrium (the saddle-point equilibrium sequence) and the game value were proposed. Subject to proper assumptions, the linear system of the game’s dynamics was transformed equivalently to a new system consisting of three modes. The first mode is controlled directly only by the maximizing player (the maximizer), the second mode is controlled directly by the maximizer and the regular coordinates of the minimizer’s control, while the third mode is controlled directly by the maximizer and the entire control of the minimizer. In Lemma 4.1 it is shown that the game, obtained by this transformation, is equivalent to the initially formulated one. This lemma has not been published before. In the sequel of the chapter, the new (transformed) game is considered as an original game. It also is singular, and it is treated by the regularization method, i.e., by its approximate replacing with an auxiliary regular game. The latter has the same equation of dynamics and a similar functional augmented by an integral of the squares of the minimizer’s singular control coordinates with a small positive weight. The regularization method was used in many works in the literature for solution of singular finite-horizon optimal control problems (see e.g. [3, 7, 8, 11, 13] and references therein). However, to the best of our knowledge, this method was applied only in few works for solution of singular finite-horizon zero-sum differential games (see [9, 12]). In both papers, linear-quadratic games were studied with different

110

4 Singular Finite-Horizon Zero-Sum Differential Game

types of singularities. Thus, in [12], the singularity of the game is due to the absence of the control cost of the minimizer in the functional. In this case, all coordinates of the minimiser’s control are singular. In [9], the more general case was studied where the weight matrix of the minimiser’s control cost, being singular, is not zero. The auxiliary (regularized) game, obtained in this chapter, is a finite-horizon zerosum differential game with a partial cheap control of the minimizer. Finite-horizon zero-sum differential games with a complete/partial minimizer’s cheap control were studied in a number of works. Thus in [6], a suboptimal (asymptotic) solution of a linear-quadratic game with the complete minimizer’s cheap control was derived. In [9, 12], minimizer’s cheap control games were obtained and analyzed due to the regularization method of the solution of the corresponding singular games. In [12] the case of the complete minimizer’s cheap control was treated, while in [9] the case of the partial minimizer’s cheap control was studied. Some classes of pursuit-evasion games with the complete minimizer’s cheap control were considered in [14, 15], where the pursuer’s state-feedback controls were derived. The asymptotic analysis of the regularized game was carried out. Based on this analysis, the saddle-point equilibrium sequence and the value of the original singular game were obtained. Note that Example 2 (see Sect. 4.8.2) is a generalization of the example considered in the paper [12]. Namely, in the example of [12] all the coefficients in the functional of the corresponding singular game are constants, while in Example 2 of the present chapter the coefficients in the integrand of the functional are functions of time. Also, it should be noted that in the papers [9, 12], all the matrices of the coefficients in the dynamics equation and in the integral part of the functional of the game are assumed to be constant. In the present chapter, we consider the case of the timedependent matrices. Moreover, here we study the singular game subject to another (than in [9, 12]) definitions of the saddle-point solution and the game value. Due to this circumstance, the proof of the main theorem (Theorem 4.1), which yields the solution of the singular game, is much simpler than the proofs of the corresponding theorems in these papers.

References 1. Abou-Kandil, H., Freiling, G., Ionescu, V., Jank, G.: Matrix Riccati Equations in Control and Systems Theory. Birkh¨auser, Basel, Switzerland (2003) 2. Basar, T., Olsder, G.J.: Dynamic Noncooparative Game Theory, 2nd edn. SIAM Books, Philadelphia, PA, USA (1999) 3. Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic Press, New York, NY, USA (1975) 4. Derevenskii, V.P.: Matrix Bernoulli equations. I. Russian Math. 52, 12–21 (2008) 5. Glizer, V.Y.: Asymptotic solution of a cheap control problem with state delay. Dynam. Control 9, 339–357 (1999) 6. Glizer, V.Y.: Asymptotic solution of zero-sum linear-quadratic differential game with cheap control for the minimizer. Nonlinear Diff. Equ. Appl. 7, 231–258 (2000)

References

111

7. Glizer, V.Y.: Solution of a singular optimal control problem with state delays: a cheap control approach. In: Reich, S., Zaslavski, A.J. (eds.) Optimization Theory and Related Topics. Contemporary Mathematics Series, vol. 568, pp. 77–107. American Mathematical Society, Providence, RI, USA (2012) 8. Glizer, V.Y.: Stochastic singular optimal control problem with state delays: regularization, singular perturbation, and minimizing sequence. SIAM J. Control Optim. 50, 2862–2888 (2012) 9. Glizer, V.Y., Kelis, O.: Solution of a zero-sum linear quadratic differential game with singular control cost of minimiser. J. Control Decis. 2, 155–184 (2015) 10. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat Mexicana 5, 102–119 (1960) 11. Kurina, G.A.: On a degenerate optimal control problem and singular perturbations. Soviet Math. Dokl. 18, 1452–1456 (1977) 12. Shinar, J., Glizer, V.Y., Turetsky, V.: Solution of a singular zero-sum linear-quadratic differential game by regularization. Int. Game Theory Rev. 16, 1440007-1–1440007-32 (2014) 13. Tikhonov A.N,. Arsenin V.Y.: Solutions of Ill-Posed Problems. Winston & Sons, Washington, DC, USA (1977) 14. Turetsky, V., Glizer, V.Y.: Robust solution of a time-variable interception problem: a cheap control approach. Int. Game Theory Rev. 9, 637–655 (2007) 15. Turetsky, V., Glizer, V.Y., Shinar, J.: Robust trajectory tracking: differential game/cheap control approach. Int. J. Syst. Sci. 45, 2260–2274 (2014) 16. Vasileva, A.B., Butuzov, V.F., Kalachev, L.V.: The Boundary Function Method for Singular Perturbation Problems. SIAM Books, Philadelphia, PA, USA (1995)

Chapter 5

Singular Infinite-Horizon Zero-Sum Differential Game

5.1 Introduction In this chapter, an infinite-horizon zero-sum linear-quadratic differential game, which cannot be solved by application of the first-order solvability conditions, presented in Sect. 3.2.2, is considered. This game is singular, and its singularity is due to the singularity of the weight matrix in the control cost of a minimizing player in the game’s functional. For this game, novel definitions of a saddle-point equilibrium (a saddle-point equilibrium sequence) and a game value are introduced. Regularization method is proposed for obtaining these saddle-point equilibrium sequence and game value. This method consists in an approximate replacement of the original singular game with an auxiliary regular infinite-horizon zero-sum linear-quadratic differential game depending on a small positive parameter. Thus, the first-order solvability conditions are applicable for this new game. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix algebraic equation, arising in these conditions, yields the solution (the saddle-point equilibrium sequence and the game value) to the original singular game. The following main notations are applied in the chapter. 1. E n is the n-dimensional real Euclidean space. 2.  ·  denotes the Euclidean norm either of a vector or of a matrix. 3. The superscript “T ” denotes the transposition of a matrix A, ( A T ) or of a vector x, (x T ). 4. L 2 [a, +∞; E n ] is the linear space of n-dimensional vector-valued real functions, square-integrable in the infinite interval [a, +∞), and  ·  L 2 [a,+∞) denotes the norm in this space. 5. On 1 ×n 2 is used for the zero matrix of the dimension n 1 × n 2 , excepting the cases where the dimension of the zero matrix is obvious. In such cases, the notation 0 is used for the zero matrix. 6. In is the n-dimensional identity matrix. 7. Re λ is the real part of a complex number λ. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_5

113

114

5 Singular Infinite-Horizon Zero-Sum Differential Game

8. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block-vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 9. diag(a1 , a2 , . . . , an ), where ai , (i = 1, 2, . . . , n) are real numbers, is the diagonal matrix with these numbers on the main diagonal. 10. Sn is the set of all symmetric matrices of the dimension n × n. 11. Sn+ is the set of all symmetric positive semi-definite matrices of the dimension n × n. 12. Sn++ is the set of all symmetric positive definite matrices of the dimension n × n.

5.2 Initial Game Formulation Consider the following differential equation controlled by two decision makers (players): dζ(t) (5.1) = Aζ(t) + Bu(t) + Cv(t), ζ(0) = ζ0 , t ≥ 0, dt where ζ(t) ∈ E n is a state vector; u(t) ∈ E r , (r ≤ n) and v(t) ∈ E s are controls of the players; A, B and C are given constant matrices of corresponding dimensions; ζ0 ∈ E n is a given vector. Along with the Eq. (5.1), we consider the following cost functional, to be minimized by the control u (the minimizer) and maximized by the control v (the maximizer): J (u, v) =

+∞ 

 ζ T (t)Dζ(t) + u T (t)G u u(t) − v T (t)G v v(t) dt,

(5.2)

0

where D, G u and G v are given constant matrices of corresponding dimensions; D ∈ Sn+ , G v ∈ Ss++ , G u has the form  Gu =

G¯ u 0

 0 , 0

(5.3)

where q G¯ u ∈ S++ ,

0 < q < r.

Remark 5.1 The form (5.3) of the matrix G u means that this matrix is not invertible. Therefore, the results of Sect. 3.2.2 are not applicable to the game (5.1)–(5.2). Thus, this game is singular which requires another approach to its solution.  of all functions u =  Consider the set U u (ζ) : E n → E r satisfying the local Lip be the set of all functions v =  v (ζ) : E n → E s schitz condition. Similarly, let V satisfying the local Lipschitz condition.

5.2 Initial Game Formulation

115

  Definition 5.1 By (U u (ζ), v (ζ) , ζ ∈ E n , such V ), we denote the set of all pairs  ,   (ii) the initial-value u (ζ) ∈ U v (ζ) ∈ V; that the following conditions are valid: (i)  n u (ζ), v(t) =  v (ζ) and any ζ0 ∈ E has the unique soluproblem (5.1) for u(t) =  (t; ζ ) in the entire interval [0, +∞); (iii) ζ0 ) ∈ L 2 [0, +∞; E n ]; (iv) ζ tion uv 0  ζuv (t;  2 r 2 v ζuv (t; ζ0 ) ∈ L [0, +∞; E s ]. Such a defined  u ζuv (t; ζ0 ) ∈ L [0, +∞; E ]; (v)   set (U V ) is called the set of all admissible pairs of the players’ state-feedback controls in the game (5.1)–(5.2).  

 :   consider the set F v  u (ζ) =  v (ζ) ∈ V u (ζ), v (ζ) ∈ For a given  u (ζ) ∈ U,    

 : F v  u = u (ζ) = ∅ .  u (ζ) ∈ U (U V ) . Let H u , the value u (ζ) ∈ H Definition 5.2 For a given control   Ju  u (ζ); ζ0 =

 u (ζ), v (ζ) J 

sup

(5.4)

v   v (ζ)∈F u (ζ)

is called the guaranteed result of  u (ζ) in the game (5.1)–(5.2).  

  consider the set F u  :  Similarly, for a given  v (ζ) ∈ V, v (ζ) =  u (ζ) ∈ U u (ζ),    

 :F u  v =  v (ζ) ∈ V v (ζ) = ∅ .  v (ζ) ∈ (U V ) . Let H v , the value v (ζ) ∈ H Definition 5.3 For a given control   Jv  v (ζ); ζ0 =

inf

 u (ζ), v (ζ) J 

u   u (ζ)∈F v (ζ)

(5.5)

is called the guaranteed result of  v (ζ) in the game (5.1)–(5.2).  ∗  Consider the sequence of the pairs  u k (ζ), v ∗ (ζ) ∈ (U V ), (k = 1, 2, . . .).

 ∗  +∞ Definition 5.4 The sequence  u k (ζ), v ∗ (ζ) k=1 is called a saddle-point equilibrium sequence (or briefly, a saddle-point sequence) of the game (5.1)–(5.2) if for any ζ0 ∈ E n :  ∗ u k (ζ); ζ0 ; (a) there exist limk→+∞ Ju  (b) the following equality is valid:  ∗  ∗ u k (ζ); Z 0 = Jv  v (ζ); Z 0 . lim Ju 

k→+∞

(5.6)

In this case, the value  ∗  ∗ J ∗ (ζ0 ) = lim Ju  u k (ζ); ζ0 = Jv  v (ζ); ζ0 k→+∞

is called a value of the game (5.1)–(5.2).

(5.7)

116

5 Singular Infinite-Horizon Zero-Sum Differential Game

5.3 Transformation of the Initially Formulated Game  Let us partition the matrix B into blocks as: B = B1 , B2 , where the matrices B1 and B2 are of the dimensions n × q and n × (r − q). We assume that: B has full column rank r . A5.I. The matrix  A5.II. det B2T DB2 = 0. Remark 5.2 Note that the assumptions A5.I–A5.II allow to construct and carry out the time-invariant version of the state transformation (3.72) in the system (5.1) and the functional (5.2) as it is presented below. This transformation converts the system (5.1) and the functional (5.2) to a much simpler form for the analysis and solution of the corresponding singular differential game. 

c = (Bc , B1 ) Let Bc be a complement matrix to B (see Definition 3.13). Then, B c and the Eq. (3.71), we construct the is a complement matrix to B2 . Using B2 , B −1    c , L = c − B2 H. Using the matrix L, we conB matrices H = B2T DB2 B2T DB struct (similarly to (3.72)) the following state transformation in the system (5.1) and the functional (5.2):   (5.8) ζ(t) = Rz(t), R = L, B2 , where z(t) ∈ E n is a new state. Due to Lemma 3.2 and Theorem 3.5, we have the following assertion. Proposition 5.1 Let the assumptions A5.I and A5.II be valid. Then, the transformation (5.8) converts the system (5.1) and the functional (5.2) to the following system and functional: dz(t) = Az(t) + Bu(t) + Cv(t), dt 

+∞ 

J (u, v) = 0

where

t ≥ 0, z(0) = z 0 ,

(5.9)

 z T (t)Dz(t) + u T (t)G u u(t) − v T (t)G v v(t) dt,

A = R−1 AR, C = R−1 C , z 0 = R−1 ζ0 ,

⎛

O(n−r )×q B = R−1 B = ⎝ Iq H2

 D = R DR = T

D1 O(r−q)×(n−r+q)

(5.11)

⎞ O(n−r )×(r −q) ⎠, Oq×(r −q) Ir −q

O(n−r +q)×(r −q) D2

(5.10)

(5.12)  ,

(5.13)

5.3 Transformation of the Initially Formulated Game

D1 = LT DL, D2 = B2T DB2 ,

117

(5.14)

H2 is the right-hand block of the matrix H of the dimension (r − q) × q, D1 ∈ n−r+q r−q , D2 ∈ S++ . S+ Similarly to the game (5.1)–(5.2), in the new game with the dynamics (5.9) and the functional (5.10) the objective of the control u is to minimize the functional, while the objective of the control v is to maximize the functional. Moreover, like the game (5.1)–(5.2), the new game (5.9)–(5.10) is singular. Remark 5.3 the set (U V ) (see Definition 3.5) to be the set of all admis We choose sible pairs u(z), v(z) , z ∈ E n of the players’ state-feedback controls in the game (5.9)–(5.10). Moreover, we keep Definitions 3.6 and 3.7 (see Sect. 3.2.2) of the guaranteed results of the controls u(z) and v(z) to be valid in the game (5.9)–(5.10).  Consider a sequence of the pairs u ∗k (z), v ∗ (z) ∈ (U V ), (k = 1, 2, . . .). +∞

 Definition 5.5 The sequence u ∗k (z), v ∗ (z) k=1 is called a saddle-point equilibrium sequence (or briefly, a saddle-point sequence) of the game (5.9)–(5.10) if for any z 0 ∈ E n :  (a) there exist limk→+∞ Ju u ∗k (z); z 0 ; (b) the following equality is valid:   lim Ju u ∗k (z); z 0 = Jv v ∗ (z); z 0 ,

k→+∞

(5.15)

  where Ju u ∗k (z); z 0 and Jv v ∗ (z); z 0 are the guaranteed results of the controls u ∗k (z) and v ∗ (z), respectively, in the game (5.9)–(5.10). The value   J ∗ (z 0 ) = lim Ju u ∗k (z); z 0 = Jv v ∗ (z); z 0 k→+∞

(5.16)

is called a value of this game. Quite similarly to Lemma 4.1, we have the following assertion. Lemma Let the assumptions A5.I and A5.II be valid. If the sequence of the

 5.1 +∞ v ∗ (ζ) k=1 is a saddle-point sequence of the game (5.1)–(5.2), then pairs  u ∗k (ζ),    +∞ the sequence of the pairs  u ∗k Rz , v ∗ Rz is a saddle-point sequence of k=1

 +∞ the game (5.9)–(5.10). Vice versa: if the sequence of the pairs u ∗k (z), v ∗ (z) k=1 is sequence of the game (5.9)–(5.10), then the sequence of the pairs a saddle-point  −1 ∗  −1 +∞ ∗ is a saddle-point sequence of the game (5.1)–(5.2). uk R ζ , v R ζ k=1

Remark 5.4 Due to Lemma 5.1, the initially formulated game (5.1)–(5.2) is equivalent to the new game (5.9)–(5.10). From the other hand, due to Proposition 5.1, the latter game is simpler than the former one. Therefore, in what follows of this chapter, we deal with the game (5.9)–(5.10). We consider this problem as an original one and call it the Singular Infinite-Horizon Game (SIHG).

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5 Singular Infinite-Horizon Zero-Sum Differential Game

5.4 Auxiliary Lemma Let the pair {A, B} be stabilizable, and M be an r × n-matrix, such that the trivial solution of the system dz(t) = (A + B M)z(t), dt

t ≥ 0, z(t) ∈ E n

(5.17)

is asymptotically stable. Consider the following state-feedback control of the minimizer: u = u M (z) = M z, z ∈ E n .

(5.18)

Along with this state-feedback control, let us consider the matrix Riccati algebraic equation (5.19) K A M + A TM K + K Sv K + D M = 0, where A M = A + B M,

T Sv = C G −1 v C ,

D M = D + M T G u M.

(5.20)

Lemma 5.2 Let the Eq. (5.19) have a solution K = K M ∈ Sn , such that the trivial solution of the system dz(t) = (A M + Sv K M )z(t), dt

t ≥ 0, z(t) ∈ E n

(5.21)

is asymptotically stable. Then: (i) u M (z) ∈ Hu ; (ii) K M ∈ Sn+ ; (iii) theguaranteed result of u M (z) in the SIHG is Ju u M (z); z 0 = supv(z)∈F u (z) J u M (z), v(z) = z 0T K M z 0 ; v

M

(iv) this supremum value is attained for 

T v(z) = v M (z) = G −1 v C K M z,

z ∈ En.

(5.22)

Proof Let us start with the item (i). Due to Definition 3.6, to prove this item, it is sufficient to show the existence of v(z) ∈ V such that u M (z), v(z) ∈ (U V ). We choose v(z) = v M (z), given by (5.22). Since v M (z) is a linear function with a constant gain, then v M (z) ∈ V. Also, it should be noted that the system (5.21) is obtained from the system in (5.9) by replacing u(t) and v(t) with u M (z) and v M (z), respectively. Let, for any given z 0 ∈ E n , z M (t; z 0 ) be the solution of (5.21) subject to the initial condition z(0) = z 0 . Since the trivial solution of the system (5.21) is asymptotically

5.4 Auxiliary Lemma

119

stable, then z M (t; z 0 ) satisfies the inequality z M (t; z 0 ) ≤ az 0  exp(−βt), t ≥ 0, where a > 0 and β > 0 are some constants. This inequality implies the inclusion due to (5.18) and (5.22), uM z M (t; z 0 ) ∈ z M (t; z 0 ) ∈ L 2 [0, +∞;E n ]. Therefore, L 2 [0, +∞; E r ] and v M z M (t; z 0 ) ∈ L 2 [0, +∞; E s ] meaning that u M (z), v M (z) ∈ (U V ). Thus, the item (i) is proven. Proceed to the item (ii). Consider the Lyapunov-like function V (z) = z T K M z.

(5.23)

 Let, for any given v(z) ∈ Fv u M (z) and any given z 0 ∈ E n , z Mv (t; z 0 ) be the solution (5.9) with u(t) = u M (z) and v(t) = v(z). Then, differentiating  of the problem V z Mv (t; z 0 ) with respect to t ≥ 0, we obtain after a routine algebra:    d V z Mv (t; z 0 ) dz Mv (t; z 0 ) T K M z Mv (t; z 0 ) =2 dt dt  T  = 2 A M z Mv (t; z 0 ) + Cv z Mv (t; z 0 ) K M z Mv (t; z 0 )   = z TMv (t; z 0 ) K M A M + A TM K M z Mv (t; z 0 )  +2v T z Mv (t; z 0 ) C T K M z Mv (t; z 0 ).

(5.24)

Due to (5.19), we can rewrite the Eq. (5.24) as  d V z Mv (t; z 0 ) = −z TMv (t; z 0 )K M Sv K M z Mv (t; z 0 ) dt  −z TMv (t; z 0 )D M z Mv (t; z 0 ) + 2v T z Mv (t; z 0 ) C T K M z Mv (t; z 0 ).

(5.25)

Using (5.22), the Eq. (5.25) can be represented in the form  d V z Mv (t; z 0 ) = −z TMv (t; z 0 )D M z Mv (t; z 0 ) dt     T    − v z Mv (t; z 0 ) − v M z Mv (t; z 0 ) G v v z Mv (t; z 0 ) − v M z Mv (t; z 0 )   +v T z Mv (t; z 0 ) G v v z Mv (t; z 0 ) . (5.26)

Since the trivial solution of the equation (5.17) is asymptotically stable, the pair   u M (z), 0 ∈ (U V ). Let z M0 (t; z 0 ) = z Mv (t; z 0 )|v(z)≡0 . Then the Eq. (5.26) yields the inequality d V z M0 (t; z 0 ) /dt ≤ 0, t ≥ 0. Integrating this inequality with respect to t from 0 to +∞ and taking into account that limt→+∞ z M0 (t; z 0 ) = 0, we obtain that V (z 0 ) ≥ 0, i.e., z 0T K M z 0 ≥ 0 ∀z 0 ∈ E n . Hence, K M ∈ Sn+ which completes the proof of the item (ii). Now, let us proceed to the proof of the items (iii) and (iv). Equation (5.26) directly yields the inequality

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5 Singular Infinite-Horizon Zero-Sum Differential Game

 d V z Mv (t; z 0 ) + z TMv (t; z 0 )D M z Mv (t; z 0 ) dt   −v T z Mv (t; z 0 ) G v v z Mv (t; z 0 ) ≤ 0, t ≥ 0.

(5.27)

Integrating this inequality with respect to t from 0 to +∞ and using the  Eqs. (5.10) and (5.23), and the inclusions z Mv (t; z 0 ) ∈ L 2 [0, +∞; E n ] and v z Mv (t; z 0 ) ∈ L 2 [0, +∞; E s ], we obtain   J u M (z), v(z) ≤ z 0T K M z 0 ∀v(z) ∈ Fv u M (z) .

(5.28)

Now, setting v(z) = v M (z) in the Eq. (5.26) and treating the obtained equation  similarly to the inequality (5.27), we have J u M (z), v M (z) = z 0T K M z 0 . Comparison of this equality and the inequality (5.28) immediately implies the validity of the items ⃞ (iii) and (iv) which completes the proof of the lemma.

5.5 Regularization of the Singular Infinite-Horizon Game 5.5.1 Partial Cheap Control Infinite-Horizon Game Like in Chap. 4, where the singular finite-horizon game is studied, the first step in the solution of the SIHG is its regularization. Namely, we replace the SIHG with a regular differential game, which is close in some sense to the SIHG. The new game has the same dynamics (5.9) as the SIHG. However, in contrast with the SIHG, the functional in the new game has the “regular” form. The latter means that this functional contains a quadratic control cost of the minimizer with a positive definite weight matrix:  Jε (u, v) = 0

+∞



 z T (t)Dz(t) + u T (t) G u + Λ(ε) u(t)  −v T (t)G v v(t) dt,

(5.29)

where ε > 0 is a small parameter, and  Λ(ε) = diag 0, . . . , 0, ε2 , . . . , ε2 .       q

(5.30)

r −q

¯ Taking into accountthe Eqs. (5.3), (5.30) and the positive definiteness of G u , we directly conclude that G u + Λ(ε) ∈ Sr++ . Remark 5.5 The functional of the game (5.9), (5.29) is minimized by the control u and maximized by the control v. Since the parameter ε > 0 is small, this game is a partial cheap control differential game, i.e., it is a differential game where a

5.5 Regularization of the Singular Infinite-Horizon Game

121

cost of some control coordinates of at least one of the players is much smaller than control coordinates and a state cost. Since, for any ε > 0, the costs of the other G u + Λ(ε) ∈ Sr++ , the game (5.9), (5.29) is a regulardifferential game. The set of all admissible pairs of players’ state-feedback controls u(z, t), v(z, t) in this game is the same as in the SIHG, i.e., it is (U V ). Also, we keep Definitions 3.6 and 3.7 of the guaranteed results of the controls u(z, t) and v(z, t) to be valid in the game (5.9), (5.29) for any given ε > 0. The above mentioned definitions can be found in Sect. 3.2.2.  Definition 5.6 For a given ε > 0, the pair u ∗ε (z), vε∗ (z) ∈ (U V ) is called a saddlepoint equilibrium solution (or briefly, solution) of the game (5.9),  ∗ a saddle-point ∗ of u u (z); z (z) and the guaranteed result if the guaranteed result J (5.29) ε,u 0 ε ε  Jε,v vε∗ (z); z 0 of vε∗ (z) in this game are equal to each other for all z 0 ∈ E n , i.e.,   Jε,u u ∗ε (z); z 0 = Jε,v vε∗ (z); z 0 ∀z 0 ∈ E n .

(5.31)

In this case, the value   Jε∗ (z 0 ) = Jε,u u ∗ε (z); z 0 = Jε,v vε∗ (z); z 0

(5.32)

is called a value of the game (5.9), (5.29).

5.5.2 Saddle-Point Solution of the Game (5.9), (5.29) Consider the following matrix Riccati algebraic equation with respect to the n × nmatrix P: P A + A T P − P[Su (ε) − Sv ]P + D = 0, where

 −1 Su (ε) = B G u + Λ(ε) B T ,

T Sv = C G −1 v C .

(5.33)

(5.34)

We assume: A5.III. For a given ε > 0, the Eq. (5.33) has a solution P = P ∗ (ε) ∈ Sn such that the trivial solution of each of the following systems:   dz(t)  = A − Su (ε) − Sv P ∗ (ε) z(t), dt dz(t) = [ A − Su (ε)P ∗ (ε)]z(t), dt

t ≥ 0,

t ≥0

is asymptotically stable. Using the above mentioned matrix P ∗ (ε), we construct the functions

(5.35)

(5.36)

122

5 Singular Infinite-Horizon Zero-Sum Differential Game T ∗ u ∗ε (z) = −[G u + Λ(ε)]−1 B T P ∗ (ε)z, vε∗ (z) = G −1 v C P (ε)z.

(5.37)

The following assertion directly follows from Theorem 3.2. Proposition5.2 Let the assumptions A5.I - A5.III be valid. Then: (a) the pair u ∗ε (z), vε∗ (z) , given by (5.37), is the saddle-point solution of the game (5.9), (5.29); (b) the value of this game is  Jε∗ (z 0 ) = Jε u ∗ε (z), vε∗ (z) = z 0T P ∗ (ε)z 0 ;

(5.38)

  (c) u(z) ∈ Fu vε∗ (z) and any v(z) ∈ Fv u ∗ε (z) , the saddle-point solution  ∗ for any u ε (z), vε∗ (z) of the game (5.9), (5.29) satisfies the following inequality:    J u ∗ε (z), v(z) ≤ J u ∗ε (z), vε∗ (z) ≤ J u(z), vε∗ (z) ;

(5.39)

(d) P ∗ (ε) ∈ Sn+ meaning that Jε∗ (z 0 ) ≥ 0. Remember that the sets Fu (·) and Fv (·) are introduced in Sect. 3.2.2, and these sets are used in Definitions 3.6 and 3.7. Remark 5.6 In the next section, an asymptotic behaviour (for ε → +0) of the solution to the game (5.9), (5.29) is analyzed. Using this analysis, ε-free conditions for the existence of the solution P ∗ (ε) to the Eq. (5.33), mentioned in the assumption A5.III, are established. All these results, will be used to derive the saddle-point sequence and the value of the SIHG.

5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29) We start this analysis with an asymptotic solution with respect to the small parameter ε > 0 of the Eq. (5.33).

5.6.1 Transformation of the Eq. (5.33) Using the block form of the matrix B and the block-diagonal form of the matrix G u + Λ(ε) (see the Eqs. (5.12) and (5.3), (5.30)), we can represent the matrix Su (ε) (see the Eq. (5.34)) in the following block form: ⎛ Su (ε) = ⎝

Su 1

Su 2

SuT2

(1/ε )Su 3 (ε) 2

⎞ ⎠,

(5.40)

5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29)

123

where the (n − r + q) × (n − r + q)-matrix Su 1 , the (n − r + q) × (r − q)-matrix Su 2 and (r − q) × (r − q)-matrix Su 3 (ε) are of the form ⎛ Su 1 = ⎝

⎛

⎞

0 0 0 G¯ −1 u

⎠,

Su 2 = ⎝

⎞

0 T G¯ −1 u H2

⎠,

T Su 3 (ε) = ε2 H2 G¯ −1 u H2 + Ir −q .

(5.41)

Due to the form of the matrices Su (ε) and Su 3 (ε), the left-hand side of the Eq. (5.33) has the singularity at ε = 0. To remove this singularity, we look for the solution of this equation in the block form ⎛ P(ε) = ⎝

P1 (ε) εP2 (ε) εP2T (ε)

εP3 (ε)

⎞ ⎠,

P1T (ε) = P1 (ε),

P3T (ε) = P3 (ε),

(5.42)

where the matrices P1 (ε), P2 (ε) and P3 (ε) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively. Let us partition the matrices A and Sv into blocks as: ⎛ A=⎝

A1 A2 A3 A4

⎞ ⎠,

⎛ Sv = ⎝

Sv1 Sv2 SvT2

⎞ ⎠,

(5.43)

Sv3

where the blocks A1 , A2 , A3 and A4 are of the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively; the blocks Sv1 , Sv2 and Sv3 have the form T −1 T −1 T Sv1 = C1 G −1 v C 1 , Sv2 = C 1 G v C 2 , Sv3 = C 2 G v C 2 ,

and C1 and C2 are the upper and lower blocks of the matrix C of the dimensions (n − r + q) × s and (r − q) × s, respectively. Now, substitution of (5.13), (5.40), (5.42), (5.43) into the Eq. (5.33) transforms the latter after a routine algebra to the following equivalent set of the equations:  P1 (ε)A1 + εP2 (ε) A3 + A1T P1 (ε) + εA3T P2T (ε) − P1 (ε) Su 1 − Sv1 P1 (ε)   −εP2 (ε) SuT2 − SvT2 P1 (ε) − εP1 (ε) Su 2 − Sv2 P2T (ε)  −P2 (ε) Su 3 (ε) − ε2 Sv3 P2T (ε) + D1 = 0, (5.44)

124

5 Singular Infinite-Horizon Zero-Sum Differential Game

 P1 (ε)A2 + εP2 (ε)A4 + εA1T P2 (ε) + εA3T P3 (ε) − εP1 (ε) Su 1 − Sv1 P2 (ε)   −ε2 P2 (ε) SuT2 − SvT2 P2 (ε) − εP1 (ε) Su 2 − Sv2 P3 (ε)  −P2 (ε) Su 3 (ε) − ε2 Sv3 P3 (ε) = 0, (5.45)

 εP2T (ε)A2 + εP3 (ε)A4 + εA2T P2 (ε) + εA4T P3 (ε) − ε2 P2T (ε) Su 1 − Sv1 P2 (ε)   −ε2 P3 (ε) SuT2 − SvT2 P2 (ε) − ε2 P2T (ε) Su 2 − Sv2 P3 (ε)  −P3 (ε) Su 3 (ε) − ε2 Sv3 P3 (ε) + D2 = 0. (5.46)

5.6.2 Asymptotic Solution of the Set (5.44)–(5.46) 

We look for the zero-order asymptotic solution P1,0 , P2,0 , P3,0 of the set (5.44)– (5.46). Setting formally ε = 0 in this set, we obtain the following equations for its zero-order asymptotic solution:  T + D1 = 0, P1,0 A1 + A1T P1,0 − P1,0 Su 1 − Sv1 P1,0 − P2,0 P2,0

(5.47)

P1,0 A2 − P2,0 P3,0 = 0,

(5.48)

 2 − P3,0 + D2 = 0.

(5.49)

Solving the Eqs. (5.49) and (5.48) with respect to P3,0 and P2,0 , we obtain  1/2 , P3,0 = D2

 −1/2 P2,0 = P1,0 A2 D2 ,

(5.50)

where the superscript “1/2” denotes the unique symmetric positive definite square root of the corresponding symmetric positive definite matrix, the superscript “– 1/2” denotes the inverse matrix of this square root. Now, substitution of the expression for P2,0 from (5.50) into (5.47) yields

where

P1,0 A1 + A1T P1,0 − P1,0 S1,0 P1,0 + D1 = 0,

(5.51)

S1,0 = Su 1 − Sv1 + A2 D2−1 A2T .

(5.52)

5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29)

125

In what follows of this chapter, we assume: A5.IV. The matrix Riccati algebraic equation (5.51) has a solution P1,0 ∈ Sn−r +q such that the trivial solution of each of the following systems: d x(t) = (A1 − S1,0 P1,0 )x(t), t ≥ 0, x(t) ∈ E n−r +q , dt d x(t) = [ A1 − (Su 1 + A2 D2−1 A2T )P1,0 ]x(t), t ≥ 0, x(t) ∈ E n−r +q dt

(5.53)

(5.54)

is asymptotically stable. Remark 5.7 Note that the assumption A5.IV is an ε-free assumption, which guarantees the existence of the stabilizing solution to the set of the Eqs. (5.44)–(5.46) for all sufficiently small ε > 0 (see Lemma 5.3, Corollary 5.1 and their proofs below). The latter provides the existence of the saddle-point solution to the game (5.9), (5.29) for all such ε > 0 (see Corollary 5.1 and its proof below). Moreover, the assumption A5.IV guarantees the existence of a saddle-point solution to the Reduced InfiniteHorizon Game (see Sect. 5.7 below). Lemma 5.3 Let the assumptions A5.I, A5.II and A5.IV be valid. Then, there exists a positive number ε0 such that, for all ε ∈ (0, ε0 ], the set of the Eqs. (5.44)–(5.46) has the solution Pi (ε) = Pi∗ (ε), (i = 1, 2, 3), satisfying the inequalities  ∗   P (ε) − Pi,0  ≤ aε, i

i = 1, 2, 3,

(5.55)

where a > 0 is some constant independent of ε. Proof Let us make the following transformation of variables in the set of the Eqs. (5.44)–(5.46): Pi (ε) = Pi,0 + δi (ε),

i = 1, 2, 3,

where δi (ε), (i = 1, 2, 3) are new unknown matrices. Consider the block-form matrix   δ1 (ε) εδ2 (ε) . δ(ε) = εδ2T (ε) εδ3 (ε)

(5.56)

(5.57)

Substituting (5.56) into the set (5.44)–(5.46) and using the Eqs. (5.47)–(5.49) and the block representations of the matrices Su (ε), P(ε), A, Sv (see the Eqs. (5.40), (5.42), (5.43)), we obtain after a routine algebra the equation for δ(ε) δ(ε)α(ε) + αT (ε)δ(ε) − δ(ε)[Su (ε) − Sv ]δ(ε) + γ(ε) = 0,

(5.58)

126

5 Singular Infinite-Horizon Zero-Sum Differential Game

where ⎛ α(ε) = A − [Su (ε) − Sv ]P0 (ε), P0 (ε) = ⎝

P1,0 εP2,0 T εP2,0 εP3,0

⎞ ⎠;

the matrix γ(ε) is expressed in a known form by the matrices P0 (ε), Su (ε) and Sv , and it satisfies the inequality γ(ε) ≤ b1 ε,

ε ∈ (0, ε1 ],

(5.59)

where b1 is some positive number independent of ε; ε1 > 0 is some sufficiently small number. Now, let us treat the matrix α(ε). We can rewrite this matrix in the block form as:  α(ε) =

α1 (ε) 1 α (ε) ε 3

 α2 (ε) , 1 α (ε) ε 4

T , α1 (ε) = A1 − (Su 1 − Sv1 )P1,0 − ε(Su 2 − Sv2 )P2,0 α2 (ε) = A2 − ε(Su 1 − Sv1 )P2,0 − ε(Su 2 − Sv2 )P3,0 , T α3 (ε) = εA3 − ε(Su 2 − Sv2 )T P1,0 − (Su 3 (ε) − ε2 Sv3 )P2,0 ,

α4 (ε) = εA4 − ε2 (Su 2 − Sv2 )T P2,0 − (Su 3 (ε) − ε2 Sv3 )P3,0 . (5.60) The characteristic equation of the matrix α(ε) is the following polynomial equation with respect to λ: det (λ, ε) = 0,

(λ, ε) = α(ε) − λIn ,

(5.61)

and this equation is a particular case of the quasipolynomial equation studied in [2] (see the Eq. (2.4)). Applying the results of this work (see Theorem 2.1) to the Eq. (5.61), we can conclude the following. If α4−1 (0) exists and the matrices    φ = α1 (ε) − α2 (ε)α4−1 (ε)α3 (ε) ε=0 and α4 (0)

(5.62)

are Hurwitz matrices, then there exists a positive number ε2 ≤ ε1 such that, for all ε ∈ (0, ε2 ], n − r + q roots λk of the Eq. (5.61) satisfy the inequality Reλk < −β1 ,

k = 1, . . . , n − r + q,

(5.63)

while the other r − q roots λ p satisfy the inequality Reλ p < −β2 /ε,

p = n − r + q + 1, . . . , n,

(5.64)

5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29)

127

where β1 > 0 and β2 > 0 are some constants independent of ε. Calculating the first matrix in (5.62) and using the expressions of αi (ε), (i = 1, . . . , 4) in (5.60), the Eqs. (5.50), (5.52) and the equality Su 3 (0) = Ir −q , we directly obtain that φ = A1 − S1,0 P1,0 . Hence, due to the assumption A5.IV, the matrix φ is a  1/2 Hurwitz matrix. The second matrix in (5.62) is α4 (0) = −P3,0 = − D2 , meaning that this matrix also is a Hurwitz matrix. Thus, both matrices in (5.62) are Hurwitz matrices and, therefore, the inequalities (5.63) and (5.64) are satisfied. Based on the inequalities (5.63), (5.64) and using the results of [1], we rewrite the Eq. (5.58) in the equivalent form 

+∞

δ(ε) =

   Ω T (σ, ε) γ(ε) − δ(ε) Su (ε) − Sv δ(ε) Ω(σ, ε)dσ, ε ∈ (0, ε2 ],

0

(5.65) where, for any given ε ∈ (0, ε2 ], the n × n-matrix-valued function Ω(σ, ε) is the unique solution of the problem dΩ(σ, ε) = α(ε)Ω(σ, ε), Ω(0, ε) = In , σ ≥ 0. dσ

(5.66)

Let Ω1 (σ, ε), Ω2 (σ, ε), Ω3 (σ, ε) and Ω4 (σ, ε) be the upper left-hand, upper right-hand, lower left-hand and lower right-hand blocks of the matrix Ω(σ, ε) of the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively. By virtue of the results of [2] (Theorem 2.3), we have the following estimates of these blocks:     Ωk (σ, ε) ≤ b2 exp(−β1 σ), k = 1, 3, Ω2 (σ, ε) ≤ b2 ε exp(−β1 σ),      Ω4 (σ, ε) ≤ b2 ε exp(−β1 σ) + exp − β2 σ/ε , σ ≥ 0, ε ∈ (0, ε3 ], (5.67) where b2 is some positive number independent of ε; 0 < ε3 ≤ ε2 is some sufficiently small number. Applying the method of successive approximations to the equation (5.65), let us +∞ consider the sequence of the matrices δ j (ε) j=0 given as:  δ j+1 (ε) =

+∞

   Ω T (σ, ε) γ(ε) − δ j (ε) Su (ε) − Sv δ j (ε) Ω(σ, ε)dσ, (5.68)

0

where ( j = 0, 1, . . . , ), ε ∈ (0, ε3 ]; δ0 (ε) = 0; the matrices δ j (ε) have the block form   δ j,1 (ε) εδ j,2 (ε) δ j (ε) = , j = 1, 2, . . . , εδ Tj,2 (ε) εδ j,3 (ε) and the dimensions of the blocks in each of these matrices are the same as the dimensions of the corresponding blocks in (5.57).

128

5 Singular Infinite-Horizon Zero-Sum Differential Game

Using the block form of all the matrices appearing in the Eq. (5.68), as well as using the inequalities (5.59) and (5.67), we obtain the +∞ of a positive number

existence ε0 ≤ ε3 such that, for any ε ∈ (0, ε0 ], the sequence δ j (ε) j=0 converges in the linear space of n × n-matrices. Moreover, the following inequalities are fulfilled: δ j,i (ε) ≤ aε,

i = 1, 2, 3

j = 1, 2, . . . ,

where a > 0 is some number independent of ε, i and j. Thus, for any ε ∈ (0, ε0 ], 

δ ∗ (ε) = lim δ j (ε) j→+∞

is a solution of the Eq. (5.65) and, therefore, of the Eq. (5.58). Moreover, this solution has the block form similar to (5.57) and satisfies the inequalities δi∗ (ε) ≤ aε,

i = 1, 2, 3.

The latter, along with the Eq. (5.56), proves the lemma.

⃞

Corollary 5.1 Let the assumptions A5.I, A5.II and A5.IV be valid. Then, there exists a positive number ε∗ ≤ ε0 such that, for any ε ∈ (0, ε∗ ], the assumption A5.III is fulfilled and 

∗

P (ε) =

 P1∗ (ε) εP2∗ (ε)  T , ε P2∗ (ε) εP3∗ (ε)

where Pi (ε) = Pi∗ (ε), (i = 1, 2, 3) is the solution of the set (5.44)–(5.46) mentioned in Lemma 5.3. Moreover, all the statements of Proposition 5.2 are valid. Proof To prove the fulfilment of the assumption A5.III, we should show the asymptotic stability of the trivial solution to the systems (5.35) and (5.36) for any sufficiently small ε > 0. We start with the system (5.35). Let us partition the matrix of the coefficients of this system into blocks as: 

 α2∗ (ε) α (ε) = A − Su (ε) − Sv P (ε) = , 1 ∗ α (ε) ε 4  T α1∗ (ε) = A1 − (Su 1 − Sv1 )P1∗ (ε) − ε(Su 2 − Sv2 ) P2∗ (ε) , α2∗ (ε) = A2 − ε(Su 1 − Sv1 )P2∗ (ε) − ε(Su 2 − Sv2 )P3∗ (ε),   T α3∗ (ε) = εA3 − ε(Su 2 − Sv2 )T P1∗ (ε) − Su 3 (ε) − ε2 Sv3 P2∗ (ε) ,  α4∗ (ε) = εA4 − ε2 (Su 2 − Sv2 )T P2∗ (ε) − Su 3 (ε) − ε2 Sv3 P3∗ (ε). ∗







∗

α1∗ (ε) 1 ∗ α (ε) ε 3

(5.69)

5.7 Reduced Infinite-Horizon Differential Game

129

Using the Eq. (5.69) and Lemma 5.3 (see the inequalities in (5.55)), one can prove (quite similarly to the proof of this lemma) that the matrix α∗ (ε) is a Hurwitz matrix for any sufficiently small ε > 0. The latter implies the asymptotic stability of the trivial solution to the system (5.35) for any such ε > 0. The asymptotic stability of the trivial solution to the system (5.36) is proven in the similar way. Thus, there exists a positive number ε∗ ≤ ε0 such that, for any ε ∈ (0, ε∗ ], the assumption A5.III is fulfilled. Moreover, since this assumption is fulfilled, then all the statements of ⃞ Proposition 5.2 are valid for any ε ∈ (0, ε∗ ].

5.6.3 Asymptotic Representation of the Value of the Game (5.9), (5.29) Let us consider the following value: 

J0 (x0 ) = x0T P1,0 x0 ,

(5.70)

where the matrix P1,0 is the solution of the Eq. (5.51) mentioned in the assumption A5.IV; x0 ∈ E n−r+q is the upper block of the vector z 0 . Lemma 5.4 Let the assumptions A5.I, A5.II and A5.IV be valid. Then, for all ε ∈ (0, ε∗ ], the following inequality is satisfied:  ∗   J (z 0 ) − J0 (x0 ) ≤ c(z 0 )ε, ε where Jε∗ (z 0 ) is the value of the game (5.9), (5.29); c(z 0 ) > 0 is some constant independent of ε, while depending on z 0 . Proof The statement of the lemma directly follows from Proposition 5.2 (item (b)), ⃞ Lemma 5.3 and Corollary 5.1.

5.7 Reduced Infinite-Horizon Differential Game Let us consider the following block-form matrices  B= B, A2 ,  B1,0 = 



O(n−r )×q Iq



 , =

Oq×(r −q) G¯ u O(r −q)×q D2

 .

(5.71)

Quite similarly to Proposition 4.3 (see Sect. 4.6), we have the following assertion. Proposition 5.3 The matrix S1,0 , defined by (5.52), can be represented as: T S1,0 = B1,0 −1 B1,0 − Sv1 .

130

5 Singular Infinite-Horizon Zero-Sum Differential Game

Let us consider the infinite-horizon zero-sum linear-quadratic differential game with the dynamics d xr (t) = A1 xr (t) + B1,0 u r (t) + C1 vr (t), xr (0) = x0 , t ≥ 0, dt

(5.72)

where xr (t) ∈ E n−r+q is a state variable; u r (t) ∈ E r and vr (t) ∈ E s are controls of the game’s players. The functional of this game, to be minimized by u r (t) and maximized by vr (t), is  Jr (u r , vr ) = 0

+∞



 xrT (t)D1 xr (t) + u rT (t)u r (t) − vrT (t)G v vr (t) dt.

(5.73)

We call the game (5.72)–(5.73) the Reduced Infinite-Horizon Game (RIHG). Since  ∈ Sr++ and G v ∈ Ss++, the RIHG is regular. The set of the admissible pairs of the state-feedback controls u r (xr ), vr (xr ) and the guaranteed results of u r (xr ) and vr (xr ) in the RIHG are defined similarly to those notions in Sect. 3.2.2 (see Definitions 3.5– 3.7). The saddle-point solution and the game value of the RIHG are defined similarly to those notions for the game (5.9), (5.29) in Sect. 5.5.1 (see Definition 5.6). Using Proposition 5.3, we have (as a direct consequence of Theorem 3.2) the following assertion. Proposition 5.4 Let the assumptions A5.I, A5.II and A5.IV be valid. Then:  (a) the pair u ∗r (xr ), vr∗ (xr ) , where T T u ∗r (xr ) = −−1 B1,0 P1,0 xr , vr∗ (xr ) = G −1 v C 1 P1,0 x r ,

(5.74)

is the saddle-point solution of the RIHG; (b) the value of this game has the form Jr∗ = J0 (x0 ), where J0 (x0 ) is defined by (5.70). Remark 5.8 Using the expressions for B1,0 and  (see the Eq. (5.71)), we can represent the minimizer’s control u ∗r (xr ) in the saddle point solution of the RIHG as: u ∗r (xr ) =



 u ∗r,1 (xr ) , u ∗r,2 (xr )

5.8 Saddle-Point Sequence of the SIHG

131

where T u ∗r,1 (xr ) = −G¯ −1 u B P1,0 x r ,

u ∗r,2 (xr ) = −D2−1 A2T P1,0 xr .

(5.75)

5.8 Saddle-Point Sequence of the SIHG 5.8.1 Main Assertions For any given ε ∈ (0, ε∗ ], consider the following function of z ∈ E n : u ∗ε,0 (z)

 =

 u ∗r,1 (x)  , T x + P3,0 y − 1ε P2,0

z = col(x, y),

(5.76)

where x ∈ E n−r+q , y ∈ E r−q ; P2,0 , P3,0 are given in (5.50); P1,0 is the solution of the Eq. (5.51) mentioned in the assumption A5.IV. Lemma 5.5 Let the assumptions A5.I, A5.II and A5.IV be valid. Then,  there exist a positive number ε∗1 ≤ ε∗ such that, for any given ε ∈ (0, ε∗1 ], the pair u ∗ε,0 (z), vr∗ (x) is pair of the players’ state-feedback controls in the SIHG, i.e.,  ∗an admissible u ε,0 (z, t), vr∗ (x) ∈ (U V ). Proof The players’ state-feedback controls u ∗ε,0 (z) and vr∗ (x) are linear in z with the constant gain matrices. Therefore, due to Definition 3.5, to prove the lemma, it is sufficient to show the asymptotic stability of the trivial solution to the closed-loop system obtained from (5.9) by replacing there u(t) and v(t) with the aforementioned B state-feedback controls. Using the block form of the matrices B, Su 1 , Su 2 , A, Sv ,  (see the Eqs. (5.12), (5.41), (5.43), (5.71)), as well as the expressions for u ∗ε,0 (z) and vr∗ (x), we obtain after a routine algebra this closed-loop system as: dz(t) = Ψ (ε)z, t ≥ 0, dt

(5.77)

where  Ψ (ε) =

Ψ1 Ψ2 1 1 Ψ (ε) Ψ 3 ε ε 4

 ,

 Ψ1 = A1 − Su 1 − Sv1 P1,0 , Ψ2 = A2 ,    T Ψ3 (ε) = ε A3 − SuT2 − SvT2 P1,0 − P2,0 , Ψ4 = −P3,0 .

(5.78)

Now, using the Eq. (5.78), we prove (quite similarly to the proof of Lemma 5.3) that the matrix Ψ (ε) is a Hurwitz matrix for any sufficiently small ε > 0. The latter means the asymptotic stability of the trivial solution to the system (5.77) for any such ε > 0. ⃞

132

5 Singular Infinite-Horizon Zero-Sum Differential Game

Lemma 5.6 Let the assumptions A5.I, A5.II and A5.IV be valid. Then, there exists ∗ ε ∈ (0, εu ], the guaranteed a positive  ∗number εu , (εu ≤ ε1 ) such that, for all result Ju u ε,0 (z), z 0 of the minimizer’s control u ∗ε,0 (z) in the SIHG satisfies the inequality    ∗  Ju u (z); z 0 − J0 (x0 ) ≤ aε, ε,0 where a > 0 is some constant independent of ε; J0 (x0 ) is defined by (5.70). Proof of the lemma is presented in Sect. 5.8.2. Lemma 5.7 Let assumptions A5.I, A5.II and A5.IV be valid. Then, the guaran the ∗ of the maximizer’s control vr∗ (x) in the SIHG is v (x); z J teed result v 0 r  ∗ Jv vr (x); z 0 = J0 (x0 ). Proof of the lemma is presented in Sect. 5.8.3. Let {εk }+∞ k=1 be a sequence of numbers, satisfying the following conditions: (i) εk ∈ (0, εu ], (k = 1, 2, . . .), where εu > 0 is defined in Lemma 5.6; (ii) εk → +0 for k → +∞. The following assertion is an immediate consequence of Definition 5.5 and Lemmas 5.5–5.7. Theorem 5.1 +∞ A5.I, A5.II and A5.IV be valid. Then the sequence

Let the assumptions of the pairs u ∗εk ,0 (z), vr∗ (x) k=1 is the saddle-point sequence of the SIHG. Moreover, J0 (x0 ) is the value of this game. Remark 5.9 Note that the upper block of the minimizer’s control sequence +∞

o∗ u εk ,0 (z) k=1 is independent of k, and it coincides with the upper block of the minimizer’s control in the saddle-point solution of the RIHG. Similarly, the maximizer’s control in the saddle-point sequence of the SIHG and the value of this game coincide with the saddle-point maximizer’s control and the game value, respectively, in the RIHG. The latter game is regular, and it is of a smaller dimension than the SIHG. Thus, in order to solve the SIHG, one has to solve the smaller dimension regular RIHG and construct two gain matrices P3,0 and P2,0 using the Eq. (5.50). Remark 5.10 If q = 0, i.e., all the coordinates of the control of the minimizing player in the SIHG (5.9)–(5.10) are singular, then Su 1 = O(n−r )×(n−r ) , Su 2 = O(n−r )×r , Su 3 = Ir and S1,0 = A2 D2−1 A2T − Sv1 . The matrix P1,0 is the solution of the Riccati algebraic equation (5.51) with this S1,0 . The upper block of the control u ∗ε,0 (z) (see the Eq. (5.76)) vanishes, while the lower block remains unchanged. Thus,  T in this case we have u ∗ε,0 (z) = − 1ε P2,0 x + P3,0 y , z = col(x, y), x ∈ E n−r , y ∈ E r . Theorem 5.1 is valid with this particular form of u ∗ε,0 (z) and the aforementioned P1,0 .

5.8 Saddle-Point Sequence of the SIHG

133

5.8.2 Proof of Lemma 5.6 Consider the following matrix: 



M = M(ε) = −

T G¯ −1 u B P1,0 Oq×(r −q) 1 T 1 P P ε 2,0 ε 3,0

 ,

ε ∈ (0, ε∗1 ].

(5.79)

Using the expression for u ∗r,1 (x) in (5.75) and the matrix M(ε), we can rewrite the control (5.76) as: u ∗ε,0 (z) = M(ε)z,

z ∈ En.

By virtue of Lemma 5.2, we can conclude the following. If for some ε ∈ (0, ε∗1 ] the trivial solution of the system (5.17), (5.79) is asymptotically stable and the Eq. (5.19), (5.79) has a solution K = K M ∈ Sn such that the trivial solution of the system (5.21), (5.79) is asymptotically stable, then u ∗ε,0 (z) ∈ Hu and its guaranteed result in the SIHG is  (5.80) Ju u ∗ε,0 (z); z 0 = z 0T K M z 0 . The asymptotic stability of the trivial solution to the system (5.17), (5.79) for all ε ∈ (0, ε∗2 ], where 0 < ε∗2 ≤ ε∗1 is sufficiently small, is shown similarly to the proof of Corollary 5.1. Now, we are going to show the existence of the above mentioned solution to the equation (5.19), (5.79) for all sufficiently small ε > 0. For this purpose, we seek this solution in the block form   K M,1 (ε) εK M,2 (ε) (5.81) , K M = K M (ε) = T (ε) εK M,3 (ε) εK M,2 where the matrices K M,1 (ε), K M,2 (ε) and K M,3 (ε) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively; T T (ε) = K M,1 (ε), K M,3 (ε) = K M,3 (ε). K M,1 Substitution of (5.3), (5.12), (5.13), (5.43), (5.79) and (5.81) into (5.19) transforms the latter after a routine algebra to the following equivalent set equations: T (ε) K M,1 (ε) A1 + εK M,2 (ε)A3 + A1T K M,1 (ε) + εA3T K M,2 T −K M,1 (ε)Su 1 P1,0 − εK M,2 (ε)SuT2 P1,0 − K M,2 (ε)P2,0 T T −P1,0 Su 1 K M,1 (ε) − εP1,0 Su 2 K M,2 (ε) − P2,0 K M,2 (ε) T +K M,1 (ε)Sv1 K M,1 (ε) + εK M,2 (ε)SvT2 K M,1 (ε) + εK M,1 (ε)Sv2 K M,2 (ε) T (ε) + D1 + P1,0 Su 1 P1,0 = 0, +ε2 K M,2 (ε)Sv3 K M,2

(5.82)

134

5 Singular Infinite-Horizon Zero-Sum Differential Game

K M,1 (ε)A2 + εK M,2 (ε)A4 + εA1T K M,2 (ε) + εA3T K M,3 (ε) −K M,2 (ε)P3,0 − εP1,0 Su 1 K M,2 (ε) + εP1,0 Su 2 K M,3 (ε) −P2,0 K M,3 (ε) + εK M,1 (ε)Sv1 K M,2 (ε) + ε2 K M,2 (ε)SvT2 K M,2 (ε) +εK M,1 (ε)Sv2 K M,3 (ε) + ε2 K M,2 (ε)Sv3 K 3 (ε) = 0,

(5.83)

T (ε) A2 + εK M,3 (ε)A4 + εA2T K M,2 (ε) + εA4T K M,3 (ε) εK M,2 T −K M,3 (ε)P3,0 − P3,0 K M,3 (ε) + ε2 K M,2 (ε)Sv1 K M,2 (ε) T +ε2 K M,3 (ε)SvT2 K M,2 (ε) + ε2 K M,2 (ε)Sv2 K M,3 (ε)

+ε2 K M,3 (ε)Sv3 K M,3 (ε) + D2 = 0,

(5.84)

where Su 1 and Su 2 are given in (5.41). 

Looking for the zero-order asymptotic solution K M,10 , K M,20 , K M,30 of the Eqs. (5.82)–(5.84), we obtain similarly to Sect. 5.6.2 the following set of equations for these terms: T T − K M,10 Su 1 P1,0 − P2,0 K M,20 K M,10 A1 + A1T K M,10 − K M,20 P2,0 −P1,0 Su 1 K M,10 + K M,10 Sv1 K M,10 + D1 + P1,0 Su 1 P1,0 = 0,

(5.85)

K M,10 A2 − K M,20 P3,0 − P2,0 K M,30 = 0,

(5.86)

K M,30 P3,0 + P3,0 K M,30 − D2 = 0.

(5.87)

Solving the algebraic equations (5.87) and (5.86) with respect to K M,30 and K M,20 , and using the Eq. (5.50), we obtain K M,30 =

1  1/2 , D2 2

 −1/2 1  −1/2 − P1,0 A2 D2 . K M,20 = K M,10 A2 D2 2

(5.88)

(5.89)

Substitution of P2,0 and K M,20 from (5.50) and (5.89), respectively, into (5.85) yields after a routine algebra the following equation with respect to K M,10 :   K M,10 A1 + A1T K M,10 − K M,10 S1,0 + Sv1 P1,0 − P1,0 S1,0 + Sv1 K M,10  +K M,10 Sv1 K M,10 + D1 + P1,0 S1,0 + Sv1 P1,0 = 0, (5.90) where S1,0 is given by (5.52).

5.8 Saddle-Point Sequence of the SIHG

135

By substitution of P1,0 into (5.90) instead of K M,10 and using (5.51), we directly have that (5.91) K M,10 = P1,0 is a solution of the Eq. (5.90). Now, based on the Eqs. (5.88), (5.89) and (5.91), it is shown (quite similarly to Lemma 5.3) the existence of a positive number ε¯u , (ε¯u ≤ ε ∗1 ) such that, for all ε ∈ (0, ε¯u ], the set of the Eqs. (5.82)–(5.84) has the solution K M,1 (ε), K M,2 (ε), K M,3 (ε) . Consequently, the equivalent Eq. (5.19), (5.79) has the solution (5.81) for all ε ∈ (0, ε¯u ]. Moreover, the solution of (5.82)–(5.84) satisfies the inequalities    K M,i (ε) − K M,i0 (ε) ≤ aε,

ε ∈ (0, ε¯u ],

(5.92)

where K M,i0 (ε), (i = 1, 2, 3) are given by (5.91), (5.89) and (5.88); a > 0 is some constant independent of ε. Also, quite similarly to the proof of Corollary 5.1, it is shown the existence of a positive number εu ≤ ε¯u such that, for any ε ∈ (0, εu ] and for the solution (5.81) of the Eq. (5.19), (5.79), the trivial solution of the system (5.21), (5.79) is asymptotically stable. Hence, for all ε ∈ (0, εu ], the guaranteed result of the minimizer’s control u ∗ε,0 (z) has the form (5.80). Let us represent the vector z 0 in the block form as: z 0 = col(x0 , y0 ), where x0 ∈ E n−r+q , y0 ∈ E r−q . Substitution of this representation and (5.81) into (5.80) yields for all ε ∈ (0, εu ]:    Ju u ∗ε,0 (z); z 0 = x0T K M,1 (ε)x0 + ε 2x0T K M,2 (ε)y0 + y0T K M,3 (ε)y0 .

(5.93)

Subtracting (5.70) from (5.93), estimating the resulting expression and taking into account (5.91), we obtain the following inequality for all ε ∈ (0, εu ]:   ∗   Ju u (z); z 0 − J0 (x0 ) ≤ K M,1 (ε) − K M,10 x0 2 ε,0  +ε 2K M,2 (0)x0 y0  + K M,3 (ε)y0 2 .

(5.94)

Due to the Eqs. (5.88)–(5.89) and the inequality (5.92) for i = 2, 3, the matrices K M,2 (ε) and K M,3 (ε) are bounded with respect to ε ∈ (0, εu ]. Now, the inequality in (5.92) for i = 1 and the inequality (5.94), along with the above mentioned boundedness of the matrices K M,2 (ε) and K M,3 (ε), directly yield the statement of the lemma.

5.8.3 Proof of Lemma 5.7  To calculate Jv vr∗ (x); z 0 , let us note that, due to Definition 3.7, this value is an optimal value of the functional in the optimal control problem, obtained from the

136

5 Singular Infinite-Horizon Zero-Sum Differential Game

SIHG by substitution of v(t) = vr∗ (x) into the equations of dynamics (5.9) and the functional (5.10). This problem has the form dz(t)  = Az(t) + Bu(t), t ≥ 0, z(0) = z 0 , dt

(5.95)

 +∞   T   J(u) = J u, vr∗ (x) = z (t) Dz(t) 0  +u T (t)G u u(t) dt → inf  ,

(5.96)

u(t)=u(z)∈Fu vr∗ (x)

where  = A(t) = D Thus,





1 A 3 A

1 0 D 2 0 D

2 A 4 A 







= 

=

 Jv vr∗ (x); z 0 =



A1 + Sv1 P1,0 A2 A3 + SvT2 P1,0 A4



D1 − P1,0 Sv1 P10 0 0 D2

,

(5.97)

 .

(5.98)

J(u)

inf 

(5.99)

u(t)=u(z)∈Fu vr∗ (x)

in the optimal control problem (5.95)–(5.96). The latter problem is singular (see e.g. [5] and references therein). The value in the right-hand side of (5.99) can be calculated in the way, similar to that proposed in [5] for solution of an infinite horizon singular linear-quadratic optimal control problem with a known disturbance. The further proof consists of three stages.

5.8.3.1

Stage 1: Regularization of (5.95)–(5.96)

Let us replace the singular optimal control problem (5.95)–(5.96) with a regular one, consisting of the equation of dynamics (5.95) and the functional Jε (u) =   + + u T (t) G u + Λ(ε) u(t) dt →



+∞



 z T (t) Dz(t)

0

inf 

,

(5.100)

u(t)=u(z)∈Fu vr∗ (x,t)

where ε > 0 is a small parameter; Λ(ε) is given by (5.30). For a given ε > 0, consider the following Riccati algebraic equation for the n × n(ε): matrix K

5.8 Saddle-Point Sequence of the SIHG

137

(ε) + D  = 0, (ε) A + A T K (ε) − K (ε)Su (ε) K K

(5.101)

where Su (ε) is given in (5.34). (ε) ∈ Sn Due to the results of [9], if for some ε > 0 the Eq. (5.101) has a solution K such that the trivial solution of the system dz(t)   (ε) z(t), = A − Su (ε) K dt

t ≥0

(5.102)

is asymptotically stable, then the optimal control problem (5.95), (5.100) is solvable. Its solution (the optimal control) has the form  −1  (ε)z, u = u ∗ε (z) = − G u + Λ(ε) B T K while the optimal value of its functional is (ε)z 0 . Jε∗ (z 0 ) = z 0T K 5.8.3.2

(5.103)

Stage 2: Asymptotic Analysis of the Problem (5.95), (5.100)

We begin with an asymptotic solution of the Eq. (5.101). Similarly to (5.42) and (ε) of the Eq. (5.101) in the block form (5.81), we seek the solution K (ε) = K



 2 (ε) 1 (ε) ε K K 3 (ε) , 2T (ε) ε K εK

(5.104)

1 (ε), K 2 (ε) and K 3 (ε) have the dimensions (n − r + q) × (n − where the blocks K 1 (ε), 1T (ε) = K r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively; K T   K 3 (ε) = K 3 (ε). Using the block representations (5.40), (5.97), (5.98) and (5.104), the Eq. (5.101) can be rewritten in the following equivalent form: 1 + ε K 2 (ε) A 3 + A 1T K 3T K 1 (ε) A 1 (ε) + ε A 2T (ε) K 2 (ε)SuT K 1 (ε)Su 2 K 1 (ε) − ε K 1 (ε) − ε K 2T (ε) 1 (ε)Su 1 K −K 2 2T (ε) + D 1 (t) = 0, 2 (ε)Su 3 (ε) K −K

2 + ε K 2 (ε) A 4 + ε A 1T K 3T K 1 (ε)Su 1 K 2 (ε) + ε A 3 (ε) − ε K 2 (ε) 1 (ε) A K 2 T  1 (ε)Su 2 K 2 (ε)Su 3 (ε) K 3 (ε) = 0, 3 (ε) − K −ε K 2 (ε)Su 2 K 2 (ε) − ε K

(5.105)

(5.106)

138

5 Singular Infinite-Horizon Zero-Sum Differential Game

2T (ε) A 2 + ε K 3 (ε) A 4 + ε A 2T K 4T K 2 (ε) + ε A 3 (ε) εK 2 T 2 T  2 T  3 (ε) −ε K 2 (ε)Su 1 K 2 (ε) − ε K 3 (ε)Su K 2 (ε) − ε K 2 (ε)Su 2 K 2

3 (ε)Su 3 (ε) K 3 (ε) + D 2 = 0. −K

(5.107)

5.6.2 and 5.8.2, we seek the zero-order asymptotic solution 

Similarly to Sects. 2,0 , K 3,0 of the set (5.105)–(5.107). Like in the above mentioned subsec1,0 , K K tions, setting formally ε = 0 in (5.105)–(5.107), we obtain the following set of equai,0 , (i = 1, 2, 3): tions for K T 1T K 1,0 Su 1 K 2,0 K 1 = 0, 1 + A 1,0 − K 1,0 − K 20 1,0 A +D K

2,0 K 1,0 A 2 − K 3,0 = 0, K 

3,0 K

2

2 = 0. −D

(5.108)

(5.109)

(5.110)

2,0 , 3,0 and K Solving the algebraic equations (5.110) and (5.109) with respect to K 2 = D2 (see the Eqs. (5.97), (5.98)), we 2 = A2 , D and taking into account that A obtain   2,0 = K 1,0 A2 D2 −1/2 . 3,0 = D2 1/2 , K (5.111) K 1 and K 2,0 (see the Eqs. (5.97), (5.98) 1 , D Now, substitution of the expressions for A and (5.111)) into (5.108) yields after a routine algebra the following equation for 1,0 : K 1,0 1,0 (A1 + Sv1 P1,0 ) + (A1 + Sv1 P1,0 )T K K 1,0 (S1,0 + Sv1 ) K 1,0 + (D1 − P1,0 Sv1 P10 ) = 0, −K

(5.112)

where S1,0 is given by (5.52). Based on the Eq. (5.51), it verifies immediately that the Eq. (5.112) has the following solution: 1,0 = P1,0 . (5.113) K Now, based on the Eqs. (5.111) and (5.113), it is shown (similarly to Lemma ε1 ], the set of the 5.3) the existence of a positive number  ε1 such that, for all ε ∈ (0,  Eqs. (5.105)–(5.107) (and, consequently, the Eqs. (5.101), (5.104)) has the solution 2 (ε), K 3 (ε)} satisfying the inequalities 1 (ε), K {K   K i,0  ≤ aε, i = 1, 2, 3, i (ε) − K

(5.114)

where a > 0 is some constant independent of ε. Moreover, similarly to the proof of Corollary 5.1, one can show the existence ε1 such that, for any ε ∈ (0,  ε2 ] and for the above menof a positive number  ε2 ≤ 

5.8 Saddle-Point Sequence of the SIHG

139

tioned solution (5.104) of the Eq. (5.101), the trivial solution of the system (5.102) is asymptotically stable. Further, using the Eqs. (5.103), (5.104), (5.111), (5.113) and the inequalities (5.114), we prove (quite similarly to the proof of Lemma 5.6) the validity of the inequality   ∗  J (z 0 ) − J0 (x0 ) ≤ aε, ε ∈ (0,  (5.115) ε2 ], ε where J0 (x0 ) is defined by (5.70); a > 0 is some constant independent of ε.

5.8.3.3

Stage 3: Deriving the Expression for the Optimal Value of the Functional in the Problem (5.95)–(5.96)

First of all, let us note that the inequality (5.115) can be rewritten as: J0 (x0 ) − aε ≤ Jε∗ (z 0 ) ≤ J0 (x0 ) + aε, ε ∈ (0,  ε2 ].

(5.116)

Based on this inequality, we obtain the following chain of inequalities and equality: inf 

 ∗  ∗ u ε (z) u ε (z) ≤ Jε  J(u) ≤ J 

u(t)=u(z)∈Fu vr∗ (x)

= Jε∗ (z 0 ) ≤ J0 (x0 ) + aε, yielding

ε ∈ (0,  ε2 ],

(5.117)

inf 

J(u) ≤ J0 (x0 ).

(5.118)

inf 

J(u) = J0 (x0 ).

(5.119)

u(t)=u(z)∈Fu vr∗ (x)

Let us show that

u(t)=u(z)∈Fu vr∗ (x)

To do this, let us assume the opposite, i.e., inf 

J(u) < J0 (x0 ).

(5.120)

u(t)=u(z)∈Fu vr∗ (x)

 The latter means the existence of  u (z) ∈ Fu vr∗ (x) , such that inf 

 u (z) < J0 (x0 ). J(u) < J 

(5.121)

u(t)=u(z)∈Fu vr∗ (x)

Since  u ∗ε (z) is an optimal control in the problem (5.95), (5.100), and (5.116) holds, we obtain for any ε ∈ (0,  ε2 ]:

140

5 Singular Infinite-Horizon Zero-Sum Differential Game

 ∗   J0 (x0 ) − aε ≤ Jε∗ (z 0 ) = Jε  u ε (z) ≤ Jε  u (z) = J  u (z) + bε2 ,

(5.122)

where  b=

+∞

  z(t) Λ(1) u z(t) dt;  uT 

0

 z(t), t ∈ [0, +∞) is the solution of (5.95) generated by the control u =  u (z); Λ(ε) is given by (5.30). The chain  of the inequalities and the equalities (5.122) yields the inequality u (z, t) . This inequality contradicts the right-hand inequality in (5.121). J0 (x0 ) ≤ J  Therefore, the inequality (5.119) is wrong, meaning the validity of the equality (5.119). The latter, along with (5.99), directly yields the statement of the lemma.

5.9 Examples 5.9.1 Example 1 Here, we consider the particular case of the SIHG with the following data: n = 2, 2, s = 2,  1, z 0 =  col(2,√1),  q=  r = 1 0 −3 4 1 − 3 , C= , B= A= , 2 1 1 2 5    6   7 0 1 0 1 0 D= , Gu = , Gv = . 0 4 0 0 0 1

(5.123)

Based on these data, and using the expression for Su 1 , the expression for Sv , the block form of this matrix and the expression for S1,0 (see the Eqs. (5.41), (5.34), (5.43) and (5.52)), we obtain (5.124) Su 1 = 1, Sv1 = 4, S1,0 = 1. Using (5.123) and (5.124), the Eq. (5.51) becomes the scalar equation 2  − 6P1,0 − P1,0 + 7 = 0. This equation has two solutions (1) P1,0 = 1,

(2) P1,0 = −7.

(1) , the Eqs. (5.53) and (5.54) are For P1,0 = P1,0

(5.125)

5.9 Examples

141

d x(t) = −4x(t), dt

t ≥ 0,

(5.126)

d x(t) = −8x(t), dt

t ≥ 0,

(5.127)

(2) , the Eqs. (5.53) and (5.54) are while for P1,0 = P1,0

d x(t) = 4x(t), dt

t ≥ 0,

(5.128)

d x(t) = 32x(t), dt

t ≥ 0.

(5.129)

The trivial solution of the Eqs. (5.126) and (5.127) is asymptotically stable, while (1) such a solution of the Eqs. (5.128) and (5.129) is not. Therefore, P1,0 = P1,0 =1 (2) satisfies the assumption A5.IV, while P1,0 = P1,0 = −7 does not. (1) Using the Eqs. (5.50), (5.123) and the equality P1,0 = P1,0 = 1, we calculate the gains P3,0 and P2,0 , appearing in the minimizer’s control (5.76), as: P3,0 = 2,

P2,0 = 2.

(5.130)

Now, by virtue of Theorem 5.1, the Eqs. (5.71), (5.74), (5.75), (5.76), (5.130) (1) = 1, we directly obtain the entries of the saddle-point and the equality P1,0 = P1,0 sequence in the SIHG with the data (5.123): u ∗εk ,0 (z) = −

 2 (x εk

 x , + y)

vr∗ (x) =



x √ − 3x

 ,

where z = col(x, y), x and y are scalar variables. (1) = 1, Finally, by virtue of Theorem 5.1, the Eq. (5.70) and the equality P1,0 = P1,0 the value of the SIHG with the data (5.123) is J0 (x0 ) = J0 (2) = 4.

5.9.2 Example 2: Solution of Infinite-Horizon Vibration Isolation Game In this example, we solve the game (2.128), (2.131), which is the model of the rejection (the isolation) of the body’s vibration caused by unknown disturbance (for details, see Sect. 2.3.4). For the sake of books reading convenience, we rewrite this game here as:

142

5 Singular Infinite-Horizon Zero-Sum Differential Game

d x1 (t) 1 = x2 (t), dt m

t ≥ 0,

x1 (0) = x0 ,

(5.131)

d x2 (t)  = v(t) + u(t), t ≥ 0, x2 (0) = mx0 , dt







+∞

J u(·), v(·) = 0

(5.132)

  β2 2 2 2 β1 x1 (t) + 2 x2 (t) − βv v (t) dt, m

(5.133)

 where col x1 (t), x2 (t) is the state vector; v(t) and u(t) are the scalar forces of the disturbance and the actuator, respectively; m > 0 is the mass of the body; β1 > 0, β2 > 0 and βv > 0 are given constant weight coefficients. The objective of the actuator is to minimize the functional (5.133) along trajectories of the system of the differential equations (5.131)–(5.132) by a proper choice of its force u(·). This minimization should be carried out by the actuator subject to its assumption on the worse case behaviour of the disturbance, i.e., that the objective of the disturbance is to maximize the functional (5.133) by a proper choice of the force v(·). The game (5.131)–(5.132), (5.133) can be considered as a particular case of the SIHG (5.9)–(5.10) with the following data: n = 2,



r = 1,

s = 1, q = 0, z 0 = col(x0 , mx0 ),       0 0 0 m1 , B= , C= , A= 1 1 0 0   β2 D = diag β1 , 2 , G u = 0, G v = βv . m

(5.134)

Using these data, as well as the expression for Sv (t), the block form of this matrix and the expression for S1,0 (t) (see the Eqs. (4.15), (4.22) and Remark 5.10), we obtain Sv1 = 0,

S1,0 =

1 . β2

For these values and the data (5.134), the Eq. (5.51) becomes the following scalar equation: 2 1 P1,0 − β1 = 0. β2 This equation has two solutions (1) P1,0 =



β1 β2 ,

 (2) P1,0 = − β1 β2 .

(5.135)

5.9 Examples

143

(1) For P1,0 = P1,0 , both Eqs., (5.53) and (5.54), have the form

 β1 d x(t) x(t), =− dt β2

t ≥ 0,

(5.136)

(2) while for P1,0 = P1,0 , both Eqs., (5.53) and (5.54), have the form

 d x(t) = dt

β1 x(t), β2

t ≥ 0.

(5.137)

The trivial solution of the Eq. (5.136) is asymptotically √ stable, while such a solu(1) = β1 β2 satisfies the assumption of the Eq. (5.137) is not. Therefore, P1,0 = P1,0 √ (2) = − β1 β2 does not. tion A5.IV, while P1,0 = P1,0 √ (1) Using the Eqs. (5.50), (5.134) and the equality P1,0 = P1,0 = β1 β2 , we calculate the gains P3,0 and P2,0 , appearing in the minimizer’s control (5.76), as follows: P3,0 =



√ β2 ,

P2,0 =

β1 . m

√ (1) = β1 β2 , as well as the Eq. (5.74), Remark Using these values, the value P1,0 = P1,0 5.10 and Theorem 5.1, we directly obtain the entries of the saddle-point sequence in the singular infinite-horizon zero-sum game (5.131)–(5.132), (5.133) u ∗εk ,0 (z) = −

1 εk

√

  β1 x + β2 y , m

vr∗ (x) = 0,

where z = col(x, y), x = x1 and y = x2 . Moreover, by virtue of Remark 5.10, Theorem 5.1 and the Eq. (5.70), the value of the game (5.131)–(5.132), (5.133) is  (1) 2 x0 = β1 β2 x02 . J o (x0 ) = P1,0 x02 = P1,0 Let us show that, for any given exponentially decaying disturbance v(t) as t → +∞ and for all sufficiently small εk > 0, the actuator’s state-feedback control u ∗εk ,0 (z) exponentially stabilizes the system (5.131)–(5.132). The latter means that the solution of this system with u(t) = u ∗εk ,0 (z) exponentially decays for t → +∞. Thus, we assume that a given disturbance v(t), t ∈ [0, +∞) satisfies the inequality |v(t)| ≤ c exp(−κt), where c > 0 and κ > 0 are some constants.

t ≥ 0,

(5.138)

144

5 Singular Infinite-Horizon Zero-Sum Differential Game

Let us introduce into the consideration the following matrix and vector: 

Γ (εk ) =



0√ 1/m √ − β1 /(εk m) − β2 /εk

 ,





f v (t) =

 0 . v(t)

(5.139)

Substituting u(t) = u ∗εk ,0 (z) into the system (5.131)–(5.132), and using the notation z = col(x1 , x2 ) and the notations (5.139), we obtain dz(t) = Γ (εk )z(t) + f v (t), dt

t ≥ 0,

z(0) = z 0 ,

(5.140)

where z 0 is given in (5.134). Let us study the behaviour of the eigenvalues of the matrix Γ (εk ). These eigenvalues are roots of the following quadratic equation with respect to λ: λ2 +

√ √ β2 β1 λ+ = 0. εk εk m 2

Solving this equation, we obtain the eigenvalues of the matrix Γ (εk )  √ 1/2  β2 4εk β1 λ1 (εk ) = − 1− 1− , 2εk β2 m 2 √ 1/2 √   β2 4εk β1 λ2 (εk ) = − 1+ 1− . 2εk β2 m 2 √

In what follows of this example, we consider the values of εk satisfying the inequality β2 m 2 0 < εk < √ . 4 β1

(5.141)

For these values of εk , the eigenvalues λ1 (εk ) and λ2 (εk ) are estimated as:  1 λ1 (εk ) < − 2 m

β1  = −ϑ, β2

√ λ2 (εk ) < −

β2 < −ϑ. 2εk

(5.142)

Let, for a given εk , the matrix-valued function Φ(t, εk ), t ∈ [0, +∞) be the solution of the following initial-value problem dΦ(t, εk ) = Γ (εk )Φ(t, εk ), dt

t ≥ 0,

Φ(0, εk ) = I2 .

Due to the estimates (5.142) and the results of [2] (see Theorem 2.3), we have the inequality

5.10 Concluding Remarks and Literature Review

145

Φ(t, εk ) ≤ a exp(−ϑt),

t ≥ 0,

(5.143)

where a > 0 is some constant independent of εk . Solving the initial-value problem (5.140), we obtain 

t

z(t, εk ) = Φ(t, εk )z 0 +

Φ(t − σ, εk ) f v (σ)dσ,

t ≥ 0.

(5.144)

0

Let us estimate the solution z(t, εk ) to the problem (5.140). To do this, we distinguish two cases: (i) ϑ = κ; (ii) ϑ = κ. Case (i). In this case, using the inequalities (5.138), (5.143) and the Eq. (5.144) yields 

   Φ(t − σ, εk ) f v (σ)dσ 0  t  ≤ az 0  exp(−ϑt) + ac exp(−ϑt) exp (ϑ − κ)σ dσ

  z(t, εk ) ≤ Φ(t, εk )z 0  +

t

0

 ac  exp(−κt) − exp(−ϑt), t ≥ 0. = az 0  exp(−ϑt) + |ϑ − κ|

(5.145)

Case (ii). In this case, using the inequalities (5.138), (5.143) and the Eq. (5.144), we obtain  t      Φ(t − σ, εk ) f v (σ)dσ   z(t, εk ) ≤ Φ(t, εk ) z 0  + 0  t  ≤ az 0  exp(−ϑt) + ac exp(−ϑt) exp (ϑ/2)σ dσ 0

  2ac  exp − (ϑ/2)t − exp(−ϑt) , t ≥ 0. (5.146) = az 0  exp(−ϑt) + ϑ Due to (5.145)–(5.146), the solution z(t, εk ) of the problem (5.140) exponentially decays for t → +∞. Thus, for any given disturbance v(t), satisfying the inequality (5.138), and for all εk > 0, satisfying the inequality (5.141), the actuator’s statefeedback control u ∗εk ,0 (z) exponentially stabilizes the system (5.140) (and, therefore, the system (5.131)–(5.132)).

5.10 Concluding Remarks and Literature Review In this chapter, the singular infinite-horizon zero-sum linear-quadratic differential game was considered. Like in the previous chapter (see Chap. 4), the singularity of this game is due to the singularity of the weight matrix in the control cost of the minimizing player (the minimizer) in the game’s functional. In the case where this

146

5 Singular Infinite-Horizon Zero-Sum Differential Game

weight matrix is non-zero, only a part of the coordinates of the minimizer’s control is singular, while the others are regular. For the considered game, the novel definitions of the saddle-point equilibrium (the saddle-point equilibrium sequence) and the game value were proposed. Based on proper assumptions, the linear system of the game’s dynamics was transformed equivalently to a new system consisting of three modes. The first mode is controlled directly only by the maximizing player (the maximizer), the second mode is controlled directly by the maximizer and the regular coordinates of the minimizer’s control, while the third mode is controlled directly by the maximizer and the entire control of the minimizer. Lemma 5.1 asserts that the game, obtained by this transformation, is equivalent to the initially formulated one. This lemma has not been published before. The result close to Lemma 5.1 was obtained in [7]. In this work the equivalence of the initially formulated game and the corresponding transformed game with respect to the guaranteed result of a minimizer’s state-feedback control was shown. The new (transformed) game also is singular, while it is simpler than the initially formulated one. Therefore, in the sequel of this chapter, the new game was considered as an original game. This singular game was solved by the regularization method, i.e., by its approximate replacing with an auxiliary regular game. The latter has the same equation of dynamics and a similar functional augmented by an infinite-horizon integral of the squares of the minimizer’s singular control coordinates with a small positive weight. The regularization method was used in the literature for solution of singular infinite-horizon optimal control problems (see e.g. [3, 5, 8] and references therein). However, to the best of our knowledge, this method was applied only in few works for analysis and solution of singular infinite-horizon zero-sum differential games (see [4, 6, 7]). In all these papers, linear-quadratic games are studied. In the first two papers the singularity of the games is due to the absence of the control cost of the minimizer in the functional. In the third paper the more general case was studied where the weight matrix of the minimisers control cost, being singular, is not zero. The auxiliary (regularized) game, introduced in this chapter, is an infinite-horizon zero-sum differential game with a partial cheap control of the minimizer. Infinitehorizon zero-sum differential games with a complete/partial minimizer’s cheap control were studied in a number of works. Thus in [10], a linear-quadratic game with the complete cheap control of the minimizer was considered. The limit behaviour of the game’s value was studied in the case where the small positive weight of the minimizer’s control cost tends to zero. In [4, 6, 7], minimizer’s cheap control games were obtained and analyzed due to the regularization method of the solution of the corresponding singular games. In [4, 6] the case of the complete minimizer’s cheap control was treated, while in [7] the case of the partial minimizer’s cheap control was studied. The asymptotic analysis of the regularized game was carried out. Based on this analysis, the saddle-point equilibrium sequence and the value of the original singular game were obtained.

References

147

It should be noted that in the present chapter we study the singular game subject to another (than in [4, 6]) definitions of the saddle-point solution and the game value. Due to this circumstance, the proof of the main theorem (Theorem 5.1), which yields the solution of the singular game, is much simpler than the proofs of the corresponding theorems in these papers.

References 1. Gajic, Z., Qureshi, M.T.J.: Lyapunov Matrix Equation in System Stability and Control. Dover Publications, Mineola, NY, USA (2008) 2. Glizer, V.Y.: Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution. J. Math. Anal. Appl. 278, 409–433 (2003) 3. Glizer, V.Y.: Singular solution of an infinite horizon linear-quadratic optimal control problem with state delays. In: Wolansky, G., Zaslavski, A.J. (eds.) Variational and Optimal Control Problems on Unbounded Domains, Contemporary Mathematics Series, vol. 619, pp. 59–98. American Mathematical Society, Providence, RI, USA (2014) 4. Glizer, V.Y.: Saddle-point equilibrium sequence in one class of singular infinite horizon zerosum linear-quadratic differential games with state delays. Optimization 68, 349–384 (2019) 5. Glizer, V.Y., Kelis, O.: Singular infinite horizon quadratic control of linear systems with known disturbances: a regularization approach, p. 36 (2016). arXiv:160301839v1 [math.OC]. Last accessed 06 Mar 2016 6. Glizer, V.Y., Kelis, O.: Singular infinite horizon zero-sum linear-quadratic differential game: saddle-point equilibrium sequence. Numer. Algebra Control Optim. 7, 1–20 (2017) 7. Glizer, V.Y., Kelis, O.: Upper value of a singular infinite horizon zero-sum linear-quadratic differential game. Pure Appl. Funct. Anal. 2, 511–534 (2017) 8. Glizer, V.Y., Kelis, O.: Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances. Numer. Algebra Control Optim. 8, 211–235 (2018) 9. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat Mexicana 5, 102–119 (1960) 10. Petersen, I.R.: Linear-quadratic differential games with cheap control. Syst. Control Lett. 8, 181–188 (1986)

Chapter 6

Singular Finite-Horizon H∞ Problem

6.1 Introduction In this chapter, we consider a system consisting of an uncertain controlled linear time-dependent differential equation and a linear time-dependent output algebraic equation. For this system, a finite-horizon H∞ problem is studied in the case where the rank of the coefficients’ matrix for the control in the output equation is smaller than the Euclidean dimension of this control. In this case, the solvability conditions, presented in Sect. 3.2.3, are not applicable to the solution of the considered H∞ problem meaning its singularity. To solve this H∞ problem, a regularization method is proposed. Namely, the original problem is replaced approximately with a regular finite-horizon H∞ problem depending on a small positive parameter. Thus, the firstorder solvability conditions are applicable to this new problem. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix differential equation, arising in these conditions, yields a controller solving the original singular H∞ problem. Properties of this controller are studied. The following main notations are applied in the chapter. 1. E n is the n-dimensional real Euclidean space. 2.  ·  denotes the Euclidean norm either of a vector or of a matrix. 3. The superscript “T ” denotes the transposition of a matrix A, (A T ) or of a vector x, (x T ). 4. L 2 [a, b; E n ] is the linear space of n-dimensional vector-valued real functions, square-integrable in the finite interval [a, b], and  ·  L 2 [a,b] denotes the norm in this space. 5. On 1 ×n 2 is used for the zero matrix of the dimension n 1 × n 2 , excepting the cases where the dimension of the zero matrix is obvious. In such cases, the notation 0 is used for the zero matrix. 6. In is the n-dimensional identity matrix. 7. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block-vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_6

149

150

6 Singular Finite-Horizon H∞ Problem

8. diag(a1 , a2 , . . . , an ), where ai , (i = 1, 2, . . . , n) are real numbers, is the diagonal matrix with these numbers on the main diagonal. 9. Sn is the set of all symmetric matrices of the dimension n × n. 10. Sn+ is the set of all symmetric positive semi-definite matrices of the dimension n × n. 11. Sn++ is the set of all symmetric positive definite matrices of the dimension n × n.

6.2 Initial Problem Formulation Consider the system dζ(t) = A(t)ζ(t) + B(t)u(t) + C(t)w(t), t ∈ [0, t f ], ζ(0) = 0, dt

(6.1)

where t f is a given final time moment; ζ(t) ∈ E n is a state vector; u(t) ∈ E r , (r ≤ n), is a control; w(t) ∈ E s is a disturbance and A(t), B(t) and C(t), t ∈ [0, t f ], are given matrix-valued functions of the corresponding dimensions. Along with the system (6.1), describing the dynamics of the H∞ problem, we consider the output equation of this problem   W(t) = col N (t)ζ(t), M(t)u(t) ,

t ∈ [0, t f ],

(6.2)

where W(t) ∈ E p is an output; N (t) and M(t), t ∈ [0, t f ], are given matrix-valued functions; the dimensions of the matrices N (t) and M(t) are p1 × n and p2 × r , respectively, and (6.3) p1 + p2 = p, r ≤ p2 . Assuming that w(t) ∈ L 2 [0, t f ; E s ], we consider the cost functional of the H∞ problem  2  2 J (u, w) = ζ T (t f )Fζ(t f ) + W(t) L 2 [0,t f ] − γ 2 w(t) L 2 [0,t f ] ,

(6.4)

where a given matrix F ∈ Sn+ ; γ > 0 is a given constant called the performance level. Let us introduce into the consideration the matrices D(t) = N T (t)N (t) and G(t) = M T (t)M(t), t ∈ [0, t f ]. Using these matrices, we can rewrite the functional (6.4) in the equivalent form as 

tf

+

J (u, w) = ζ T (t f )Fζ(t f )  T  ζ (t)D(t)ζ(t) + u T (t)G(t)u(t) − γ 2 w T (t)w(t) dt.

(6.5)

0

In what follows in this chapter, we treat the case where the matrix M(t) has the form

6.3

Transformation of the Initially Formulated H∞ Problem

  ¯ ¯ = q, t ∈ [0, t f ]. M(t) = M(t), O p2 ×(r −q) , 0 < q < r, rank M(t) Thus,

G(t) =

¯ G(t) 0 ¯ ¯ t ∈ [0, t f ], , G(t) = M¯ T (t) M(t), 0 0

151

(6.6)

(6.7)

¯ ∈ S++ for all t ∈ [0, t f ]. and due to (6.3) and (6.6), G(t) q

Remark 6.1 Due to (6.7), the matrix G(t) is not invertible, meaning that the results of Sect. 3.2.3 are not applicable to the H∞ problem with the dynamics (6.1) and the cost functional (6.5). Thus, this problem is singular which requires another approach to its solution. H,t f , we denote the set of all functions u =  u (ζ, t) : E n × Definition 6.1 By U u (ζ, t) is measurable w.r.t. [0, t f ] → E r satisfying the following conditions: (i)  t ∈ [0, t f ] for any fixed ζ ∈ E n and satisfies the local Lipschitz condition w.r.t. u (ζ, t) ζ ∈ E n uniformly in t ∈ [0, t f ]; (ii) the initial-value problem (6.1) for u(t) =  continuous solution ζuw (t) and any w(t) ∈ L 2 [0, t f ; E s ] has the unique absolutely  u ζuw (t), t ∈ L 2 [0, t f ; E r ]. Such a defined set in the entire interval t ∈ [0, t f ]; (iii)  H,t f is called the set of all admissible controllers for the H∞ problem (6.1), (6.5). U H,t f is called the solution of the H∞ problem Definition 6.2 Controller  u ∗ (ζ, t) ∈ U (6.1), (6.5), if it ensures the validity of the inequality J ( u ∗ , w) ≤ 0

(6.8)

along trajectories of (6.1) with u(t) =  u ∗ (ζ, t) and any w(t) ∈ L 2 [0, t f ; E s ].

6.3 Transformation of the Initially Formulated H∞ Problem   We partition the matrix B(t) into blocks as B(t) = B1 (t), B2 (t) , t ∈ [0, t f ], where the matrices B1 (t) and B2 (t) are of dimensions n × q and n × (r − q). We assume that A6.I. The matrix B(t) has full  column rank r for all t ∈ [0, t f ].  A6.II. det B2T (t)D(t)B2 (t) = 0, t ∈ [0, t f ]. A6.III. FB2 (t f ) = 0. ¯ A6.IV. The matrix-valued functions A(t), C(t) and G(t) are continuously differentiable in the interval [0, t f ]. A6.V. The matrix-valued functions B(t) and D(t) are twice continuously differentiable in the interval [0, t f ]. Remark 6.2 Note that the assumptions A6.I–A6.V allow to construct and carry out the state transformation (3.72) in the system (6.1) and the functional (6.5) as it is

152

6 Singular Finite-Horizon H∞ Problem

presented below. This transformation converts the system (6.1) and the functional (6.5) to a much simpler form for the analysis and solution of the corresponding singular H∞ problem. Let Bc (t) be a complement matrix-valued function to B(t) in the interval [0, t f ]   c (t) = Bc (t), B1 (t) is a complement matrix-valued (see Definition 3.13). Then, B function to B2 (t) in the interval [0, t f ]. Due to the assumptions A6.I and A6.V, as well c (t) to be twice continuously differas Remark 3.5 and Lemma 3.1, we can choose B c (t) and entiable in the interval [0, t f ]. Using such a chosen matrix-valued function B the above introduced matrix-valued function B2 (t), we construct the matrix-valued functions H(t) and L(t), t ∈ [0, t f ] (see Eq. (3.71)). Then, using these matrices, we make the state transformation (3.72) in the system (6.1) and the functional (6.5). As a direct consequence of Lemma 3.2 and Theorem 3.5, we have the following assertion. Proposition 6.1 Let the assumptions A6.I–A6.V be valid. Then, the transformation (3.72) converts the system (6.1) and the functional (6.5) to the following system and functional: dz(t) = A(t)z(t) + B(t)u(t) + C(t)v(t), dt



tf

+

t ∈ [0, t f ], z(0) = 0,

J (u, v) = z T (t f )F z(t f )   T z (t)D(t)z(t) + u T (t)G(t)u(t) − γ 2 w T (t)w(t) dt,

(6.9)

(6.10)

0

where the matrix-valued coefficients A(t), B(t), C(t), F and D(t) are given by (3.76)–(3.80). The coefficients A(t), B(t), C(t), D(t) and G(t) are continuously n−r +q , D2 (t) ∈ differentiable in the interval [0, t f ]. Moreover, in (3.80), D1 (t) ∈ S+ r −q n−r+q S++ for all t ∈ [0, t f ], F1 ∈ S+ . Remark 6.3 Similarly to the H∞ problem (6.1), (6.5), the new H∞ problem (6.9)– (6.10) is singular. We choose the set U H,t f (see Definition 3.9) to be the set of all admissible controllers u(z, t), (z, t) ∈ E n × [0, t f ] for the H∞ problem (6.9)–(6.10). Moreover, we keep Definition 3.10 (see Sect. 3.2.3) for the solution u ∗ (z, t) to this problem.  u ∗ (ζ, t) solves Lemma 6.1 Let the assumptions A6.I–A6.V be valid.  If the controller  ∗ u R(t)z, t solves the H∞ problem the H∞ problem (6.1), (6.5), then the controller  (6.9)–(6.10) (the n × n-matrix R(t), t ∈ [0, t f ] is given by (3.71)–(3.72)). Vice versa, ∗ if the controller  −1  u (z, t) solves the H∞ problem (6.9)–(6.10), then the controller ∗ u R (t)ζ, t solves the H∞ problem (6.1), (6.5). Proof Let us start with the first statement of the lemma. Since the controller  u ∗ (ζ, t) ∗  u (ζ, t) ∈ U H,t f , and the inequality (6.8) is solves the H∞ problem (6.1), (6.5), then 

6.4

Regularization of the Singular Finite-Horizon H∞ Problem

153

satisfied along trajectories of (6.1) with u(t) =  u ∗ (ζ, t) and any w(t) ∈ L 2 [0, t f ; E s ]. Now, let us make the state transformation (3.72) in (6.1) and (6.5). Due to this transformation and Proposition 6.1, the system (6.1) and the functional (6.5) become the system (6.9) and the functional (6.10), respectively. Remember that the transH,t f , ]. Therefore, since  u ∗ (ζ, t) ∈ U formation  (3.72) is invertible for all t ∈ [0, t f ∗ then  u R(t)z, t ∈ U H,t f . Moreover, the inequality (6.8) becomes the inequality       J  u ∗ R(t)z, t , w(t) ≤ 0 along trajectories of (6.9) with u(t) =  u ∗ R(t)z, t and   u ∗ R(t)z, t any w(t) ∈ L 2 [0, t f ; E s ]. This inequality means that the controller  solves the H∞ problem (6.9)–(6.10). This completes the proof of the first statement. The second statement is proven similarly. 

Remark 6.4 Due to Lemma 6.1, the initially formulated H∞ problem (6.1), (6.5) is equivalent to the new H∞ problem (6.9)–(6.10). On the other hand, due to Proposition 6.1, the latter problem is simpler than the former one. Therefore, in the sequel of this chapter, we deal with the H∞ problem (6.9)–(6.10). We consider this problem as an original one and call it the Singular Finite-Horizon H∞ Problem (SFHP).

6.4 Regularization of the Singular Finite-Horizon H∞ Problem 6.4.1 Partial Cheap Control Finite-Horizon H∞ Problem We study the SFHP by its regularization. Namely, we replace the SFHP with a regular H∞ problem. This regular H∞ problem has the same dynamics (6.9) as the SFHP has. However, the functional in this problem has the form 

tf

+

Jε (u, w) = z T (t f )F z(t f )  T    z (t)D(t)z(t) + u T (t) G(t) + (ε) u(t) − γ 2 w T (t)w(t) dt, (6.11)

0

where ε > 0 is a small parameter, and   (ε) = diag 0, . . . , 0, ε2 , . . . , ε2 .     q

(6.12)

r −q

Thus, taking into account Eqs. and the positive definiteness of  (6.7) and (6.12)  ¯ we have immediately that G(t) + (ε) ∈ Sr++ for all t ∈ [0, t f ] and ε > 0. G(t), Therefore, the control cost in the functional (6.11) is regular for all ε > 0. On the other hand, for ε = 0, the functional (6.11) becomes the functional (6.10) of the SFHP. Remark 6.5 For the H∞ problem with the dynamics (6.9) and the functional (6.11), we chose the same set of admissible controllers U H,t f as it is for the SFHP. Solution of

154

6 Singular Finite-Horizon H∞ Problem

  the H∞ problem (6.9), (6.11) is a controller u ∗ε z, t ∈ U H,t f which ensures the valid  ity of the inequality Jε (u ∗ε , w) ≤ 0 along trajectories of (6.9) with u(t) = u ∗ε z, t and any w(t) ∈ L 2 [0, t f ; E s ]. The problem (6.9), (6.11) is regular for all ε > 0. Moreover, due to the smallness of the parameter ε and the structure of the matrix G(t) + (ε), this H∞ problem is a partial cheap control problem, i.e., the problem where the cost only of some (but not all) control coordinates is much smaller than the costs of the other control coordinates and a state cost.

6.4.2 Solution of the H∞ Problem (6.9), (6.11) Let us consider the following terminal-value problem with respect to the unknown n × n-matrix K (t) in the time interval [0, t f ]:   d K (t) = −K (t)A(t) − A T (t)K (t) + K (t) Su (t, ε) − Sw (t) K (t) − D(t), dt K (t f ) = F, (6.13) where  −1 Su (t, ε) = B(t) G(t) + (ε) B T (t),

Sw (t) = γ −2 C(t)C T (t).

(6.14)

We assume the following: A6.VI. For a given ε > 0, the terminal-value problem (6.13) has the solution K (t) = K ∗ (t, ε) in the entire interval [0, t f ]. The following assertion is a direct consequence of Theorem 3.3. Proposition 6.2 Let the assumptions A6.I–A6.VI be valid. Then (a) the controller  −1 u ∗ε (z, t) = − G(t) + (ε) B T (t)K ∗ (t, ε)z

(6.15)

solves the H∞ problem (6.9), (6.11); (b) for any t ∈ [0, t f ], K ∗ (t, ε) ∈ Sn+ .

6.5 Asymptotic Analysis of the H∞ Problem (6.9), (6.11) We start the asymptotic analysis of the H∞ problem (6.9), (6.11) with an asymptotic solution of the terminal-value problem (6.13).

6.5

Asymptotic Analysis of the H∞ Problem (6.9), (6.11)

155

6.5.1 Transformation of the Problem (6.13) Using the block form of the matrices B(t) and G(t) and the diagonal form of the matrix (ε) (see Eqs. (3.77), (6.7) and (6.12)), we can represent the matrix Su (t, ε) (see Eq. (6.14)) in the block form ⎛ Su (t, ε) = ⎝

Su 1 (t)

Su 2 (t)

SuT2 (t) (1/ε2 )Su 3 (t, ε)

⎞ ⎠,

(6.16)

where the (n − r + q) × (n − r + q)-matrix Su 1 (t), the (n − r + q) × (r − q)matrix Su 2 (t) and the (r − q) × (r − q)-matrix Su 3 (t, ε) have the form ⎛ Su 1 (t) = ⎝

⎞

0 0 0 G¯ −1 (t)

⎠,

⎛ Su 2 (t) = ⎝

0 G¯ −1 (t)H2T (t)

⎞ ⎠,

Su 3 (t, ε) = ε2 H2 (t)G¯ −1 (t)H2T (t) + Ir −q .

(6.17)

The block form of the matrix Su (t, ε) and the form of the matrix Su 3 (t, ε) directly show that the right-hand side of the differential equation in (6.13) has the singularity at ε = 0. To remove this singularity and to express this differential equation in an explicit singular perturbation form, we look for the solution of the terminal-value problem (6.13) as follows: ⎛ K (t, ε) = ⎝

K 1 (t, ε) εK 2 (t, ε) εK 2T (t, ε)

εK 3 (t, ε)

⎞ ⎠ , K 1T (t, ε) = K 1 (t, ε), K 3T (t, ε) = K 3 (t, ε),

(6.18) where the matrices K 1 (t, ε), K 2 (t, ε) and K 3 (t, ε) are of the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively. We also represent the matrices A(t) and Sw (t) in the block form ⎛ A(t) = ⎝

A1 (t) A2 (t) A3 (t) A4 (t)

⎞ ⎠,

⎛ Sw (t) = ⎝

Sw1 (t) Sw2 (t) SwT 2 (t) Sw3 (t)

⎞ ⎠,

(6.19)

where the blocks A1 (t), A2 (t), A3 (t) and A4 (t) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively; the blocks Sw1 (t), Sw2 (t) and Sw3 (t) have the form Sw1 (t)=γ −2 C1 (t)C1T (t), Sw2 (t) = γ −2 C1 (t)C2T (t) and Sw3 (t) = γ −2 C2 (t)C2T (t) and C1 (t) and C2 (t) are the upper and lower blocks of the matrix C(t) of the dimensions (n − r + q) × s and (r − q) × s, respectively.

156

6 Singular Finite-Horizon H∞ Problem

Now, substituting (3.78), (3.79), (6.16), (6.18) and (6.19) into the problem (6.13) and using a routine algebra, we transform this problem into the following equivalent problem: d K 1 (t, ε) = −K 1 (t, ε)A1 (t) − εK 2 (t, ε)A3 (t) − A1T (t)K 1 (t, ε) dt   −εA3T (t)K 2T (t, ε) + K 1 (t, ε) Su 1 (t) − Sw1 (t) K 1 (t, ε)     +εK 2 (t, ε) SuT2 (t) − SwT 2 (t) K 1 (t, ε) + εK 1 (t, ε) Su 2 (t) − Sw2 (t) K 2T (t, ε)   +K 2 (t, ε) Su 3 (t, ε) − ε2 Sw3 (t) K 2T (t, ε) − D1 (t), K 1 (t f , ε) = F1 , (6.20)

d K 2 (t, ε) = −K 1 (t, ε) A2 (t) − εK 2 (t, ε) A4 (t) − εA1T (t)K 2 (t, ε) dt   −εA3T (t)K 3 (t, ε) + εK 1 (t, ε) Su 1 (t) − Sw1 (t) K 2 (t, ε)     +ε2 K 2 (t, ε) SuT2 (t) − SwT 2 (t) K 2 (t, ε) + εK 1 (t, ε) Su 2 (t) − Sw2 (t) K 3 (t, ε)   +K 2 (t, ε) Su 3 (t, ε) − ε2 Sw3 (t) K 3 (t, ε), K 2 (t f , ε) = 0, ε

(6.21)

d K 3 (t, ε) = −εK 2T (t, ε)A2 (t) − εK 3 (t, ε) A4 (t) − εA2T (t)K 2 (t, ε) dt   −εA4T (t)K 3 (t, ε) + ε2 K 2T (ε) Su 1 (t) − Sw1 (t) K 2 (t, ε)     +ε2 K 3 (t, ε) SuT2 (t) − SwT 2 (t) K 2 (t, ε) + ε2 K 2T (t, ε) Su 2 (t) − Sw2 (t) K 3 (t, ε)   +K 3 (t, ε) Su 3 (t, ε) − ε2 Sw3 (t) K 3 (t, ε) − D2 (t), K 3 (t f , ε) = 0. ε

(6.22) The problem (6.20)–(6.22) is a singularly perturbed terminal-value problem for a set of Riccati-type matrix differential equations. In the next subsection, based on the Boundary Function Method (see, e.g., [6]), an asymptotic solution of this problem is constructed and justified.

6.5.2 Asymptotic Solution of the Problem (6.20)–(6.22) First of all let us note the following. The problem (6.20)–(6.22) is similar to the terminal-value problem (4.23)–(4.25) (see Sect. 4.5.1), and the asymptotic solution (the zero-order one) of (6.20)–(6.22) is constructed and justified quite similarly to the zero-order asymptotic solution of (4.23)–(4.25) (see Sect. 4.5.2). Therefore, the

6.5

Asymptotic Analysis of the H∞ Problem (6.9), (6.11)

157

results of this subsection are presented in a concise form, while all the necessary details can be found in Sect. 4.5.2. Using the Boundary Function Method [6], we look for the zero-order asymptotic solution to the problem (6.20)–(6.22) in the form o b (t) + K i,0 (τ ), i = 1, 2, 3, τ = (t − t f )/ε, K i,0 (t, ε) = K i,0

(6.23)

where the terms with the upper index “o” constitute the so-called outer solution, while the terms with the upper index “b” are the boundary correction terms in a left-hand neighbourhood of t = t f . Equations and conditions for these terms are obtained by substitution of (6.23) into the problem (6.20)–(6.22) instead of K i (t, ε), (i = 1, 2, 3), and equating coefficients for the same power of ε on both sides of the resulting equations, separately the coefficients depending on t and on τ . Note that in the construction of the asymptotic solution, the variable τ is considered as a new independent variable. For any t ∈ [0, t f ), τ → −∞ as ε → +0. Therefore, τ is called the stretched time. Similarly to Sect. 4.5.2 (see Eqs. (4.27)–(4.28)), we have b (τ ) = 0, K 1,0

τ ≤ 0.

o For the outer solution terms K i,0 (t), (i = 1, 2, 3), we have the equations

 o o o (t)A1 (t) − A1T (t)K 1,0 (t) + K 1,0 (t) Su 1 (t) = −K 1,0 dt  o T o o o (t) + K 2,0 (t) K 2,0 (t) − D1 (t), K 1,0 (t f ) = F1 , −Sw1 (t) K 1,0 o d K 1,0 (t)

(6.24)

o o o (t)A2 (t) + K 2,0 (t)K 3,0 (t), 0 = −K 1,0

(6.25)

2  o (t) − D2 (t). 0 = K 3,0

(6.26)

Algebraic equations (6.26) and (6.25) with respect to K 3,0 (t) and K 2,0 (t) yield the solutions  1/2  −1/2 o o o (t) = D2 (t) , K 2,0 (t) = K 1,0 (t)A2 (t) D2 (t) , K 3,0

(6.27)

where the superscript “1/2” denotes the unique positive definite square root of the corresponding positive definite matrix and the superscript “–1/2” denotes the inverse matrix of this square root. Remark 6.6 Since D2 (t) is continuously differentiable in the interval [0, t f ], then  1/2 −1/2  and D2 (t) are continuously differentiable for t ∈ [0, t f ]. D2 (t) o Substituting the expression for K 2,0 (t) into (6.24), we obtain the terminal-value o problem for K 1,0 (t)

158

where

6 Singular Finite-Horizon H∞ Problem o d K 1,0 (t) o o (t)A1 (t) − A1T (t)K 1,0 (t) = −K 1,0 dt o o o +K 1,0 (t)S1,0 (t)K 1,0 (t) − D1 (t), K 1,0 (t f ) = F1 ,

(6.28)

S1,0 (t) = Su 1 (t) − Sw1 (t) + A2 (t)D2−1 (t)A2T (t).

(6.29)

In what follows, we assume: o (t) in the entire interval [0, t f ]. A6.VII. The problem (6.28) has the solution K 1,0 Remark 6.7 Note that the assumption A6.VII is an ε-free assumption, guaranteeing the existence of the solution to the problem (6.20)–(6.22) in the entire interval [0, t f ] for all sufficiently small ε > 0 (see Lemma 6.2 below). The latter provides the existence of the controller solving the finite-horizon H∞ problem (6.9), (6.11) for all such ε > 0 (see Corollary 6.1 below). Moreover, the assumption A6.VII guarantees the existence of a controller solving the Reduced Finite-Horizon H∞ Problem (see Sect. 6.6 below). n−r +q

Remark 6.8 By virtue of the results of [1] (Chap. 6), if S1,0 (t) ∈ S+ t ∈ [0, t f ], then the assumption A6.VII is valid.

for all

b b (τ ) and K 3,0 (τ ), we have the terminalFor the boundary correction terms K 2,0 value problem b (τ ) d K 2,0 o b b o b b (t f )K 3,0 (τ ) + K 2,0 (τ )K 3,0 (t f ) + K 2,0 (τ )K 3,0 (τ ), τ ≤ 0, = K 2,0 dτ b  b 2 (τ ) d K 3,0 o b b o (t f )K 3,0 (τ ) + K 3,0 (τ )K 3,0 (t f ) + K 3,0 (τ ) , τ ≤ 0, = K 3,0 dτ b o b o K 2,0 (0) = −K 2,0 (t f ), K 3,0 (0) = −K 3,0 (t f ).

Similarly to Sect. 4.5.2 (see Eqs. (4.35)–(4.39)), we obtain the unique solution of this problem    −1/2 1/2  b (τ ) = −2F1 A2 (t f ) D2 (t f ) exp 2 D2 (t f ) τ Ir −q K 2,0   1/2 −1 + exp 2 D2 (t f ) τ ,       1/2 1/2 b (τ ) = −2 D2 (t f ) exp 2 D2 (t f ) τ Ir −q K 3,0   1/2 −1 + exp 2 D2 (t f ) τ , satisfying the inequality   b b (τ ), K 3,0 (τ ) ≤ c exp(βτ ), max K 2,0 where c > 0 and β > 0 are some constants.

τ ≤ 0,

6.6

Reduced Finite-Horizon H∞ Problem

159

Similarly to Sect. 4.5.2 (see Lemma 4.2 and Corollary 4.2), we have the following two assertions. Lemma 6.2 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, there exists a positive number ε0 such that, for all ε ∈ (0, ε0 ], the problem (6.20)–(6.22) has the unique solution K i (t, ε) = K i∗ (t, ε), (i = 1, 2, 3), in the entire interval [0, t f ]. This solution satisfies the inequalities   ∗  K (t, ε) − K i,0 (t, ε) ≤ aε, i

t ∈ [0, t f ],

where K i,0 (t, ε), (i = 1, 2, 3), are given by (6.23); a > 0 is some constant independent of ε. Corollary 6.1 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, for any ε ∈ (0, ε0 ], the assumption A6.VI is fulfilled and ∗



K (t, ε) =

K 1∗ (t, ε) εK 2∗ (t, ε)  T . ε K 2∗ (t, ε) εK 3∗ (t, ε)

Moreover, all the statements of Proposition 6.2 are valid. Remark 6.9 The ε-free conditions of Corollary 6.1 provide the solvability of the ε-dependent H∞ problem (6.9), (6.11) for all ε ∈ (0, ε0 ], i.e., robustly with respect to ε ∈ (0, ε0 ].

6.6 Reduced Finite-Horizon H∞ Problem First of all let us note that, similarly to the results of Sect. 4.6 (see Proposition 4.3), the matrix S1,0 (t) (see Eq. (6.29)) can be rewritten in the form T S1,0 (t) = B1,0 (t)Θ −1 (t)B1,0 (t) − Sw1 (t),

(6.30)

where 

 B1,0 (t) =  B, A2 (t) ,  B=



O(n−r )×q Iq



, Θ(t) =

¯ G(t) Oq×(r −q) O(r −q)×q D2 (t)

.

(6.31) ¯ Due to the positive definiteness of the matrices G(t) and D2 (t) for all t ∈ [0, t f ], we obtain that Θ(t) ∈ Sr++ , t ∈ [0, t f ]. Now, consider the H∞ problem for the following system and functional: d xr (t) = A1 (t)xr (t) + B1,0 (t)u r (t) + C1 (t)w(t), t ∈ [0, t f ], xr (0) = 0, (6.32) dt

160

6 Singular Finite-Horizon H∞ Problem

 + 0

tf

Jr (u r , w) = xrT (t f )F1 xr (t f )  T  xr (t)D1 (t)xr (t) + u rT (t)Θ(t)u r − γ 2 w T (t)w(t) dt,

(6.33)

where xr (t) ∈ E n−r +q is a state variable, u r (t) ∈ E r is a control and w(t) ∈ E s is a disturbance. We call the problem (6.32)–(6.33) the Reduced Finite-Horizon H∞ Problem (RFHP). The set of the admissible controllers and the solution of the RFHP are defined similarly to those notions in Sect. 3.2.3 (see Definitions 3.9–3.10). Based on Theorem 3.3 and Eqs. (6.30)–(6.31), we have the following assertion. Proposition 6.3 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, the controller T o (t)K 1,0 (t)xr (6.34) u ∗r (xr , t) = −Θ −1 (t)B1,0 o solves the RFHP, where K 1,0 (t), t ∈ [0, t f ], is the solution of the terminal-value problem (6.28) mentioned in the assumption A6.VII. Moreover, the controller (6.34) can be represented in the block form

u ∗r (xr , t) =



u ∗r,1 (xr , t) , u ∗r,2 (xr , t)

o o u ∗r,1 (xr , t) = −G¯ −1 (t)  B T K 1,0 (t)xr , u ∗r,2 (xr , t) = −D2−1 (t) A2T (t)K 1,0 (t)xr .

(6.35)

6.7 Controller for the SFHP 6.7.1 Formal Design of the Controller Lemma 6.3 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, for all ε ∈ (0, ε0 ], the controller (6.15), solving the H∞ problem (6.9), (6.11), solves also the SFHP (6.9)–(6.10). Proof First of all, let us note that both problems have the same set of the admissible controllers U H,t f . Now, comparing the functionals (6.10) and (6.11) of the problem (6.9), (6.11) and the SFHP, respectively, we directly obtain the inequality     J u(z, t), w(t) ≤ Jε u(z, t), w(t) ,

(6.36)

where u(z, t) ∈ U H,t f is any admissible controller in both problems; w(t) is any function from L 2 [0, t f ; E s ]. By virtue of Corollary 6.1, the controller (6.15) is admissible in the H∞ problem (6.9), (6.11) for all ε ∈ (0, ε0 ]. Therefore, it is admissible in the SFHP for all ε ∈ (0, ε0 ]. Moreover, since this controller solves the problem (6.9), (6.11), then

6.7

Controller for the SFHP

161

  Jε u ∗ε (z, t), w(t) ≤ 0 for all ε ∈ (0, ε0 ] and w(t) ∈ L 2 [0, t f ; E s ]. The latter, along 

with the inequality (6.36), yields immediately the statement of the lemma. Remark 6.10 The design of the controller (6.15) is a complicated task, because it requires the solution of the high dimension singularly perturbed system of nonlinear differential equations (6.20)–(6.22). To overcome this difficulty, in the sequel of this section, another (simplied) controller for the SFHP is designed and its properties are studied. Consider the matrix o K 1,0 (t) o  o T K 0 (t, ε) = ε K 2,0 (t)

o εK 2,0 (t) , o εK 3,0 (t)

t ∈ [0, t f ], ε ∈ (0, ε0 ].

(6.37)

Also, let us partition the vector z ∈ E n into blocks as   z = col x, y , x ∈ E n−r +q , y ∈ E r −q .

(6.38)

Using the matrix K 0o (t, ε), we consider the following auxiliary controller, obtained from the controller u ∗ε (z, t) (see Eq. (6.15)) by replacing K (t, ε) with K 0o (t, ε):  −1 u aux (z, t) = − G(t) + (ε) B T (t)K 0o (t, ε)z. Substitution of the block representations for the matrices B(t), G(t), (ε), and K 0o (t, ε) and the vector z (see Eqs. (3.77), (6.7), (6.12), (6.37) and (6.38)) into this B (see Eq. expression for u aux (z, t), and the use of the block form of the matrix  (6.31)) yield after a routine algebra the block representation for u aux (z, t)  ⎞ G¯ −1 (t) Q 1 (t, ε)x + εQ 2 (t, ε)y ⎟ ⎜ ⎟, u aux (z, t) = − ⎜ ⎠   ⎝  T 1 o o (t) x + K (t)y K 2,0 3,0 ε  o T o Q 1 (t, ε) =  B T K 1,0 (t) + εH2T (t) K 2,0 (t) , ⎛

o o Q 2 (t, ε) =  B T K 2,0 + εH2T (t)K 3,0 (t).

Calculating the point-wise (with respect to (z, t) ∈ E n × [0, t f ]) limit of the upper block in u aux (z, t) for ε → 0+ , we obtain the simplified controller for the SFHP ⎛ ⎜ u ∗ε,0 (z, t) = ⎝

o −G¯ −1 (t)  B T K 1,0 (t)x

− 1ε



⎞

⎟ ⎠.  T o o (t) x + K 3,0 (t)y K 2,0

(6.39)

162

6 Singular Finite-Horizon H∞ Problem

Remark 6.11 Since the controller (6.39) is linear with respect to z with the gain matrices continuous in the interval [0, t f ], this controller is admissible in the SFHP for all ε ∈ (0, ε0 ]. Comparing the simplified controller u ∗ε,0 (z, t) with the controller u ∗r (xr , t), which solves the RFHP (see Eq. (6.35)), we can see that the upper blocks of both controllers coincide with each other. Thus, to construct the controller u ∗ε,0 (z, t), one has to solve the lower dimension regular H∞ problem (the RFHP) and calculate o o (t) and K 3,0 (t) using Eq. (6.27). two gain matrices K 2,0

6.7.2 Properties of the Simplified Controller (6.39)   Let for any given ε ∈ (0, ε0 ] and w(t) ∈ L 2 [0, +∞; E s ], z 0∗ t, ε; w(·) , t ∈ [0, t f ] be the solution of the initial-value problem (6.9) with u(t) = u ∗ε,0 (z, t). Let

Q o0 (t) =



 T o o K 2,0 (t) , K 3,0 (t) ,

t ∈ [0, t f ].

(6.40)

Theorem 6.1 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, there exists a positive number ε∗1 ≤ ε0 such that, for all ε ∈ (0, ε∗1 ] and w(t) ∈ L 2 [0, +∞; E s ], the following inequality is satisfied along trajectories of the system (6.9):   J u ∗ε,0 (z, t), w(t) ≤ −



tf 0

 ∗ T  o T o   z 0 t, ε; w(·) Q 0 (t) Q 0 (t)z 0∗ t, ε; w(·) dt. (6.41)

Proof of the theorem is presented in Sect. 6.7.3. Remark 6.12 Due to Theorem 6.1, the controller u ∗ε,0 (z, t) solves the original SFHP (6.9)–(6.10). Theorem 6.2 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, there exists a positive number ε∗2 ≤ ε∗1 such that, for all ε ∈ (0, ε∗2 ] and w(t) ∈ L 2 [0, +∞; E s ], the integral in the right-hand side of (6.41), being nonnegative, satisfies the inequality 

tf 0

 2  ∗ T  o T o   z 0 t, ε; w(·) Q 0 (t) Q 0 (t)z 0∗ t, ε; w(·) dt ≤ aε w(t) L 2 [0,t f ] ,

where a > 0 is some constant independent of ε and w(·). Proof of the theorem is presented in Sect. 6.7.4. Remark 6.13 If q = 0, i.e., all the coordinates of the control in the SFHP (6.9)– (6.10) are singular, then Su 1 (t) ≡ O(n−r )×(n−r ) , Su 2 (t) ≡ O(n−r )×r , Su 3 (t) ≡ Ir and o (t) is the S1,0 (t) = A2 (t)D2−1 (t) A2T (t) − Sw1 (t). The matrix-valued function K 1,0

6.7

Controller for the SFHP

163

solution of the terminal-value problem (6.28) with this S1,0 (t). The upper block of the simplified controller u ∗ε,0 (z, t) (see Eq. (6.39)) vanishes, while the lower  o T (t) x + block remains unchanged. Thus, in this case, we have u ∗ε,0 (z, t) = − 1ε K 2,0  o (t)y , z = col(x, y), x ∈ E n−r , y ∈ E r . Theorems 6.1–6.2 and Remark 6.12 are K 3,0 o (t). valid with this particular form of u ∗ε,0 (z, t) and the aforementioned K 1,0

6.7.3 Proof of Theorem 6.1 The proof consists of four stages. Stage 1. At this stage, we transform the H∞ problem (6.9), (6.11) with u(t) = u ∗ε,0 (z, t) to an equivalent H∞ problem. Remember that the controller u ∗ε,0 (z, t), given by (6.39), solves this problem if the inequality   Jε u ∗ε,0 (z, t), w(t) ≤ 0

(6.42)

is fulfilled along trajectories of thesystem (6.9) for all w(t) ∈ L 2 [0, t f ; E s ]. Substituting u(t) = u ∗ε,0 z(t), t into the system (6.9) and the functional (6.11), as well as using the block representations for the matrices B(t), D(t), G(t), (ε) and A(t) (see Eqs. (3.77), (3.79), (6.7), (6.12) and (6.19)) and taking into account (6.27), (6.31), we obtain after a routine matrix algebra the following system and functional: dz(t)  ε)z(t) + C(t)w(t), t ∈ [0, t f ], z(0) = 0, = A(t, dt



tf

= z (t f )F z(t f ) + T

      Jε w(t) = Jε u ∗ε,0 z(t), t , w(t)    − γ 2 w T (t)w(t) dt, z T (t) D(t)z(t)

(6.43)

(6.44)

0

where



1 (t) 1 (t) D 2 (t) 2 (t) A D A  4 (t, ε) , D = D 3 (t) , 3 (t, ε) (1/ε) A 2T (t) D (1/ε) A o 2 (t) = A2 (t), 1 (t) = A1 (t) −  B G¯ −1 (t)  B T K 1,0 (t), A A   −1/2 T o o 3 (t, ε) = εA3 (t) − εH2 (t)G¯ −1 (t)  A B T K 1,0 (t) − D2 (t) A2 (t)K 1,0 (t),  1/2 4 (t, ε) = εA4 (t) − D2 (t) , A o T o 1 (t) = D1 (t) + K 1,0 (t)B1,0 (t)Θ −1 (t)B1,0 (t)K 1,0 (t), D  ε) = A(t,

o 2 (t) = K 1,0 (t)A2 (t), D

3 (t) = 2D2 (t). D (6.45)

164

6 Singular Finite-Horizon H∞ Problem

Due to (6.44), the inequality (6.42) is equivalent to the inequality   Jε w(t) ≤ 0

(6.46)

along trajectories of the system (6.43) for all w(t) ∈ L 2 [0, t f ; E s ]. The latter means that the solvability of the H∞ problem (6.9), (6.11) by u(t) = u ∗ε,0 (z, t) is equivalent to the solvability of the H∞ problem (6.43)–(6.44), (6.46). Stage 2. At this stage, we derive solvability conditions of the H∞ problem (6.43)– (6.44), (6.46). Consider the following terminal-value problem for the n × n-matrix(t) in the interval [0, t f ]: valued function K (t) dK (t) A(t,  ε) − = −K dt (t) − D(t),  (t)Sw (t) K −K

(t) T (t, ε) K A (t f ) = F. K

(6.47)

(t, ε), t ∈ [0, t f ], to (6.47) for a given Let us show that the existence of the solution K ε ∈ (0, ε0 ] yields the fulfilment of the inequality (6.46). For a given ε ∈ (0, ε0 ], consider the Lyapunov-like function (t, ε)z, z ∈ E n , t ∈ [0, t f ], V (z, t, ε) = z T K

(6.48)

(t, ε) is the above-mentioned solution to the problem (6.47). Based on (6.48), where K       consider the function V z 0∗ t, ε; w(·) , t, ε . Since z 0∗ t, ε; w(·) is the solution of the initial-value problem (6.9) with u(t) = u ∗ε,0 (z, t), then it is the solution of the     initial-value problem (6.43). Differentiating the function V z 0∗ t, ε; w(·) , t, ε with respect to t and using (6.43), (6.47) and (6.48) yield    T       dz 0∗ t, ε; w(·) d V z 0∗ t, ε; w(·) , t, ε (t, ε)z 0∗ t, ε; w(·) =2 K dt dt   (t, ε) ∗   T d K z 0 t, ε; w(·) + z 0∗ t, ε; w(·) dt T      = 2 A(t, ε)z(t) + C(t)w(t) K (t, ε)z 0∗ t, ε; w(·)    T  (t) (t) A(t,  ε) − A T (t, ε) K −K + z 0∗ t, ε; w(·)    (t) − D(t)  (t)Sw (t) K z 0∗ t, ε; w(·) −K   (t, ε)z 0∗ t, ε; w(·) = 2w T (t)C T (t) K   T   (t, ε)z 0∗ t, ε; w(·) (t, ε)Sw (t) K − z 0∗ t, ε; w(·) K    T   ∗  D(t)z − z 0∗ t, ε; w(·) 0 t, ε; w(·) . (6.49)

6.7

Controller for the SFHP

165

    (t, ε)z 0∗ t, ε; w(·) and the Using the function w ∗ t, ε; w(·) = γ −2 C T (t) K expression for Sw (t) (see Eq. (6.14)), we can rewrite (6.49) as        T    d V z 0∗ t, ε; w(·) , t, ε = −γ 2 w(t) − w∗ t, ε; w(·) w(t) − w∗ t, ε; w(·) dt    T   ∗ 2 T  − z 0∗ t, ε; w(·) D(t)z 0 t, ε; w(·) + γ w (t)w(t). From this equation, we directly obtain the following inequality for all t ∈ [0, t f ]:        T  d V z 0∗ t, ε; w(·) , t, ε ∗ 2 T  + z 0∗ t, ε; w(·) D(t)z 0 t, ε; w(·) − γ w (t)w(t) ≤ 0. dt

Integration of this inequality from t = 0 to t = t f and the use of (6.48) yield



tf

+ 0

   T ∗   F z 0 t f , ε; w(·) z 0∗ t f , ε; w(·)    T   ∗ 2 T  t, ε; w(·) − γ w (t)w(t) dt ≤ 0, z 0∗ t, ε; w(·) D(t)z 0

meaning the fulfilment of the inequality (6.46) along trajectories of the system (6.43) for all w(t) ∈ L 2 [0, t f ; E s ]. Thus, we have proven that the existence of the solution (t, ε) to the problem (6.47) in the entire interval [0, t f ] guarantees the fulfilment K of the inequality (6.46) along trajectories of (6.43) for all w(t) ∈ L 2 [0, t f ; E s ]. (t, ε) to the problem Stage 3. At this stage, we show the existence of the solution K (6.47) in the entire interval [0, t f ] for all sufficiently small ε > 0. Similarly to the solution of the problem (6.13), we look for the solution of the terminal-value problem (6.47) in the block form ⎛ (t, ε) = ⎝ K

2 (t, ε) 1 (t, ε) ε K K 2T (t, ε) ε K 3 (t, ε) εK

⎞ lT (t, ε) = K l (t, ε), l = 1, 3, ⎠, K

(6.50)

1 (t, ε), K 2 (t, ε) and K 3 (t, ε) have the dimensions (n − r + where the matrices K q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively.  ε), D(t)  Substitution of the block representations for the matrices F, Sw (t), A(t, (t, ε) (see Eqs. (3.78), (6.14), (6.45) and (6.50)) into the problem (6.47) conand K verts this problem into the following equivalent problem: 1 (t, ε) dK 1 (t) − K 2 (t, ε) A 3 (t, ε) − A 1T (t) K 1 (t, ε) 1 (t, ε) A = −K dt 3T (t, ε) K 2T (t, ε) − K 1 (t, ε)Sw1 (t) K 1 (t, ε) − ε K 2 (t, ε)SwT (t) K 1 (t, ε) −A 2 2T (t, ε) − ε2 K 2T (t, ε) 1 (t, ε)Sw2 (t) K 2 (t, ε)Sw3 (t) K −ε K 1 (t f , ε) = F1 , (6.51) 1 (t), K −D

166

6 Singular Finite-Horizon H∞ Problem

2 (t, ε) dK 2 (t) − K 2 (t, ε) A 4 (t, ε) − ε A 1T (t) K 2 (t, ε) 1 (t, ε) A = −K dt 3T (t, ε) K 3 (t, ε) − ε K 1 (t, ε)Sw1 (t) K 2 (t, ε) − ε2 K 2 (t, ε) 2 (t, ε)SwT (t) K −A 2 1 (t, ε)Sw2 (t) K 3 (t, ε) − ε2 K 3 (t, ε) 2 (t, ε)Sw3 (t) K −ε K ε

2 (t), −D

2 (t f , ε) = 0, (6.52) K

3 (t, ε) dK 2 (t) − K 3 (t, ε) A 4 (t, ε) − ε A 2T (t) K 2 (t, ε) 2T (t, ε) A = −ε K dt 4T (t, ε) K 3 (t, ε) − ε2 K 2 (t, ε) − ε2 K 2 (t, ε) 2T (t, ε)Sw1 (t) K 3 (t, ε)SwT (t) K −A 2 3 (t, ε) − ε2 K 3 (t, ε) 2T (t, ε)Sw2 (t) K 3 (t, ε)Sw3 (t) K −ε2 K 3 (t), K 3 (t f , ε) = 0. (6.53) −D ε

Looking for the zero-order asymptotic solution of the problem (6.51)–(6.53) in the form o b i,0 i,0 i,0 (t, ε) = K (t) + K (τ ), K

i = 1, 2, 3,

τ = (t − t f )/ε,

we obtain similarly to Sect. 6.5.2 b 1,0 (τ ) = 0, τ ≤ 0. K

(6.54)

o i,0 (t), (i = 1, 2, 3), satisfy the followFurthermore, the terms of the outer solution K ing set of equations for t ∈ [0, t f ]: o 1,0 dK (t) o o o 1 (t) − K 2,0 3 (t, 0) − A 1T (t) K 1,0 1,0 (t) A (t) A (t) = −K dt   T o o o o 2,0 1,0 1,0 1 (t), K 1,0 3T (t, 0) K (t) − K (t)Sw1 (t) K (t) − D (t f ) = F1 , −A o o T o 1,0 (t) A 2 (t) − K 2,0 (t) A 4 (t, 0) − A 3 (t, 0) K 3,0 (t) − D 2 (t), 0 = −K o o 3,0 4 (t, 0) − A 4T (t, 0) K 3,0 3 (t). 0 = −K (t) A (t) − D

(6.55) o 3,0 Solving the third and second equations of the set (6.55) with respect to K (t) o 2 (t), A 3 (t, 0), A 4 (t, 0), D 2 (t) and D 3 (t) 2,0 (t) and using the expressions for A and K (see Eq. (6.45)), we obtain

 1/2 o 3,0 (t) = D2 (t) , K

 −1/2 o o 2,0 1,0 K (t) = K (t)A2 (t) D2 (t) .

(6.56)

o 2,0 (t) into the differential equation of the set Substitution of the expression for K (6.55) and the use of (6.17), (6.29), (6.30) and (6.45) yield after a routine matrix o 1,0 (t): algebra the following terminal-value problem for K

6.7

Controller for the SFHP

167

o 1,0    o  dK (t) o 1,0 (t) A1 (t) − S1,0 (t) + Sw1 (t) K 10 (t) = −K dt   T  o o o o 1,0 1,0 1,0 − A1 (t) − K 10 (t) S1,0 (t) + Sw1 (t) K (t) − K (t)Sw1 (t) K (t)   o o o  −D1 (t) − K 1,0 (t) S1,0 (t) + Sw1 (t) K 1,0 (t), K 1,0 = F1 .

(6.57)

o o 1,0 (t) into (6.57) instead of K (t) that this It is verified directly by substitution of K 1,0 terminal-value problem has a solution on the entire interval [0, t f ] and this solution o (t), i.e., equals to K 1,0 o o 1,0 (t) = K 1,0 (t), K

t ∈ [0, t f ].

(6.58)

Moreover, due to the linear and quadratic dependence of the right-hand side of the o 1,0 (t), this solution is unique. differential equation in (6.57) on K b b   For the terms K 2,0 (τ ) and K 3,0 (τ ), similarly to Sect. 6.5.2, we have the problem for τ ≤ 0 b 2,0 (τ ) dK

dτ

b b b o 4 (t f , 0) − A 3T (t f , 0) K 3,0 2,0 2,0 2,0 (τ ) A (τ ), K (0) = − K (t f ), = −K

b 3,0 (τ ) dK b b b o 4 (t f , 0) − A 4 (t f , 0) K 3,0 3,0 3,0 3,0 (τ ) A (τ ), K (0) = − K (t f ). = −K dτ

Using (6.45) and (6.56), we obtain the unique solution of this problem    −1/2 1/2  b 2,0 K (τ ) = −F1 A2 (t f ) D2 (t f ) exp 2 D2 (t f ) τ , τ ≤ 0,    1/2 1/2  b 3,0 (τ ) = − D2 (t f ) exp 2 D2 (t f ) τ , τ ≤ 0, K

(6.59)

satisfying the inequality  b  b 2,0 (τ ),  K 3,0 max  K (τ ) ≤ c exp(βτ ), τ ≤ 0,

(6.60)

where c > 0 and β > 0 are some constants. Now, based on Eqs. (6.54), (6.56), (6.58) and (6.59) and the inequality (6.60), we obtain similarly to Lemma 6.2 (see also Lemma 4.2 and its proof in Sect. 4.5.2) the existence of a positive number ε∗1 ≤ ε0 such that, for all ε ∈ (0, ε∗1 ], the prob2 (t, ε), K 3 (t, ε) in the entire 1 (t, ε), K lem (6.51)–(6.53) has the unique solution K interval [0, t f ]. Since the problem (6.47) is equivalent to (6.51)–(6.53), then it has the unique solution (6.50) in the entire interval [0, t f ] for all ε ∈ (0, ε∗1 ]. The latter, along with the results of the Stages 1 and 2 of this proof, means that the controller u ∗ε,0 (z, t), given by (6.39), solves the H∞ problem (6.9), (6.11), i.e., the inequality (6.42) is fulfilled.

168

6 Singular Finite-Horizon H∞ Problem

Stage 4. Using Eqs. (6.10), (6.11), (6.12), (6.39) and (6.40), we obtain 

tf

+ 0

    Jε u ∗ε,0 (z, t), w(t) = J u ∗ε,0 (z, t), w(t)  ∗ T  o T o   z 0 t, ε; w(·) Q 0 (t) Q 0 (t)z 0∗ t, ε; w(·) dt.

This equation, along with the inequality (6.42), directly yields the inequality (6.41). This completes the proof of Theorem 6.1.

6.7.4 Proof of Theorem 6.2 The proof is based on an auxiliary proposition.

6.7.4.1

Auxiliary Proposition

Let, for a given ε ∈ (0, ε∗1 ], the n × n-matrix-valued function Φ(t, σ, ε), 0 ≤ σ ≤  ε)z(t). This means t ≤ t f , be the fundamental solution of the system dz(t)/dt = A(t, that Φ(t, σ, ε) satisfies the initial-value problem ∂Φ(t, σ, ε)  ε)Φ(t, σ, ε), 0 ≤ σ ≤ t ≤ t f , Φ(σ, σ, ε) = In . = A(t, ∂t

(6.61)

 ε) is defined in (6.45). Remember that the block matrix A(t, Let us partition the matrix Φ(t, σ, ε) into blocks as Φ(t, σ, ε) =

Φ1 (t, σ, ε) Φ2 (t, σ, ε) , Φ3 (t, σ, ε) Φ4 (t, σ, ε)

(6.62)

where the matrices Φ1 (t, σ, ε), Φ2 (t, σ, ε), Φ3 (t, σ, ε) and Φ4 (t, σ, ε) are of the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively. Along with the problem (6.61), we consider the following problem with respect to the (n − r + q) × (n − r + q)-matrix-valued function Φ0 (t, σ): ∂Φ0 (t, σ) = A0 (t)Φ0 (t, σ), 0 ≤ σ ≤ t ≤ t f , Φ0 (σ, σ) = In−r +q , ∂t where

(6.63)

1 (t) − A 2 (t) A −1  A0 (t) = A 4 (t, 0) A3 (t, 0),

1 (t), A 2 (t), A 3 (t, ε) and A 4 (t, ε) are the blocks of the matrix A(t,  ε); and A  −1/2 −1  A4 (t, 0) = − D2 (t) .

6.7

Controller for the SFHP

169

It is clear that the problem (6.63) has the unique solution Φ0 (t, σ), 0 ≤ σ ≤ t ≤ t f . By virtue of the results of [2] (Lemma 3.1), we have the following assertion. Proposition 6.4 Let the assumptions A6.I–A6.V and A6.VII be valid. Then, there exists a positive number ε∗2 ≤ ε∗1 such that, for all ε ∈ (0, ε∗2 ], the following inequalities are satisfied:   Φ1 (t, σ, ε) − Φ0 (t, σ) ≤ aε,

  Φ2 (t, σ, ε) ≤ aε,

      Φ3 (t, σ, ε) + A −1  4 (t, 0) A3 (t, 0)Φ0 (t, σ) ≤ a ε + exp − β(t − σ)/ε ,      Φ4 (t, σ, ε) ≤ a ε + exp − β(t − σ)/ε , where 0 ≤ σ ≤ t ≤ t f ; a > 0 and β > 0 are some constants independent of ε. 6.7.4.2

Main Part of the Proof

  Due to the proof of Theorem 6.1, the vector-valued function z 0∗ t, ε; w(·) , t ∈ [0, t f ], being the solution of the initial-value problem (6.9) with u(t) = u ∗ε,0 (z, t) also is the solution of the initial-value problem (6.43). Since Φ(t, σ, ε) is the fundamental matrix solution of the system dz(t)/dt =  ε)z(t), then the solution of (6.43) can be represented in the form A(t, z 0∗



 t, ε; w(·) =



t

Φ(t, σ, ε)C(σ)w(σ)dσ,

t ∈ [0, t f ].

(6.64)

0

  Let us partition the vector z 0∗ t, ε; w(·) into blocks as        z 0∗ t, ε; w(·) = col x0∗ t, ε; w(·) , y0∗ t, ε; w(·) ,

(6.65)

    where x0∗ t, ε; w(·) ∈ E n−r +q , y0∗ t, ε; w(·) ∈ E r −q . Substitution of (6.62) and (6.65) into (6.64) yields after a routine algebra x0∗



 t, ε; w(·) =

  y0∗ t, ε; w(·) =



t



 Φ1 (t, σ, ε)C1 (σ) + Φ2 (t, σ, ε)C2 (σ) w(σ)dσ, t ∈ [0, t f ],

0

 0

(6.66) t

 Φ3 (t, σ, ε)C1 (σ) + Φ4 (t, σ, ε)C2 (σ) w(σ)dσ, t ∈ [0, t f ].



(6.67) Remember that C1 (t) and C2 (t) are the upper and lower blocks of the matrix C(t) of the dimensions (n − r + q) × s and (r − q) × s, respectively.

170

6 Singular Finite-Horizon H∞ Problem

Using Proposition 6.4, we obtain the following inequalities for all 0 ≤ σ ≤ t ≤ t f and ε ∈ (0, ε∗2 ]:     Φ1 (t, σ, ε)C1 (σ) + Φ2 (t, σ, ε)C2 (σ) − Φ0 (t, σ)C1 (σ) ≤ aε,    Φ3 (t, σ, ε)C1 (σ) + Φ4 (t, σ, ε)C2 (σ)       −1 +A 4 (t, 0) A3 (t, 0)Φ0 (t, σ)C 1 (σ) ≤ a1 ε + exp − β(t − σ)/ε , (6.68) where a1 > 0 is some constant independent of ε. Consider the following vector-valued functions of the dimensions n − r + q and r − q, respectively:  t   Φ0 (t, σ)C1 (σ)w(σ)dσ, t ∈ [0, t f ], ϕx t; w(·) = 0

    −1  ϕ y t; w(·) = − A 4 (t, 0) A3 (t, 0)ϕx t; w(·) ,

t ∈ [0, t f ].

We have that     4 (t, 0)ϕ y t; w(·) = 0, 3 (t, 0)ϕx t; w(·) + A A

t ∈ [0, t f ].

 o T o 3 (t, 0) = − K 2,0 4 (t, 0) = −K 3,0 Using the equalities A (t) and A (t) (see Eqs. (6.27) and (6.45)), we can rewrite this equation as 

   T  o o (t) ϕx t; w(·) + K 3,0 (t)ϕ y t; w(·) = 0, t ∈ [0, t f ]. K 2,0

(6.69)

Moreover, using the notations       Δx t, ε; w(·) = x0∗ t, ε; w(·) − ϕx t; w(·) ,       Δy t, ε; w(·) = y0∗ t, ε; w(·) − ϕ y t; w(·) , Eqs. (6.66)–(6.67), the inequalities (6.68) and the Cauchy-Bunyakovsky-Schwarz integral inequality [5], we directly obtain the following inequality for all t ∈ [0, t f ] and ε ∈ (0, ε∗2 ]:       max Δx t, ε; w(·) , Δy t, ε; w(·)  ≤ a2 ε1/2 w(t) L 2 [0,t f ] ,

(6.70)

where a2 > 0 is some constant independent of ε and w(·). Now, the use of Eqs. (6.40), (6.65) and (6.69) yields after a routine rearrangement

6.8

Example

171

 ∗ T  o T o   z 0 t, ε; w(·) Q 0 (t) Q 0 (t)z 0∗ t, ε; w(·) =      T o     o T   (t) K 2,0 (t) ϕx t; w(·) + Δx t, ε; w(·) ϕx t; w(·) + Δx t, ε; w(·) K 2,0        T o    o +2 ϕx t; w(·) + Δx t, ε; w(·) K 2,0 (t)K 3,0 (t) ϕ y t; w(·) + Δy t, ε; w(·)      T  o    2   + ϕ y t; w(·) + Δy t, ε; w(·) K 3,0 (t) ϕ y t; w(·) + Δy t, ε; w(·)   T o  o T   = Δx t, ε; w(·) K 2,0 (t) K 2,0 (t) Δx t, ε; w(·)      T o o K 2,0 (t)K 3,0 (t)Δy t, ε; w(·) +2 Δx t, ε; w(·)   2    T  o K 3,0 (t) Δy t, ε; w(·) . + Δy t, ε; w(·) This equation, along with the inequality (6.70), yields the inequality  ∗          z t, ε; w(·) T Q o (t) T Q o (t)z ∗ t, ε; w(·)  ≤ a3 ε w(t) L 2 [0,t ] 2 , 0 0 0 0 f

(6.71)

where t ∈ [0, t f ]; ε ∈ (0, ε∗2 ]; a3 > 0 is some constant independent of ε and w(·). The inequality (6.71) immediately yields the statement of Theorem 6.2.

6.8 Example Consider the particular case of the SFHP where n = 2, r = 2, s = 2, q = 1 and





1 0 −1 2 −1.5 1 , , B(t) = , C(t) = A(t) = −t 1 t 2 cos(t) t exp(t)





2.2 0 2 0 0.5 0 F= , D(t) = D = , G(t) = G = . 0 0 0 0.5 0 0

(6.72) Also in this example, γ = 1 and t f = 1. Using these data, as well as the expression for Su 1 (t), the expression for Sw (t), the block form of this matrix and the expression for S1,0 (t) (see Eqs. (6.17), (6.14), (6.19) and (6.29)), we obtain Su 1 (t) ≡ 2,

Sw1 (t) ≡ 5,

S1,0 (t) ≡ −1, t ∈ [0, 1].

Due to these values and the data (6.72), the terminal-value problem (6.28) becomes as o  o 2 d K 1,0 (t) o o (t) − K 1,0 (t) − 2, K 1,0 (1) = 2.2. = 3K 1,0 dt

172

6 Singular Finite-Horizon H∞ Problem

Solving this problem, we have  −1 o K 1,0 (t) = 2 + 6 exp(t − 1) − 1 ,

t ∈ [0, 1].

o o Using this solution and the data (6.72), we calculate the gains K 3,0 (t) and K 2,0 (t), given by (6.27),

1 o (t) = √ , K 3,0 2

o K 2,0 (t) =

√   −1  , t ∈ [0, 1]. 2 2 + 6 exp(t − 1) − 1

o (t), (i = 1, 2, 3), we design the simplified Finally, using (6.72) and the gains K i,0 controller (6.39) for the singular H∞ problem (6.9)–(6.10), (6.72)

2K o (t)x  u ∗ε,0 (z, t) = − √2 1,0 o , z = col(x, y). K 1,0 (t)x + 21 y ε

The extensive computer simulation has shown that this controller solves the abovementioned singular H∞ problem for all ε ∈ (0, 0.002]. Moreover, this controller solves another singular H∞ control problems with the same matrices of the coefficients, the same t f and with the smaller performance level γ ∈ [0.998, 1) for ε ∈ (0, 0.002].

6.9 Concluding Remarks and Literature Review In this chapter, the finite-horizon H∞ control problem for a linear time-varying system was considered. The feature of this problem is that the matrix of coefficients for the control in the output equation, being non-zero, has the rank smaller than the dimension of the control vector. This feature means that the approach, based on the matrix Riccati differential equation, is not applicable to the solution of the considered problem. Thus, this H∞ problem is singular. However, since the abovementioned matrix is non-zero, only a part of the control coordinates is singular, while the others are regular. Subject to proper assumptions, the initially formulated H∞ control problem was converted to a new singular H∞ control problem. The dynamics of the new problem consists of three modes with different types of the dependence on the control. The first mode does not contain the control at all, the second mode contains only the regular control coordinates, while the third mode contains the entire control. Lemma 6.1 proves that the transformed H∞ problem is equivalent to the initially formulated one. The result, similar to this lemma, was obtained in [4] for a particular case of the singular H∞ control problem and subject to an additional condition. In the sequel of the chapter, the transformed problem is analysed as an original problem. The solution of this new problem was obtained by the regularization method. Namely, this problem was approximately replaced by auxiliary regular H∞ control problem with the same dynamics and a similar functional, augmented by a

References

173

finite-horizon integral of the squares of the singular control coordinates with a small weight ε2 , (ε > 0). The regularization method for solution of singular finite-horizon H∞ control problems was used only in two works in the literature. Thus, in [3], this method was applied for the solution of the constant coefficients problem with time delay dynamics and functional not containing the control variable at all. In [4], the problem, similar to the one considered in the present chapter but with a particular case of the weight matrix for the control in the functional, was solved by the regularization method. The auxiliary (regularized) problem is a finite-horizon H∞ partial cheap control problem. Finite-horizon H∞ partial cheap control problem was considered in [4], while finite-horizon H∞ complete cheap control problem was studied in [3]. The zero-order asymptotic solution of the singularly perturbed matrix Riccati differential equation, associated with the regularized H∞ problem by the solvability conditions, was constructed and justified. Based on this asymptotic solution, a simplified controller for the regularized H∞ problem was designed. It was established that this controller also solves the original singular H∞ control problem. Moreover, it was shown in the example that this controller also solves a finite-horizon singular H∞ control problem with a smaller (than the original) performance level, depending on ε. It should be noted that the example, presented in Sect. 6.8, was considered for the first time in [4] in a brief form. In the present chapter, we analyse this example in a much more detailed form.

References 1. Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory, 2nd edn. SIAM Books, Philadelphia, PA, USA (1999) 2. Glizer, V.Y.: Correctness of a constrained control Mayer’s problem for a class of singularly perturbed functional-differential systems. Control Cybernet. 37, 329–351 (2008) 3. Glizer, V.Y.: Finite horizon H∞ cheap control problem for a class of linear systems with state delays. In: Proceedings of the 6th International Conference on Integrated Modeling and Analysis in Applied Control and Automation, pp. 1–10. Vienna, Austria (2012) 4. Glizer, V.Y., Kelis, O.: Finite-horizon H∞ control problem with singular control cost. In: Gusikhin, O., Madani, K. (eds.) Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, vol. 495, pp. 23–46. Springer Nature, Switzerland (2020) 5. Kolmogorov, A.N., Fomin, S.V.: Introductory Real Analysis. Dover Publications Inc., New York, NY, USA (1970) 6. Vasileva, A.B., Butuzov V.F., Kalachev L.V.: The Boundary Function Method for Singular Perturbation Problems. SIAM Books, Philadelphia, PA, USA (1995)

Chapter 7

Singular Infinite-Horizon H∞ Problem

7.1 Introduction In this chapter, we consider a system consisting of an uncertain controlled linear timeinvariant differential equation and a linear time-invariant output algebraic equation. For this system, an infinite-horizon H∞ problem is studied in the case where the rank of the coefficients’ matrix for the control in the output equation is smaller than the Euclidean dimension of this control. In this case, the solvability conditions, presented in Sect. 3.2.4, are not applicable to the solution of the considered H∞ problem meaning its singularity. To solve this H∞ problem, a regularization method is proposed. Namely, the original problem is replaced approximately with a regular infinite-horizon H∞ problem depending on a small positive parameter. Thus, the firstorder solvability conditions are applicable to this new problem. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix algebraic equation, arising in these conditions, yields a controller solving the original singular H∞ problem. Properties of this controller are studied. The following main notations are applied in the chapter. 1. E n is the n-dimensional real Euclidean space. 2.  ·  denotes the Euclidean norm either of a vector or of a matrix. 3. The superscript “T ” denotes the transposition of a matrix A, ( A T ) or of a vector x, (x T ). 4. L 2 [a, +∞; E n ] is the linear space of n-dimensional vector-valued real functions, square-integrable in the infinite interval [a, +∞), and  ·  L 2 [a,+∞) denotes the norm in this space. 5. On 1 ×n 2 is used for the zero matrix of the dimension n 1 × n 2 , excepting the cases where the dimension of the zero matrix is obvious. In such cases, the notation 0 is used for the zero matrix. 6. In is the n-dimensional identity matrix. 7. col(x, y), where x ∈ E n , y ∈ E m , denotes the column block-vector of the dimension n + m with the upper block x and the lower block y, i.e., col(x, y) = (x T , y T )T . 8. diag(a1 , a2 , . . . , an ), where ai , (i = 1, 2, . . . , n) are real numbers, is the diagonal matrix with these numbers on the main diagonal.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & Dynamic Game Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8_7

175

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7 Singular Infinite-Horizon H∞ Problem

9. Sn is the set of all symmetric matrices of the dimension n × n. 10. Sn+ is the set of all symmetric positive semi-definite matrices of the dimension n × n. 11. Sn++ is the set of all symmetric positive definite matrices of the dimension n × n.

7.2 Initial Problem Formulation We consider the following controlled system: dζ(t) = Aζ(t) + Bu(t) + Cw(t), t ≥ 0, ζ(0) = 0, dt

(7.1)

W(t) = col{N ζ(t), Mu(t)}, t ≥ 0,

(7.2)

where ζ(t) ∈ E n is a state vector; u(t) ∈ E r , (r ≤ n) is a control; w(t) ∈ E s is a disturbance; W(t) ∈ E p is an output and A, B, C, N and M are given constant matrices of the dimensions n × n, n × r , n × s, p1 × n and p2 × r , ( p1 + p2 = p, r ≤ p2 ), respectively. Based on the assumption that w(t) ∈ L 2 [0, +∞; E s ], we consider the following cost functional of the H∞ problem:  2  2 J (u, w) = W(t) L 2 [0,+∞) − γ 2 w(t) L 2 [0,+∞) ,

(7.3)

where γ > 0 is a given constant called the performance level. Consider the matrices D = N T N and G = M T M. Thus, the functional (7.3) can be rewritten in the equivalent form as  J (u, w) =

+∞



 ζ T (t)Dζ(t) + u T (t)Gu(t) − γ 2 w T (t)w(t) dt.

(7.4)

0

In what follows in this chapter, we analyse the case where the matrix M has the form   ¯ O p2 ×(r−q) , 0 < q < r, rank M¯ = q. (7.5) M = M, Due to this form of M, we obtain the block form for the matrix G G=

G¯ 0

0 , 0

¯ G¯ = M¯ T M.

q Moreover, using (7.5) and the inequality r ≤ p2 , we obtain that G¯ ∈ S++ .

(7.6)

7.3

Transformation of the Initially Formulated H∞ Problem

177

Remark 7.1 Due to (7.6), the matrix G is not invertible. The latter means that the results of Sect. 3.2.4 are not applicable to the H∞ problem with the dynamics (7.1) and the cost functional (7.4), i.e., this problem is singular which requires another approach to its solution. H , we denote the set of all functions u =  Definition 7.1 By U u (ζ) : E n → E r such u (ζ) satisfies the local Lipschitz condition; that the following conditions are valid: (i)  u (ζ) and any w(t) ∈ L 2 [0, +∞; E s ] (ii) the initial-value problem (7.1) for u(t) =  has the unique solution  ζuw (t)  in the entire interval t ∈ [0, +∞); (iii) ζuw (t) ∈ H is called u ζuw (t) ∈ L 2 [0, +∞; E r ]. Such a defined set U L 2 [0, +∞; E n ]; (iv)  the set of all admissible controllers for the H∞ problem (7.1), (7.4). H is called the solution of the H∞ problem (7.1), Definition 7.2 Controller u ∗ (ζ) ∈ U (7.4), if it guarantees the fulfilment of the inequality J ( u ∗ , w) ≤ 0

(7.7)

along trajectories of (7.1) with u(t) =  u ∗ (ζ) and any w(t) ∈ L 2 [0, +∞; E s ].

7.3 Transformation of the Initially Formulated H∞ Problem   Let us partition the matrix B into blocks as B = B1 , B2 , where the matrices B1 and B2 have the dimensions n × q and n × (r − q), respectively. In what follows of this chapter, we assume the following: Bhas full column rank r . A7.I. The matrix  A7.II. det B2T DB2 = 0. Remark 7.2 Note that the assumptions A7.I–A7.II allow to construct and carry out the time-invariant version of the state transformation (3.72) in the system (7.1) and the functional (7.4) as it is presented below. This transformation converts the system (7.1) and the functional (7.4) to a much simpler form for the analysis and solution of the corresponding singular H∞ problem.

c = (Bc , B1 ) is Let Bc be a complement matrix to B (see Definition 3.13). Then, B c , we construct (similarly to (3.71), (3.72)) a complement matrix to B2 . Using B2 , B the following state transformation in the system (7.1) and the functional (7.4):

ζ(t) = Rz(t),



 R = L, B2 ,

(7.8)

−1

 T c − B2 H, H = c and z(t) ∈ E n is a new state vector. where L = B B2 DB2 B2T DB By virtue of Lemma 3.2 and Theorem 3.5, we have the following assertion.

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7 Singular Infinite-Horizon H∞ Problem

Proposition 1 Let the assumptions A7.I and A7.II be valid. Then, the transformation (7.8) converts the system (7.1) and the functional (7.4) to the following system and functional: dz(t) = Az(t) + Bu(t) + Cw(t), dt  J (u, v) =

+∞



t ≥ 0, z(0) = 0,

(7.9)

 z T (t)Dz(t) + u T (t)Gu(t) − γ 2 w T (t)w(t) dt,

(7.10)

A = R−1 AR,

(7.11)

0

where

C = R−1 C,

⎛

O(n−r )×q B = R−1 B = ⎝ Iq H2 D = R DR = T

D1 O(r−q)×(n−r+q)

D1 = LT DL,

⎞ O(n−r )×(r −q) ⎠, Oq×(r −q) Ir −q O(n−r +q)×(r −q) D2

(7.12)

D2 = B2T DB2 ,

,

(7.13)

(7.14)

H2 is the right-hand block of the matrix H of the dimension (r − q) × q, D1 ∈ n−r +q r −q , D2 ∈ S++ . S+ Remark 7.3 Like the H∞ problem (7.1), (7.4), the H∞ problem (7.9)–(7.10) also is singular. We choose the set U H (see Definition 3.11) as the set of all admissible controllers u(z), z ∈ E n , for the H∞ problem (7.9)–(7.10). Also, we keep Definition 3.12 (see Sect. 3.2.4) for the solution u ∗ (z) to this problem. Quite similarly to Lemma 6.1, we have the following assertion. Lemma 7.1 Let the assumptions A7.I–A7.II be valid. If the u ∗ (ζ) solves  controller  ∗ u Rz solves the H∞ problem the H∞ problem (7.1), (7.4), then the controller  ∗ (z) solves the H∞ problem (7.9)–(7.10), (7.9)–(7.10). Vice versa, if the controller u  −1  ∗ then the controller u R ζ solves the H∞ problem (7.1), (7.4). Remark 7.4 Due to Lemma 7.1, the initially formulated H∞ problem (7.1), (7.4) and the new H∞ problem (7.9)–(7.10) are equivalent to each other. On the other hand, due to Proposition 1, the problem (7.9)–(7.10) is simpler than the one (7.1), (7.4). Therefore, in the sequel of this chapter, we analyse the H∞ problem (7.9)–(7.10) as an original one and call it the Singular Infinite-Horizon H∞ Problem (SIHP).

7.4

Regularization of the Singular Infinite-Horizon H∞ Problem

179

7.4 Regularization of the Singular Infinite-Horizon H∞ Problem 7.4.1 Partial Cheap Control Infinite-Horizon H∞ Problem To study the SIHP, we replace it with a regular H∞ problem, which is close in a proper sense to the SIHP. This regular H∞ problem has the same equation of dynamics (7.9). However, the functional in this problem has the form  Jε (u, w) =

+∞



   z T (t)Dz(t) + u T (t) G + (ε) u(t) − γ 2 w T (t)w(t) dt,

0

(7.15)

where ε > 0 is a small parameter, and   (ε) = diag 0, . . . , 0, ε2 , . . . , ε2 .       q

(7.16)

r −q

¯ we and (7.16) and the positive definiteness of G, Eqs. (7.6) Taking into account   directly have that G + (ε) ∈ Sr++ for all ε > 0. Therefore, the control cost in the functional (7.15) is regular for all ε > 0. On the other hand, for ε = 0, the functional (7.15) becomes the functional (7.10) of the SIHP. Remark 7.5 For the H∞ problem with the dynamics (7.9) and the functional (7.15), we chose the same set of admissible controllers   U H as it is for the SIHP. Solution of the H∞ problem (7.9), (7.15) is a controller u ∗ε z ∈ U H which guarantees the fulfilment  of the inequality Jε (u ∗ε , w) ≤ 0 along trajectories of (7.9) with u(t) = u ∗ε z and any w(t) ∈ L 2 [0, +∞; E s ]. The problem (7.9), (7.15) is regular for all ε > 0. Moreover, due to the smallness of the parameter ε and the structure of the matrix G + (ε), this H∞ problem is a partial cheap control problem, i.e., the problem where the cost only of some (but not all) control coordinates is much smaller than the costs of the other control coordinates and a state cost.

7.4.2 Solution of the H∞ Problem (7.9), (7.15) We consider the following Riccati algebraic equation with respect to the unknown n × n-matrix K :   K A + A T K − K Su (ε) − Sw K + D = 0,  −1 T (7.17) Su (ε) = B G + (ε) B , Sw = γ −2 CC T . Let us assume the following:

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7 Singular Infinite-Horizon H∞ Problem

A7.III. For a given ε > 0, Eq. (7.17) has a solution K = K ∗ (ε) ∈ Sn such that 

 (i) the matrix Q K ∗ (ε), ε = D + K ∗ (ε)Su (ε)K ∗ (ε) ∈ Sn++ ; (ii) the trivial solution of the following system is asymptotically stable:  dz(t)  = A − Su (ε)K ∗ (ε) z(t), dt

t ≥ 0.

(7.18)

As a direct consequence of Theorem 3.4, we have the following assertion. Proposition 7.1 Let the assumptions A7.I–A7.III be valid. Then (a) K ∗ (ε) ∈ Sn++ ; (b) the following controller solves the H∞ problem (7.9), (7.15):  −1 u ∗ε (z) = − G + (ε) B T K ∗ (ε)z.

(7.19)

7.5 Asymptotic Analysis of the H∞ Problem (7.9), (7.15) The first step in this analysis is an asymptotic solution of Eq. (7.17).

7.5.1 Transformation of Eq. (7.17) Due to the block form of the matrices B and G and the diagonal form of the matrix (ε) (see Eqs. (7.12), (7.6) and (7.16)), we can represent the matrix Su (ε) (see Eq. (7.17)) in the block form ⎛ Su (ε) = ⎝

Su 1

Su 2

SuT2 (1/ε2 )Su 3 (ε)

⎞ ⎠,

(7.20)

where the (n − r + q) × (n − r + q)-matrix Su 1 , the (n − r + q) × (r − q)-matrix Su 2 and (r − q) × (r − q)-matrix Su 3 (ε) have the form ⎛ Su 1 = ⎝

⎞

0 0 0 G¯ −1

⎠,

⎛ Su 2 = ⎝

0 G¯ −1 H2T

⎞ ⎠,

Su 3 (ε) = ε2 H2 G¯ −1 H2T + Ir −q .

(7.21)

Due to the block form of the matrix Su (ε) and the form of the matrix Su 3 (ε), the left-hand side of the Riccati algebraic equation in (7.17) has the singularity at ε = 0. To remove this singularity, we seek the solution of this equation in the block form

7.5

Asymptotic Analysis of the H∞ Problem (7.9), (7.15)

⎛ K (ε) = ⎝

K 1 (ε) εK 2 (ε) εK 2T (ε)

εK 3 (ε)

181

⎞ ⎠ , K 1T (ε) = K 1 (ε), K 3T (ε) = K 3 (ε),

(7.22)

where the matrices K 1 (ε), K 2 (ε) and K 3 (ε) have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively. We also partition the matrices A and Sw into blocks as ⎛ A=⎝

A1 A2 A3 A4

⎞ ⎠,

⎛ Sw = ⎝

Sw1 Sw2 SwT 2 Sw3

⎞ ⎠,

(7.23)

where the blocks A1 , A2 , A3 and A4 have the dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively; the blocks Sw1 , Sw2 and Sw3 have the form Sw1 = γ −2 C1 C1T , Sw2 = γ −2 C1 C2T and Sw3 = γ −2 C2 C2T and C1 and C2 are the upper and lower blocks of the matrix C of the dimensions (n − r + q) × s and (r − q) × s, respectively. Substitution of Eqs. (7.13), (7.20), (7.22) and (7.23) into the Riccati algebraic equation in (7.17) converts the latter after a routine algebra to the following equivalent set of equations: K 1 (ε)A1 + εK 2 (ε) A3 + A1T K 1 (ε) + εA3T K 2T (ε) −K 1 (ε)(Su 1 − Sw1 )K 1 (ε) − εK 2 (ε)(SuT2 − SwT 2 )K 1 (ε)   −εK 1 (ε)(Su 2 − Sw2 )K 2T (ε) − K 2 (ε) Su 3 (ε) − ε2 Sw3 K 2T (ε) + D1 = 0, K 1 (ε)A2 + εK 2 (ε)A4 + εA1T K 2 (ε) + εA3T K 3 (ε) −εK 1 (ε)(Su 1 − Sw1 )K 2 (ε) − ε2 K 2 (ε)(SuT2 − SwT 2 )K 2 (ε)   −εK 1 (ε)(Su 2 − Sw2 )K 3 (ε) − K 2 (ε) Su 3 (ε) − ε2 Sw3 K 3 (ε) = 0, εK 2T (ε)A2 + εK 3 (ε)A4 + εA2T K 2 (ε) + εA4T K 3 (ε) −ε2 K 2T (ε)(Su 1 − Sw1 )K 2 (ε) − ε2 K 3 (ε)(SuT2 − SwT 2 )K 2 (ε)   −ε2 K 2T (ε)(Su 2 − Sw2 )K 3 (ε) − K 3 (ε) Su 3 (ε) − ε2 Sw3 K 3 (ε) + D2 = 0. (7.24)

7.5.2 Asymptotic Solution of the Set (7.24) First of all, we note the following. The set (7.24) is similar to the set of Eqs. (5.44)– (5.46) (see Sect. 5.6.1), and the asymptotic solution (the zero-order one) of (7.24) is constructed and justified quite similarly to the zero-order asymptotic solution of (5.44)–(5.46) (see Sect. 5.6.2). Therefore, the results of this subsection are presented in a concise form, while all the necessary details can be found in Sect. 5.6.2.

182

7 Singular Infinite-Horizon H∞ Problem

  Equations for the zero-order asymptotic solution K 1,0 , K 2,0 , K 3,0 to the set (7.24) are obtained by setting formally ε = 0 in this set, which yields T + D1 = 0, K 1,0 A1 + A1T K 1,0 − K 1,0 (Su 1 − Sw1 )K 1,0 − K 2,0 K 2,0 K 1,0 A2 − K 2,0 K 3,0 = 0, 2  K 3,0 − D2 = 0.

(7.25)

Eliminating K 3,0 and K 2,0 from (7.25), we obtain the explicit expressions for these matrices  −1/2  1/2 , K 2,0 = K 1,0 A2 D2 (7.26) K 3,0 = D2 and the Riccati algebraic equation for the matrix K 1,0 K 1,0 A1 + A1T K 1,0 − K 1,0 S1,0 K 1,0 + D1 = 0,

(7.27)

S1,0 = Su 1 − Sw1 + A2 D2−1 A2T .

(7.28)

where

In (7.26), the superscript “1/2” denotes the unique symmetric positive definite square root of the corresponding symmetric positive definite matrix, and the superscript “−1/2” denotes the inverse matrix for this square root. Let us assume the following: A7.IV. Equation (7.27) has a solution K 1,0 ∈ Sn−r +q for which the matrix A1 − S1,0 K 1,0 is a Hurwitz matrix. Remark 7.6 Note that the assumption A7.IV is an ε-free assumption, which guarantees the existence of the solution to the set of Eqs. (7.24) for all sufficiently small ε > 0 (see Lemma 7.2 below). Similarly to Lemma 5.3 (see Sect. 5.6.2), we have the following assertion. Lemma 7.2 Let the assumptions A7.I, A7.II and A7.IV be valid. Then, there exists a positive number ε0 such that, for all ε ∈ (0, ε0 ], the set of Eqs. (7.24) has the solution K i (ε) = K i∗ (ε), (i = 1, 2, 3), satisfying the inequalities  ∗   K (ε) − K i,0  ≤ aε, i

(i = 1, 2, 3),

(7.29)

where a > 0 is some constant independent of ε. Consider the following two matrices of the dimension n × n:

K¯ 0 =



K 1,0 0 T K 3,0 K 2,0



, S¯ =



Su 1 0 0 Ir −q

,

(7.30)

7.5

Asymptotic Analysis of the H∞ Problem (7.9), (7.15)

183

where K 1,0 is the solution of the equation (7.27), mentioned in the assumption A7.IV; K 3,0 and K 2,0 are given by (7.26). In the addition to the assumption A7.IV, we assume the following: A7.V. (i) The matrix D + K¯ 0T S¯ K¯ 0 ∈ Sn++ . (ii) The trivial solution of the following system is asymptotically stable:  d x(t)  = A1 − (Su 1 + A2 D2−1 A2T )K 1,0 x(t), dt

t ≥ 0, x(t) ∈ E n−r +q . (7.31)

Remark 7.7 Note that the assumption A7.V, along with the assumption A7.IV, provides the existence of the controller solving the H∞ problem (7.9), (7.15) for all sufficiently small ε > 0 (see Lemma 7.3 and its proof below). Moreover, the assumptions A7.IV–A7.V guarantee the existence of a controller solving the Reduced InfiniteHorizon H∞ Problem (see Sect. 7.6 below). Lemma 7.3 Let the assumptions A7.I–A7.II and A7.IV–A7.V be valid. Then, there exists a positive number ε∗ ≤ ε0 such that, for any ε ∈ (0, ε∗ ], the assumption A7.III is fulfilled and

K ∗ (ε) εK 2∗ (ε)  1∗ T , (7.32) K ∗ (ε) = ε K 2 (ε) εK 3∗ (ε) where K i (ε) = K i∗ (ε), (i = 1, 2, 3), is the solution of the set (7.24) mentioned in Lemma 7.2. Moreover, all the statements of Proposition 7.1 are valid. A7.III, we should show the positive Proof To prove the fulfilmentof the assumption  of definiteness of the matrix Q K ∗ (ε), ε and the asymptotic stability  the trivial  solution to the system (7.18). Let us start with the matrix Q K ∗ (ε), ε . Using the form of this matrix (see the item (i) of the assumption A7.III), as well as Eqs. (7.20), (7.21), (7.30) and (7.32) and Lemma 7.2, we obtain after a routine calculation the following limit equality:   lim+ Q K ∗ (ε), ε = D + K¯ 0T S¯ K¯ 0 ∈ Sn++ ,

ε→0

(7.33)

  meaning that Q K ∗ (ε), ε ∈ Sn++ for any sufficiently small ε > 0. Proceed to the system (7.18). To prove the asymptotic stability of the trivial solution to this system for any sufficiently small ε > 0, it is sufficient to show that the matrix of the coefficients of (7.18) is a Hurwitz matrix. Using Eqs. (7.20) and (7.32), we can represent this matrix in the block form

184

7 Singular Infinite-Horizon H∞ Problem



ψ1 (ε) ψ2 (ε) , 1 ψ (ε) 1ε ψ4 (ε) ε 3 T  ψ1 (ε) = A1 − Su 1 K 1∗ (ε) − εSu 2 K 2∗ (ε) , ψ2 (ε) = A2 − εSu 1 K 2∗ (ε) − εSu 2 K 3∗ (ε),  T ψ3 (ε) = εA3 − εSuT2 K 1∗ (ε) − Su 3 (ε) K 2∗ (ε) ,

ψ(ε) = A − Su (ε)K ∗ (ε) =

ψ4 (ε) = εA4 − εSuT2 K 2∗ (ε) − Su 3 (ε)K 3∗ (ε). (7.34) Similarly to the proof of Lemma 5.3 and Corollary 5.1 (see Sect. 5.6.2), to prove that the block matrix in (7.34) is a Hurwitz one for any sufficiently small ε > 0, it is sufficient to show (by virtue of Theorem 2.1 in [2]) that the matrices ψ4 (0) and 

 ω = ψ1 (ε) − ψ2 (ε)ψ4−1 (ε)ψ3 (ε) |ε=0 are Hurwitz matrices. Using Lemma 7.2, Eqs. (7.26) and (7.34) and the equality Su 3 (0) = Ir−q , we obtain ψ4 (0) = −(D2 )1/2 and ω = A1 − (Su 1 + A2 D2−1 A2T )K 1,0 , meaning that both matrices are Hurwitz matrices. Thus, the trivial solution to the system (7.18) is asymptotically stable for any sufficiently small ε > 0. Hence, there exists a positive number ε∗ ≤ ε0 such that, for any ε ∈ (0, ε∗ ], the assumption A7.III is fulfilled. Moreover, due to the fulfilment of this assumption, all the statements of Proposition 7.1 are valid for any ε ∈ (0, ε∗ ]. ⃞ Remark 7.8 The ε-free conditions of Lemma 7.3 provide the solvability of the εdependent H∞ problem (7.9), (7.15) for all ε ∈ (0, ε∗ ], i.e., robustly with respect to ε ∈ (0, ε∗ ].

7.6 Reduced Infinite-Horizon H∞ Problem Similarly to Proposition 5.3 (see Sect. 5.7), we can represent the matrix S1,0 , given by Eq. (7.28), in the form T − Sw1 , S1,0 = B1,0 −1 B1,0

(7.35)

where   B1,0 =  B= B, A2 , 



O(n−r)×q Iq



, =

G¯ Oq×(r −q) O(r −q)×q D2

.

(7.36)

Since G¯ ∈ S++ and D2 ∈ S++ , the matrix  ∈∈ Sr++ . Using the matrices B1,0 and , we consider the H∞ problem for the following system and functional: q

r−q

d xr (t) = A1 xr (t) + B1,0 u r (t) + C1 w(t), t ≥ 0, xr (0) = 0, dt

(7.37)

7.7

Controller for the SIHP



+∞

Jr (u r , w) = 0

185



 xrT (t)D1 xr (t) + u rT (t)u r − γ 2 w T (t)w(t) dt,

(7.38)

where xr (t) ∈ E n−r +q is a state variable, u r (t) ∈ E r is a control and w(t) ∈ E s is a disturbance. We call the problem (7.37)–(7.38) the Reduced Infinite-Horizon H∞ Problem (RIHP). The set of the admissible controllers and the solution of the RIHP are defined similarly to those notions in Sect. 3.2.4 (see Definitions 3.11–3.12). To derive the controller, solving the RIHP, let us observe the following. Using Eqs. (7.13), (7.26), (7.30) and (7.36), we directly obtain that the upper left-hand block of the matrix D + K¯ 0 S¯ K¯ 0T of the dimension (n − r + q) × (n − r + q) can be represented in the T . Since (D + K¯ 0 S¯ K¯ 0T ) ∈ Sn++ (see the assumption A7.V), form D1 + B1,0 −1 B1,0 n−r +q −1 T then (D1 + B1,0  B1,0 ) ∈ S++ . Using this observation, as well as the positive definiteness of the matrix  and Theorem 3.4 (see Sect. 3.2.4), we obtain the following assertion. Proposition 7.2 Let the assumptions A7.I–A7.II and A7.IV–A7.V be valid. Then (a) the solution K 1,0 of Eq. (7.27), mentioned in the assumptions A7.IV–A7.V, is a positive definite matrix; (b) the controller T K 1,0 xr (7.39) u ∗r (xr ) = −−1 B1,0 solves the RIHP. This controller can be represented in the block form u ∗r (xr ) =



u ∗r,1 (xr ) , u ∗r,2 (xr )

u ∗r,1 (xr ) = −G¯ −1  B T K 1,0 xr , u ∗r,2 (xr ) = −D2−1 A2T K 1,0 xr . (7.40)

7.7 Controller for the SIHP 7.7.1 Formal Design of the Controller Lemma 7.4 Let the assumptions A7.I–A7.II and A7.IV–A7.V be valid. Then, for all ε ∈ (0, ε∗ ], the controller (7.19), solving the H∞ problem (7.9), (7.15), solves also the SIHP (7.9)–(7.10). Proof First of all, let us observe that both problems have the same set of the admissible controllers U H . Now, comparing the functionals (7.10) and (7.15) of these problems, we directly obtain the inequality     J u(z), w(t) ≤ Jε u(z), w(t) ,

∀u(z) ∈ U H , w(t) ∈ L 2 [0, +∞; E s ]. (7.41)

186

7 Singular Infinite-Horizon H∞ Problem

Due to Lemma 7.3, the controller (7.19) is admissible in the H∞ problem (7.9), (7.15) for all ε ∈ (0, ε∗ ], meaning that this controller is admissible in the SIHP for ∗ since the controller (7.19) solves the problem (7.9), (7.15), all ε ∈ (0,  ∗ε ]. Moreover,  then Jε u ε (z), w(t) ≤ 0 for all ε ∈ (0, ε∗ ] and w(t) ∈ L 2 [0, +∞; E s ]. The latter, along with the inequality (7.41), yields immediately the statement of the lemma. ⃞ Remark 7.9 The design of the controller (7.19) is a complicated task, because it requires the solution of the high dimension ε-dependent nonlinear algebraic matrix equation (7.17). To overcome this difficulty, in the sequel of this section, we design another (simplified) controller for the SIHP, and we study properties of this controller. Let us partition the vector z ∈ E n into blocks as   z = col x, y , x ∈ E n−r +q , y ∈ E r −q ,

(7.42)

and consider the matrix K 0 (ε) =

εK 2,0 εK 3,0

K 1,0 T εK 2,0

,

ε ∈ (0, ε∗ ].

(7.43)

Replacing the matrix K (ε) with the matrix K 0 (ε) in the controller u ∗ε (z) (see Eq. (7.19)), we obtain the following auxiliary controller:  −1 u aux (z) = − G + (ε) B T K 0 (ε)z. Substitution of the block representations for the matrices G, B, (ε) and K 0 (ε) and the vector z (see Eqs. (7.6), (7.12), (7.16), (7.43) and (7.42)) into this expression for B (see Eq. (7.36)) yield after a u aux (z) and the use of the block form of the matrix  routine algebra the block representation for u aux (z) u aux (z) = −

T B T K 1,0 + εH2T K 2,0 , Q 1 (ε) = 

 

G¯ −1 Q 1 (ε)x + εQ 2 (ε)y  , 1 T K 2,0 x + K 3,0 y ε

Q 2 (ε) =  B T K 2,0 + εH2T K 3,0 .

(7.44)

Calculating the point-wise (with respect to (z, t) ∈ E n × [0, +∞)) limit of the upper block in (7.44) for ε → 0+ , we obtain the simplified controller for the SIHP ⎛ u ∗ε,0 (z) = ⎝

⎞

−G¯ −1  B T K 1,0 x − 1ε



T K 2,0 x

+ K 3,0 y



⎠.

(7.45)

Remark 7.10 Comparing the simplified controller u ∗ε,0 (z) with the controller u ∗r (xr ), which solves the RIHP (see Eq. (7.40)), we can see that the upper blocks of both controllers coincide with each other. Thus, to construct the controller u ∗ε,0 (z), one

7.7

Controller for the SIHP

187

has to solve the lower dimension regular H∞ problem (the RIHP) and calculate two gain matrices K 2,0 and K 3,0 using Eq. (7.26).

7.7.2 Properties of the Simplified Controller (7.45) Consider the block matrix



 T , K 3,0 . Q 0 = K 2,0

(7.46)

Theorem 7.1 Let the assumptions A7.I–A7.II and A7.IV–A7.V be valid. Then, there exists a positive number ε∗1 ≤ ε∗ such that, for any ε ∈ (0, ε∗1 ], the simplified controller u ∗ε,0 (z) is an admissible controller in the SIHP, i.e., u ∗ε,0 (z) ∈ U H . Moreover, for all ε ∈ (0, ε∗1 ] and w(t) ∈ L 2 [0, +∞; E s ], the following inequality is satisfied:   J u ∗ε,0 (z), w(t) ≤ −



+∞ 0



 T   ∗ ∗ z ε,0 t; w(·) Q 0T Q 0 z ε,0 t; w(·) dt,

(7.47)

  ∗ where z ε,0 t; w(·) , t ∈ [0, +∞), is the solution of the initial-value problem (7.9) with u(t) = u ∗ε,0 (z). Proof of the theorem is presented in Sect. 7.7.3. Remark 7.11 Due to Theorem 7.1, the controller u ∗ε,0 (z) solves the original SIHP (7.9)–(7.10). Theorem 7.2 Let the assumptions A7.I–A7.II and A7.VI–A7.V be valid. Then, there exist positive numbers a and ε∗2 ≤ ε∗1 such that, for all ε ∈ (0, ε∗2 ] and w(t) ∈ L 2 [0, +∞; E s ], the integral in the right-hand side of (7.47), being nonnegative, satisfies the inequality 

+∞ 0



 2  T   ∗ ∗ z ε,0 t; w(·) Q 0T Q 0 z ε,0 t; w(·) dt ≤ aε w(t) L 2 [0,+∞) .

(7.48)

Proof of the theorem is presented in Sect. 7.7.4. Remark 7.12 If q = 0, i.e., all the coordinates of the control in the SIHP (7.9)– (7.10) are singular, then Su 1 = O(n−r )×(n−r ) , Su 2 = O(n−r )×r , Su 3 = Ir and S1,0 = A2 D2−1 A2T − Sw1 . The matrix K 1,0 is the solution of the Riccati algebraic equation (7.27) with this S1,0 . The upper block of the simplified controller u ∗ε,0 (z) (see Eq. (7.45)) vanishes, while Thus, in this case,  T the lower block remains unchanged. x + K 3,0 y , z = col(x, y), x ∈ E n−r , y ∈ E r . Theorems we have u ∗ε,0 (z) = − 1ε K 2,0 7.1–7.2 and Remark 7.11 are valid with this particular form of u ∗ε,0 (z) and the aforementioned K 1,0 .

188

7 Singular Infinite-Horizon H∞ Problem

7.7.3 Proof of Theorem 7.1 First of all, we rewrite the expression (7.45) for u ∗ε,0 (z) as

u ∗ε,0 (z) = −(ε) K¯ 0 z, (ε) =



G¯ −1  Oq×(r −q) BT O(r −q)×(n−r +q) 1ε Ir −q

,

(7.49)

where K¯ 0 is given in (7.30); z = col(x, y). Substituting (7.49) instead of u(t) into the system (7.9) and the functional (7.15), as well as using the block representations for the matrices G, (ε) and S¯ (see Eqs. (7.6), (7.16) and (7.30)), we obtain after a routine matrix algebra the following system and functional: dz(t)  = A(ε)z(t) + Cw(t), t ≥ 0, z(0) = 0, dt





Jε u ∗ε,0 (z), w(t)



+∞

=



  − γ 2 w T (t)w(t) dt, z T (t) Dz(t)

(7.50)

(7.51)

0

where    = D + K¯ 0T  T (ε) G + (ε) (ε) K¯ 0 = D + K¯ 0T S¯ K¯ 0 .  = A − B(ε) K¯ 0 , D A(ε) (7.52) Using the block form of the matrices B, D, A, K¯ 0 , S¯ and (ε) (see Eqs. (7.12), (7.13),  (7.23), (7.30) and (7.49)) and taking into account (7.26) and (7.36), the matrices A(ε)  and D can be represented in block form as



  2 1 A A  = D1T D2 , , D 3 (ε) (1/ε) A 4 (ε) 2 D 3 (1/ε) A D 2 = A2 , A 3 (ε) = εA3 − εH2 G¯ −1  1 = A1 −  B G¯ −1  B T K 1,0 , A B T K 1,0 A  −1/2 T  1/2 4 (ε) = εA4 − D2 − D2 A2 K 1,0 , A , −1 T    D1 = D1 + K 1,0 B1,0  B1,0 K 1,0 , D2 = K 1,0 A2 , D3 = 2D2 .  = A(ε)



(7.53) For the sake of the further analysis, let us observe the following. Using the block  it is shown (quite similarly to the proof of Lemma 7.3) the form of the matrix A(ε),  ˆ the matrix A(ε) existence of a positive number εˆ ≤ ε∗ such that, for any ε ∈ (0, ε], is a Hurwitz matrix. Using the matrices (7.52), consider the Riccati algebraic equation with respect to  the matrix K + K Sw K + D  = 0.  A(ε)  +A T (ε) K (7.54) K

7.7

Controller for the SIHP

189

The further proof consists of three stages. ˆ Eq. (7.54) has a Stage 1. At this stage, we assume that, for a given ε ∈ (0, ε], = K (ε) ∈ Sn . Based on this assumption, we prove the admissibility of solution K   the controller u ∗ε,0 z in the SIHP.  is a Hurwitz matrix for any ε ∈ (0, ε], Using the above made observation that A(ε) ˆ (ε) we can represent (similarly to the Eqs. (3.61)–(3.62) in Sect. 3.2.4) the solution K to (7.54) as (ε) = K



+∞

 T      (ε)χ D + K (ε)Sw K  (ε) exp A(ε)χ exp A dχ.

(7.55)

0

 is positive definite. The Due to Eq. (7.52) and the assumption A7.V, the matrix D (ε). latter, along with (7.55), directly implies the positive definiteness of the matrix K T  n  Consider the Lyapunov-like function V (z) = z K (ε)z, z ∈ E . Also let us note that, due to the linear dependence of u ∗ε,0 (z) on z with the constant gain   ∗ t; w(·) , t ≥ 0, of the initial-value problem (7.9) with matrix, the solution z ε,0 u(t) = u ∗ε,0 (z) and any given w(t) ∈ L 2 [0, +∞; E s ] exists and is unique. Let   

∗ ∗ ∗ ε,0  z ε,0 ε,0 t; w(·) . Then, differentiating V (t) = V (t) and using Eq. (7.54), we V obtain after routine algebraic transformations ∗   ε,0 T ∗   dV (t) ∗  ε,0 t; w(·) + γ 2 w T (t)w(t) t; w(·) = − z ε,0 Dz dt  T   ε,0 (t) w(t) − w ε,0 (t) , t ≥ 0, −γ 2 w(t) − w

(7.56)

  ∗ (ε)z ε,0 t; w(·) . Using Eq. (7.56), we derive similarly to where w ε,0 (t) = γ −2 C T K the proof of Theorem 3.4 (see Eqs. (3.64)–(3.67) in Sect. 3.2.4) the following inequality:  t 0

 2  T ∗   ∗  ε,0 σ; w(·) dσ ≤ γ 2 w(t) L 2 [0,+∞) , σ; w(·) z ε,0 Dz

t ≥ 0. (7.57)

 ∈ Sn++ , the integral in the left-hand side of the inequality (7.57) is a nonSince D decreasing function of t ∈ [0, +∞). Therefore, the integral 

+∞ 0



  T ∗   ∗  ε,0 σ; w(·) dσ σ; w(·) z ε,0 Dz

(7.58)

  ∗ converges, meaning that z ε,0 t; w(·) ∈ L 2 [0, +∞; E n ]. The latter, along with Eq.    ∗ t; w(·) ∈ L 2 [0, +∞; E r ]. Thus, due to Definition 3.11 (7.49), yields u ∗ε,0 z ε,0 (see Sect. 3.2.4), u ∗ε,0 (z) ∈ U H . (ε) ∈ Sn to Eq. Stage 2. At this stage, we prove the existence of a solution K (7.54) for all sufficiently small ε > 0. Similarly to Sect. 7.5.1 (see Eq. (7.22)), we look for such a solution in the block form

190

7 Singular Infinite-Horizon H∞ Problem

⎛ (ε) = ⎝ K

1 (ε) ε K 2 (ε) K 2T (ε) ε K 3 (ε) εK

⎞ 1T (ε) = K 1 (ε), K 3T (ε) = K 3 (ε), ⎠, K

(7.59)

1 (ε), K 2 (ε) and K 3 (ε) are of the dimensions (n − r + q) × where the matrices K (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively.   and K (ε) D, Substituting the block representations for the matrices Sw , A(ε), (see Eqs. (7.23), (7.53) and (7.59)) into (7.54), we obtain after a routine algebra the following equivalent set of equations: 1 + K 2 (ε) A 3 (ε) + A 1T K 3T (ε) K 2T (ε) + K 1 (ε)Sw1 K 1 (ε) + A 1 (ε) 1 (ε) A K T  T 2 T      +ε K 2 (ε)Sw2 K 1 (ε) + ε K 1 (ε)Sw2 K 2 (ε) + ε K 2 (ε)Sw3 K 2 (ε) + D1 = 0, 2 + K 2 (ε) A 4 (ε) + ε A 1T K 3T (ε) K 3 (ε) + ε K 1 (ε)Sw1 K 1 (ε) A 2 (ε) + A 2 (ε) K 1 (ε)Sw2 K 2 = 0, 2 (ε)SwT K 2 (ε) + ε K 3 (ε) + ε2 K 2 (ε)Sw3 K 3 (ε) + D +ε2 K 2

2 + K 3 (ε) A 4 (ε) + ε A 2T K 4T (ε) K 3 (ε) + ε2 K 2T (ε) A 2 (ε) + A 2T (ε)Sw1 K 2 (ε) εK 3 = 0. 3 (ε)SwT K 2 (ε) + ε2 K 2T (ε)Sw2 K 3 (ε) + ε2 K 3 (ε)Sw3 K 3 (ε) + D +ε2 K 2 (7.60) Similarly to Sect. 7.5.2, we obtain the  following set of equations for the zero-order 2,0 , K 3,0 to the set (7.60): 1,0 , K asymptotic solution K T 1,0 A 2,0 A 1T K 3T (0) K 2,0 1,0 Sw1 K 1 = 0, 1 + K 3 (0) + A 1,0 + A 1,0 + D K +K 2,0 A 3T (0) K 3,0 + D 2 = 0, 2 + K 4 (0) + A 1,0 A K T 4 (0) K 3,0 + D 3 = 0. 4 (0) + A 3,0 A K

(7.61) 2,0 from this set and taking into account (7.28) and (7.53), 3,0 and K Eliminating K we obtain the explicit expressions for these matrices   3,0 = D2 1/2 , K

  2,0 = K 1,0 A2 D2 −1/2 K

(7.62)

1,0 and the Riccati algebraic equation for the matrix K        1,0 1,0 A1 − S1,0 + Sw1 K 10 + A1T − K 10 S1,0 + Sw1 K K   1,0 Sw1 K 1,0 + D1 + K 1,0 S1,0 + Sw1 K 1,0 = 0. +K

(7.63)

1,0 = K 1,0 , where K 1,0 ∈ Sn−r +q is the solution of the The latter has the solution K Eq. (7.27), mentioned in the assumptions A7.IV and A7.V. 1,0 = K 1,0 , we obtain (simiNow, based on Eqs. (7.61)–(7.63) and the equality K larly to Lemmas 7.2–7.3) the existence of a positive number ε∗1 ≤ εˆ such that, for all ε ∈ (0, ε∗1 ], Eq. (7.54) has the symmetric solution of the block form (7.59), where

7.7



Controller for the SIHP

191

 1 (ε), K 2 (ε), K 3 (ε) is the solution of the set (7.60) satisfying the inequality K   K i,0  ≤ aε, i (ε) − K

i = 1, 2, 3,

(7.64)

a > 0 is some constant independent of ε. This result, along with the result of Stage 1 of the proof, yields the first statement of the theorem. Stage 3. At this stage, the inequality (7.47) will be proven. Using the equation (7.51), the inequality (7.57) and the existence of the integral (7.58) directly yields the inequality     Jε u ∗ε,0 z(t) , w(t) ≤ 0

∀ε ∈ (0, ε∗1 ], w(t) ∈ L 2 [0, +∞; E s ].

(7.65)

  ∗ Now, using Eqs. (7.10), (7.15), (7.16), (7.45) and (7.46) and the inclusion z ε,0 t; w(·) ∈ L 2 [0, +∞; E n ], we obtain     Jε u ∗ε,0 (z), w(t) = J u ∗ε,0 (z), w(t)  +∞  ∗  T   ∗ z ε,0 t; w(·) Q 0T Q 0 z ε,0 t; w(·) dt. + 0

This equality, along with the inequality (7.65), yields immediately the inequality (7.47). This completes the proof of Theorem 7.1.

7.7.4 Proof of Theorem 7.2  Consider the Riccati algebraic equation with respect to the matrix K + 1 K  A(ε)  +A CC T K T (ε) K  + Q 0T Q 0 = 0, K aε

(7.66)

 is given in (7.52)–(7.53); where the matrix Q 0 is given by (7.46); the matrix A(ε) ε ∈ (0, ε∗1 ]; a > 0 is some constant independent of ε. Below, a proper condition will be imposed on the constant a. = K (ε) ∈ Let us assume that, for a given ε ∈ (0, ε∗1 ], Eq. (7.66) has a solution K n S . Based on this assumption, it is proven (quite similarly to the inequality (7.57)) the validity of the inequality 

t 0



 2  T   ∗ ∗ z ε,0 t; w(·) Q 0T Q 0 z ε,0 t; w(·) dt ≤ aε w(t) L 2 [0,+∞) , t ≥ 0,

which directly yields the inequality (7.48) for the above-mentioned ε. Thus, to com(ε) ∈ Sn plete the proof of the theorem, we should prove the existence of a solution K

192

7 Singular Infinite-Horizon H∞ Problem

to Eq. (7.66) for all sufficiently small ε > 0. We look for such a solution in the block form ⎛√ ⎞ 1 (ε) ε K 2 (ε) εK 1T (ε) = K 1 (ε), K 3T (ε) = K 3 (ε), (7.67) (ε) = ⎝ ⎠, K K T   ε K 2 (ε) ε K 3 (ε) 1 (ε), K 2 (ε) and K 3 (ε) are of the dimensions (n − r + q) × where the matrices K (n − r + q), (n − r + q) × (r − q) and (r − q) × (r − q), respectively.

C1  and , Q 0 , A(ε) Substituting the block representations for the matrices C = C2 (ε) into (7.66), we obtain after a routine algebra the following equivalent set of K equations: √

√ T 1 (ε) A 1 + K 2 (ε) A 3 (ε) + ε A 1 K 3T (ε) K 2T (ε) 1 (ε) + A εK √ 1 (ε)C1 C1T K 2 (ε)C2 C1T K 1 (ε) 1 (ε) + (1/a) ε K +(1/a) K √ T 1 (ε)C1 C2T K 2 (ε)C2 C2T K 2T (ε) + (1/a)ε K 2T (ε) + K 2,0 K 2,0 = 0, +(1/a) ε K √ T  T        ε K 1 (ε) A2 + K 2 (ε) A4 (ε) + ε A1 K 2 (ε) + A3 (ε) K 3 (ε) √ 1 (ε)C1 C1T K 2 (ε)C2 C1T K 2 (ε) + (1/a)ε K 2 (ε) +(1/a) ε K √ T T 1 (ε)C1 C2 K 2 (ε)C2 C2 K 3 (ε) + (1/a)ε K 3 (ε) + K 2,0 K 3,0 = 0, +(1/a) ε K

2 + K 3 (ε) A 4 (ε) + ε A 2T K 4T (ε) K 3 (ε) 2 (ε) + A 2T (ε) A εK 2T (ε)C1 C1T K 3 (ε)C2 C1T K 2 (ε) + (1/a)ε K 2 (ε) +(1/a)ε K T T T 2 3 (ε)C2 C2 K 3 (ε) + (1/a)ε K 3 (ε) + K 3,0 = 0. 2 (ε)C1 C2 K +(1/a)ε K (7.68)

1,0 , K 2,0 , K 3,0 } with respect Let us construct the zero-order asymptotic solution { K √ (7.68). The equations for the terms of this solution are obtained by to ε of the set √ setting formally ε = 0 in the equations of this set. Thus, taking into account (7.26) and (7.53), we have T T T 1,0 C1 C1T K 2,0 1,0 + K 2,0 K 2,0 2,0 K 2,0 − K 2,0 K + (1/a) K = 0, −K   − K 2,0 K 3,0 − K 2,0 K 3,0 + K 2,0 K 3,0 = 0, 2 3,0 + K 3,0 3,0 K 3,0 − K 3,0 K = 0. −K

(7.69)

It is directly verified that the following matrices satisfy the set (7.69): 1,0 = 0, K

2,0 = 0.5K 2,0 , K

3,0 = 0.5K 3,0 . K

(7.70)

Now, we are going to show the existence of a solution to the set (7.68) by justification of the asymptotic solution (7.70) to this set. We make the following transformation of the variables in the set (7.68):

7.7

Controller for the SIHP

193

i,0 +  i (ε) = K δi (ε), K

i = 1, 2, 3,

where  δi (ε), (i = 1, 2, 3), are new unknown matrices. Consider the block-form matrix √

ε δ1 (ε) ε δ2 (ε)  δ(ε) = . δ3 (ε) ε δ2T (ε) ε

(7.71)

(7.72)

Substitution of (7.71) into the set (7.68) and the use of Eqs. (7.69)–(7.70) and the  and K (ε) yield after a routine algebra the block form of the matrices C, Q 0 , A(ε)  equation for δ(ε) 1  δ(ε) δ(ε) +  χ(ε) = 0, α(ε) +  αT (ε) δ(ε) +  δ(ε)CC T  aε

(7.73)

where  + 1 CC T K 0 (ε),  α(ε) = A(ε) aε

0 (ε) = K

√

1,0 ε K 2,0 εK T 2,0 3,0 ; εK εK

0 (ε), C and A(ε),  the matrix  χ(ε) is expressed in a known form by the matrices K and it satisfies the inequality  χ(ε) ≤ b˜1 ε,

ε ∈ (0, ε˜1 ],

(7.74)

b˜1 is some positive number independent of ε; ε˜1 > 0 is some sufficiently small number. Now, let us analyse the matrix  α(ε). First of all, taking into account (7.70), we represent this matrix in the block form as

 α2 T 1 + 1 C1 C2T K 2,0 ,  α(ε) = ,  α1 = A 1  α (ε) 2a 4 ε T 2 + 1 C1 (C1T K 2,0 + C2T K 3,0 ),  3 (ε) + ε C2 C2T K 2,0  α2 = A α3 (ε) = A , 2a 2a 4 (ε) + ε C2 (C1T K 2,0 + C2T K 3,0 ).  α4 (ε) = A 2a (7.75)  α1 1  α(ε) ε

α(ε) is a Hurwitz matrix for some given constant a > 0 and Let us show that  all sufficiently small ε > 0. Similarly to the proofs of Lemma 5.3, Corollary 5.1 (see Sect. 5.6.2) and Lemma 7.3, to prove that the block matrix in (7.75) is a Hurwitz one for a given a > 0 and any sufficiently small ε > 0, it is sufficient

ω (a) = virtue of Theorem 2.1 in [2]) that the matrices  α4 (0) and  to  show (by α2  α4−1 (ε) α3 (ε) |ε=0 are Hurwitz matrices. Using (7.53), we obtain after a  α1 −  ω (a) = A1 − (Su1 + A2 D2−1 A2T )K 1,0 − direct calculation that  α4 (0) = −(D2 )1/2 , 

194

7 Singular Infinite-Horizon H∞ Problem

−1 T K 2,0 K 3,0 K 2,0 . Thus,  α4 (0) is a Hurwitz matrix. Proceed to the matrix  ω (a). −1 T ω (a) = A1 − (Su1 + A2 D2 A2 )K 1,0 . Hence, by virtue of the We have lima→+∞  assumption A7.V, this limit matrix is a Hurwitz matrix. The latter means the exisa such that, for all a >  a , the matrix  ω (a) is a Hurwitz tence of a positive number  a , there exists a matrix. Therefore, due to the results of [2], for a given constant a >  α(ε) is a Hurwitz matrix for all ε ∈ (0, ε˜2 (a)]. positive number ε˜2 (a) ≤ ε˜1 such that  Based on this proven fact and using the results of [1], we can rewrite Eq. (7.73) with a in the equivalent form a given a >  1 C CT 2a 1 1

 δ(ε) =



+∞ 0

  1 (σ, ε)dσ, ε ∈ (0, ε˜2 (a)],  T (σ, ε)  δ(ε)  χ(ε) +  δ(ε)CC T   aε (7.76)

(σ, ε) is the where, for any given ε ∈ (0, ε˜2 (a)], the n × n-matrix-valued function  unique solution of the problem (σ, ε) d (σ, ε),  (0, ε) = In , σ ≥ 0. = α(ε) dσ

(7.77)

2 (σ, ε),  3 (σ, ε) and  4 (σ, ε) be the upper left-hand, upper right1 (σ, ε),  Let  (σ, ε) of the hand, lower left-hand and lower right-hand blocks of the matrix  dimensions (n − r + q) × (n − r + q), (n − r + q) × (r − q), (r − q) × (n − r + q) and (r − q) × (r − q), respectively. By virtue of the results of [2] (Theorem 2.3), we have the following estimates of these blocks:      2 (σ, ε) ≤ b˜2 ε exp(−β˜1 σ), k (σ, ε) ≤ b˜2 exp(−β˜1 σ), k = 1, 3,        4 (σ, ε) ≤ b˜2 ε exp(−β˜1 σ) + exp − β˜2 σ/ε , σ ≥ 0, ε ∈ (0, ε˜3 ], (7.78) where b˜2 , β˜1 and β˜2 are some positive numbers independent of ε; 0 < ε˜3 ≤ ε˜2 (a) is some sufficiently small number. Applying the method of successive  +∞approximations to Eq. (7.76), let us consider δ j (ε) j=0 given as the sequence of the matrices   δ j+1 (ε) =



+∞ 0

  1 (σ, ε)dσ,  T (σ, ε)  δ j (ε)  χ(ε) +  δ j (ε)CC T   aε

(7.79)

δ0 (ε) = 0; the matrices  δ j (ε) have the block where ( j = 0, 1, . . . , ), ε ∈ (0, ε˜3 ];  form

√ ε δ j,1 (ε) ε δ j,2 (ε)  , j = 1, 2, . . . , δ j (ε) = ε δ Tj,2 (ε) ε δ j,3 (ε) and the dimensions of the blocks in each of these matrices are the same as the δ j (ε), dimensions of the corresponding blocks in (7.72). Using the block form of 

7.8

Examples

195

( j = 1, 2, . . .), as well as using the inequalities (7.74) and (7.78), we obtain the existence of a positive number ε∗2 ≤ ε˜3 such that, for any ε ∈ (0, ε∗2 ], the sequence +∞   δ j (ε) j=0 converges in the linear space of n × n-matrices. Moreover, the following inequalities are fulfilled:   √  δ j,1 (ε) ≤ b˜3 ε,

   δ j,i (ε) ≤ b˜3 ε,

i = 2, 3,

j = 1, 2, . . . ,

where b˜3 > 0 is some number independent of ε, j and i. Thus, for any ε ∈ (0, ε∗2 ],

 δ(ε) = lim  δ j (ε) j→+∞

is a symmetric solution of Eq. (7.76)  therefore,  (7.73). Moreover, this  and, √  of Eq. δi (ε) ≤ b˜3 ε, (i = 2, 3). The δ1 (ε) ≤ b˜3 ε,  solution satisfies the inequalities  existence of the solution to Eq. (7.73), along with Eq. (7.71), proves the existence (ε) ∈ Sn to Eq. (7.66) with a given a >  a for all ε ∈ (0, ε∗2 ]. Thus, of the solution K the theorem is proven.

7.8 Examples 7.8.1 Example 1 To illustrate the theoretical results of this chapter, we consider a particular case of the SIHP (7.9)–(7.10) with the data:



1 0 0 2 , , B= n = r = s = 2, q = 1, A = 2 1 6 1



1 0 2 0 1 2 . , G= , D= C= 0 0 0 1 4 3

(7.80)

√

Here, we take the performance level γ = 25 . Using this value of γ, the data (7.80), as well as the expression for Su 1 , the expression for Sw , the block form of Sw and the expression for S1,0 (see Eqs. (7.21), (7.17), (7.23) and (7.28)), we obtain Su 1 = 1,

Sw1 = 4,

S1,0 = 1.

2 Due to these values and the data (7.80), Eq. (7.27) becomes as −K 1,0 + 2 = 0, √ √ √ yielding two solutions K 1,0 = 2 and K 1,0 = − 2. The first solution K 1,0 = 2 satisfies the assumption A7.IV. Let us find out whether this solution satisfies the assumption A7.V. Using this solution and (7.26), we have

196

7 Singular Infinite-Horizon H∞ Problem

Table 7.1 Minimum H∞ performance level ε γε∗

0.04 1.1324

0.03 1.1218

0.02626 √ 5/2 ∼ = 1.11803

√ K 2,0 = 2 2,

0.02 1.1121

0.01 1.1034

K 3,0 = 1.

Then, using these K i,0 , (i = 1, 2, 3) and (7.30), we obtain √

2 0 ¯ √ , S¯ = I2 . K0 = 2 2 1 Finally, we calculate the matrix D + K¯ 0T S¯ K¯ 0 =



√

12√ 2 2 . 2 2 1

√ the This matrix is positive definite. Hence, K 1,0 = 2 satisfies the item (i) of √ assumption A7.V. The fulfilment of the item (ii) of this assumption with K 1,0 = 2 directly follows from √ the data (7.80) and the following calculation: A1 − (Su 1 + A22 /D2 )K 1,0 = −5 2. Now, using the above calculated values K i,0 , (i = 1, 2, 3) and Eq. (7.45), we derive the controller √



− 2x√ x  u ∗ε,0 (z) = , , z = y − 1ε (2 2x + y √

which solves the SIHP (7.9)–(7.10) with the data (7.80) and γ = 25 for all sufficiently small ε > 0. In Table 7.1, the minimum H∞ performance level γε∗ , for which the above derived controller u ∗ε,0 (z) solves the SIHP (7.9)–(7.10) with the data (7.80) and γ = γε∗ , is presented for various values of ε. It is seen that γε∗ decreases for the decreasing ε > 0. Moreover, ε = 0.02626 is the√largest value of ε, for which γε∗ does not exceed the given performance level γ = 25 of the SIHP considered in this example.

7.8.2 Example 2: H∞ Control in Infinite-Horizon Vibration Isolation Problem Here, we consider and analyse another, than in Sect. 2.3.4 and in [6], mathematical model of the infinite-horizon robust vibration isolation problem. Namely, the

7.8

Examples

197

dynamics equation in this model is similar to that in Sect. 2.3.4 and in [6], i.e., m

d 2 x(t) = w(t) + u(t), dt 2

t ≥ 0,

(7.81)

where x(t) is the deviation of the body’s position from 0 at the time moment t; u(t) is the actuator force; w(t) is the disturbance force; m is the mass of the body. However, the initial conditions are the following: x(0) = 0,

d x(t)   = 0. dt t=0

(7.82)

Let us rewrite the initial-value problem (7.81)–(7.82) in the equivalent form as d x1 (t) 1 = x2 (t), dt m

t ≥ 0,

x1 (0) = 0,

(7.83)

d x2 (t) (7.84) = w(t) + u(t), t ≥ 0, x2 (0) = 0, dt   where z(t) = col x1 (t), x2 (t) , t ∈ [0, +∞), is the state vector. Assuming that the disturbance force w(t) is square-integrable in the interval [0, +∞), i.e., w(t) ∈ L 2 [0, +∞; E 1 ], we consider the following functional along trajectories of the system (7.83)–(7.84):   T β z(t) L 2 [0,+∞)     , J u(·), w(·) =  w(t) 2 L [0,+∞)

(7.85)

√ √   = col β1 , β2 /m , β1 > 0 and β2 > 0 are given constants. where β The problem is to find a controller u = u ∗ (z) asymptotically stabilizing the system (7.83)–(7.84) for any w(t) ∈ L 2 [0, +∞; E 1 ] and guaranteeing the fulfilment of the inequality     J u ∗ (z), w(t) ≤ γ ∀w(t) ∈ L 2 [0, +∞; E 1 ] : w(t) L 2 [0,+∞) = 0, (7.86) along trajectories of (7.83)–(7.84), where γ > 0 is a given performance level. The problem (7.83)–(7.84), (7.85) and (7.86) is equivalent to the SIHP (7.9)– (7.10) with the data:

0 0

1 m



n = 2, r = s = 1, q = 0, A = 0

  0 β2 C= , D = diag β1 , m 2 , G = 0, 1



0 , B= , 1

(7.87)

198

7 Singular Infinite-Horizon H∞ Problem

and the performance level γ. Using the data (7.87), as well as the expression for Sw , the block form of Sw and the expression for S1,0 (see Eqs. (7.17) and (7.23) and Remark 7.12), we obtain Sw1 = 0, S1,0 = 1/β2 . Due to these values and the data (7.87), Eq. (7.27) becomes the scalar quadratic equation 1 2 K − β1 = 0. β2 1,0 √ √ (2) (1) = − β1 β2 . The solution This equation has two solutions K 1,0 = β1 β2 and K 1,0 √ (1) K 1,0 = β1 β2 satisfies the assumption A7.IV. Let us find out whether this solution satisfies the assumption A7.V. Using this solution and (7.26), we obtain K 2,0 =



β1 ,

K 3,0 =

(1) Then, using these values, the value K 1,0 =

K¯ 0 =

√

√ β2 . m

β1 β2 and (7.30), we have

√

β1 β2 √0 √ , β2 β1 m

S¯ =



0 0 . 0 1

Finally, we calculate the following matrix mentioned in the assumption A7.V:  D + K¯ 0T S¯ K¯ 0 =

√

2β √ 1

β1 β2 m

β1 β2 m 2β2 2 m

 .

√ (1) This matrix is positive definite. Hence, K 1,0 = β1 β2 satisfies the item (i) of the assumption A7.V. The fulfilment of the item (ii) of this assumption with √ (1) K 1,0 = β1 β2 directly follows from the data (7.87), Remark 7.12 and the following √ (1) calculation: A1 − (Su 1 + A22 /D2 )K 1,0 = − β1 /β2 . Now, using the above calculated values K i,0 , (i = 2, 3) and Remark 7.12, we derive the controller √

β2 1  ∗ (7.88) u ε,0 (z) = − β1 x1 + x2 . m ε      ∗   ∗ ∗ Let z ε,0 t; w(·) = col x1ε,0 t; w(·) , x2ε,0 t; w(·) , t ∈ [0, +∞), be the solution of the initial-value problem (7.83)–(7.84) generated by u(t) = u ∗ε,0 (z) and any given w(t) ∈ L 2 [0, +∞; E 1 ]. Then, using the data (7.87), the inclusion D ∈ S2++ , Remark 7.12 and Theorem 7.1, we can conclude the following. There exists a positive level γ, such that for number ε∗1 , depending on the data (7.87) and the performance   ∗ t; w(·) ∈ L 2 [0, +∞; E 2 ]. all ε ∈ (0, ε∗1 ], the following inclusion is satisfied: z ε,0 The latter yields the limit equality

7.9

Concluding Remarks and Literature Review

199

  ∗ t; w(·) = 0, lim z ε,0

t→+∞

valid for any w(t) ∈ L 2 [0, +∞; E 1 ] and any ε ∈ (0, ε∗1 ]. Thus, the controller (7.88) asymptotically stabilizes the system (7.83)–(7.84) for any w(t) ∈ L 2 [0, +∞; E 1 ] and any ε ∈ (0, ε∗1 ]. Moreover, by virtue of Remark 7.12 and Theorem 7.1, the following inequality is valid: 

+∞ 0

    +∞ 2  2 β2  ∗  2 ∗ w 2 (t)dt + 2 x2ε,0 t; w(·) β1 x1ε,0 t; w(·) dt ≤ γ m 0

for any w(t) ∈ L 2 [0, +∞; E 1 ] and any ε ∈ (0, ε∗1 ]. This inequality directly yields the validity of the inequality (7.86) for u ∗ (z) = u ∗ε,0 (z) and any ε ∈ (0, ε∗1 ], meaning that, for any such ε, the controller (7.88) solves the problem (7.83)–(7.84), (7.85) and (7.86).

7.9 Concluding Remarks and Literature Review In this chapter, the infinite-horizon H∞ control problem for a linear system was considered. In this problem, the matrix of coefficients for the control in the output equation, being non-zero, has a rank smaller than the dimension of the control vector. This means that the approach, based on the matrix Riccati algebraic equation, is not applicable to the solution of the considered problem, i.e., this H∞ problem is singular. However, since the above-mentioned matrix is non-zero, only a part of the control coordinates is singular, while the others are regular. Based on proper assumptions, the initially formulated H∞ control problem was converted to a new singular H∞ control problem which dynamics consists of three modes with different types of the dependence on the control. The first mode does not contain the control at all, the second mode contains only the regular control coordinates, while the third mode contains the entire control. Lemma 7.1 states the equivalence of the transformed H∞ problem and the initially formulated one. The result, similar to this lemma, was obtained in [5] for a particular case of the singular H∞ control problem. In the sequel of the chapter, the transformed problem is analysed as an original problem. To derive the solution of this new problem, the regularization method was used. Namely, this problem was approximately replaced by auxiliary regular H∞ control problem with the same dynamics and a similar functional, augmented by an infinite-horizon integral of the squares of the singular control coordinates with a small weight ε2 , (ε > 0). The regularization method for solution of singular infinite-horizon H∞ control problems was used only in a few works in the literature. Thus, in the short conference paper [4], this method was applied for the solution of the problem with time delay dynamics and functional not containing the control variable at all. In [5], the problem, similar to the one considered in the present chapter but with a particular case of the weight matrix for the control in the functional, was solved by the regularization method. In [7],

200

7 Singular Infinite-Horizon H∞ Problem

the H∞ problem with no control in the output was considered. For this problem, an “extended” (regularized) matrix Riccati algebraic equation was constructed. Based on the assumption of the existence of a proper solution to this equation, the solution of the considered problem was derived. The regularized H∞ problem, constructed in this chapter, is an infinite-horizon H∞ partial cheap control problem. Infinite-horizon H∞ partial cheap control problem was considered in [5], while infinite-horizon H∞ complete cheap control problem was studied in [3, 4, 7]. The zero-order asymptotic solution of the perturbed matrix Riccati algebraic equation, associated with the regularized H∞ problem by the solvability conditions, was constructed and justified. Based on this asymptotic solution, a simplified controller for the regularized H∞ problem was designed. It was established that this controller also solves the original singular H∞ control problem. Moreover, it was shown in the example that this controller also solves an infinite-horizon singular H∞ control problem with a smaller (than the original) performance level, depending on ε.

References 1. Gajic, Z., Qureshi, M.T.J.: Lyapunov Matrix Equation in System Stability and Control. Dover Publications, Mineola, NY, USA (2008) 2. Glizer, V.Y.: Blockwise estimate of the fundamental matrix of linear singularly perturbed differential systems with small delay and its application to uniform asymptotic solution. J. Math. Anal. Appl. 278, 409–433 (2003) 3. Glizer, V.Y.: H∞ cheap control for a class of linear systems with state delays. J. Nonlinear Convex Anal. 10, 235–259 (2009) 4. Glizer, V.Y.: Solution of a singular H∞ control problem for linear systems with state delays. Proc. 2013 European Control Conference, pp. 2843–2848, Zurich, Switzerland (2013) 5. Glizer, V.Y., Kelis, O.: Solution of a singular H∞ control problem: a regularization approach. In: Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics, pp. 25–36. Madrid, Spain (2017) 6. Hampton, R.D., Knospe, C.R., Townsend, M.A.: A Practical solution to the deterministic nonhomogeneous LQR problem. J. Dyn. Syst. Meas. Control 118, 354–360 (1996) 7. Petersen, I.R.: Disturbance attenuation and H∞ optimization: a design method based on the algebraic Riccati equation. IEEE Trans. Automat. Control 32, 427–429 (1987)

Index

A Academic examples, 8, 44 Acceleration, 38, 39, 41, 48 Actuator, 42–44, 48, 49, 142, 143, 145, 197 Actuator force, 42, 43, 48, 49, 197 Actuator objective, 44, 49, 142 Admissible control, 11–15, 23, 32 Admissible control function, 11 Admissible controller, 61, 63, 151–153, 160, 177–179, 185, 187 Admissible force of actuator, 43, 44 Admissible force of disturbance, 43, 44 Admissible pair of controls, 32–34, 131 Asymptotic analysis, 2, 73, 80, 81, 97, 110, 113, 122, 137, 146, 149, 154, 175, 180 Asymptotic representation, 88, 129 Asymptotic solution, 3, 81–83, 92, 93, 97, 98, 110, 122, 124, 134, 137, 138, 154, 156, 157, 166, 173, 180–182, 190, 192, 200 Asymptotic stability, 128, 129, 131, 133, 183 Auxiliary controller, 161, 186 Auxiliary regular H∞ control problem, 173, 199 Auxiliary regular game, 109, 146

B Bellman-Isaacs equation, 1 Block form, 69, 81, 86–89, 92, 95, 97, 101, 103, 106, 108, 122, 123, 126–128, 131, 133, 135, 137, 140, 142, 155, 160, 161, 165, 171, 176, 180, 183, 185, 186, 188–190, 192–195, 198

Block matrix, 67–69, 168, 184, 187, 193 Block of control, 132 Block of vector, 88 Block representation, 86, 98, 125, 137, 161, 163, 165, 186, 188, 190, 192 Block-diagonal form of matrix, 81, 122 Block-form matrix, 67, 86, 125, 193 Block-form matrix-valued function, 67, 86 Block-vector, 8, 73, 114, 149, 175

C Cauchy-Bunyakovsky-Schwarz integral inequality, 28, 29, 170 Complement matrix, 67, 76, 116, 152, 177 Complete cheap control, 146, 173, 200 Complete/partial cheap control, 79, 110, 120, 146, 153, 154, 173, 179, 200 Continuous control function, 38, 40, 41, 48 Continuous function, 7, 12, 25, 38, 40–42, 86 Control coordinate, 79, 109, 121, 146, 154, 172, 173, 179, 199 Control coordinate regular, 172, 199 Control coordinate singular, 109, 146, 172, 173, 199 Control cost, 1, 48, 73, 79, 109, 110, 113, 120, 145, 146, 153, 179 Control function, 11, 13, 15, 17, 23, 24, 38, 40, 41 Control objective, 77, 117 Control variable, 2, 45–47, 173, 199 Control vector, 105, 108, 172, 199 Controlled Differential system, 51, 71 Controlled system, 2, 3, 60, 176

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 V. Y. Glizer and O. Kelis, Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems, Static & DynamicGame Theory: Foundations & Applications, https://doi.org/10.1007/978-3-031-07051-8

201

202 Controller, 2, 61–64, 71, 149, 151–154, 158, 160–163, 167, 172, 173, 175, 177– 180, 183, 185–187, 189, 196–200 Controller simplified, 161–163, 172, 173, 186, 187, 200 Cost functional, 1, 52, 56, 74, 114, 150, 151, 176, 177

D Decision maker, 23, 52, 56, 114 Decision maker objective, 23 Differential equation linear, 48 Differential system, 71 Differential system controlled, 38, 40, 41, 71 Differential system time-dependent, 51, 71 Dirac delta-function, 7, 14, 34 Disturbance, 2, 43, 66, 71, 142, 143, 145, 150, 160, 176, 185 Disturbance behaviour, 44, 142 Disturbance exponentially decaying, 143 Disturbance force, 42, 43, 48, 49, 197 Disturbance known, 43, 49, 136 Disturbance objective, 44, 142 Disturbance unknown, 43, 61, 63, 141 Double integral, 27, 28 Duration of engagement, 37, 39, 41 Duration of game, 102, 105 Dynamics first-order, 7, 39 Dynamics zero-order, 7, 37, 40, 102, 104

E Equality, 7, 9, 13, 14, 19–21, 24, 26, 28, 30– 34, 46, 47, 53, 57, 64, 67, 75, 77, 78, 83, 89, 115, 117, 120, 127, 139–141, 143, 183, 184, 190, 191, 198 Equation algebraic, 2, 58, 83, 94, 99, 118, 121, 132, 134, 136, 138, 149, 157, 175, 179–182, 187, 188 Equation algebraic output, 62, 149, 175 Equation controlled, 52, 56, 114 Equation differential, 2, 11, 15, 23, 24, 34, 35, 43–46, 48, 53, 62, 74, 81, 88, 95, 99, 107, 114, 142, 155, 166, 167, 172 Equation output, 2, 61, 149, 150, 172, 175, 199 Equation scalar, 22, 140, 142 Equation solution, 11, 13, 16, 23, 30, 33, 36, 64, 88, 119, 122, 123, 128, 129, 131, 133, 135, 141, 183, 190, 195 Evader, 37–42, 48, 103, 105 Evader control, 38, 48

Index Evader objective, 38, 40, 41, 48, 103, 105, 108

F Flying vehicle, 7, 37, 39, 40, 48, 102, 104 Function absolutely continuous, 12, 25 Functional, 1, 3, 11–15, 17, 23–25, 27, 34, 35, 38–49, 51, 52, 56, 57, 61–64, 66– 71, 74, 76, 77, 79, 89, 92, 96, 97, 100, 103, 105, 107–110, 113, 114, 116, 117, 120, 130, 135–137, 139, 142, 145, 146, 150–153, 159, 160, 163, 173, 176–179, 184, 185, 188, 197, 199 Functional infimum, 44–46 Functional minimization, 14, 43 Functional minimization problem, 13, 14 Functional minimum value, 14 Functional optimal value, 11, 92, 96, 97, 100, 135, 137, 139 Functional quadratic, 3, 48, 51, 66, 71 Function continuous, 7, 12, 38, 40–42, 48, 86, 103, 105 Function continuously differentiable, 108 Function generalized, 45 Function impulse-like, 17 Function infimum, 44–46 Function linear, 54, 58, 62, 118 Function matrix-valued, 52, 53, 61, 66–70, 74, 76, 79, 84–87, 91, 92, 95, 97, 99, 106, 127, 144, 150–152, 162, 164, 168, 194 Function minimization, 8–11, 22 Function minimum point, 10, 22 Function minimum value, 9, 22 Function non-decreasing, 65, 189 Function of two scalar variables, 18 Function positive, 42 Function regular, 45, 47 Function smooth, 44, 46 Function time realization, 54, 55, 59, 60 Function twice continuously differentiable, 67 Function vector-valued, 42, 67, 169, 170

G Gain constant, 118, 131, 189 Gain matrix, 62, 71 Game differential finite-horizon, 51, 52 Game differential finite-horizon reduced, 89 Game differential infinite-horizon, 1, 56

Index Game differential infinite-horizon reduced, 129 Game differential linear-quadratic, 3, 73, 109, 113, 130, 145 Game differential regular, 79, 120, 121 Game differential singular, 1, 76, 116 Game differential zero-sum, 1, 3, 23, 24, 34, 35, 38, 40, 41, 43, 44, 46–49, 66, 67, 70, 71, 109, 110, 146 Game infinite-horizon vibration isolation, 141 Guaranteed result, 18, 19, 24, 35, 37, 53, 57, 75, 77, 79, 89–91, 115, 117, 118, 121, 130, 132, 133, 135, 146 H High dimension singularly perturbed system, 161 H∞ control problem, 51, 61–64, 67, 71, 172, 173, 199, 200 H∞ problem finite-horizon, 149, 153, 158, 172, 173 H∞ problem finite-horizon reduced, 158– 160 H∞ problem infinite-horizon, 51, 62, 71, 175, 178, 179, 199, 200 H∞ problem infinite-horizon reduced, 183, 184 H∞ problem linear-quadratic, 71 H∞ problem regular, 71, 153, 162, 172, 179, 187 H∞ problem singular, 149, 152, 172, 173, 175, 177, 199, 200 Hurwitz matrix, 127, 129, 131, 182, 183, 188, 189, 193, 194 I Ideal linear dynamics, 38, 48 Identity matrix, 51, 73, 113, 149 Ill-posed problem, 8, 13, 20, 45, 46 Inequality, 1, 2, 12, 13, 16, 19, 26–33, 54, 56, 58, 60–63, 65, 68, 80, 85, 86, 88, 90, 95, 96, 100, 101, 106, 119, 120, 122, 126, 129, 135, 139, 140, 143– 145, 151–154, 158, 160–165, 167, 168, 171, 176, 177, 179, 185–187, 189, 191, 193, 197, 199 Infinite-horizon integral, 49, 146, 199 Information pattern, 24, 34, 38, 40, 41, 43, 44 Integral constraint, 49 Integral form, 86

203 Integral functional, 45, 46, 49 Integral part of functional, 110 Integrand, 29, 48, 49, 65, 110 Isaacs MinMax principle, 1

K Kinematics linearized, 39 Kinematics of engagement, 40 Kinematics planar, 40

L Lebesgue integral, 11, 12, 23, 43 Linear algebraic equation, 67 Linear differential system, 51 Linear dynamics, 39, 48 Linear planar model of engagement, 37 Linear space, 51, 73, 87, 113, 128, 149, 175, 195 Lipschitz condition local, 52, 57, 61, 63, 74, 75, 88, 114, 151, 177 Lower block, 8, 73, 82, 91, 114, 123, 132, 149, 155, 163, 169, 175, 181, 187 Lyapunov-like function, 54, 58, 64, 71, 119, 164, 189

M Mathematical model, 7, 37–40, 44, 47–49, 196 Matrix column rank, 66, 68, 76, 116, 151, 177 Matrix inverse, 84, 124, 157, 182 Matrix positive definite, 51, 70, 74, 84, 114, 124, 150, 157, 159, 176, 182, 183, 185, 189, 196, 198 Matrix positive semi-definite, 51, 74, 114, 150, 176 Matrix rank, 2, 149, 151, 175, 177 Matrix symmetric, 51, 74, 114, 150, 176 Maximizer, 23, 24, 30, 32, 34, 47, 52, 57, 74, 91, 109, 114, 132, 146 Minimizer, 23, 24, 30, 32, 34, 47, 52, 57, 74, 79, 90, 91, 102, 107–110, 114, 118, 120, 130, 132, 135, 141, 143, 145, 146 Minimizing sequence, 9–11, 14–17, 44–46 Model of infinite-horizon robust vibration isolation problem, 196 Model of planar pursuit-evasion engagement, 7, 39 Model of rejection of body’s vibration, 141

204 O Optimal control, 1, 11, 13, 15, 45, 97, 101, 137, 139 Optimal control problem, 7, 11, 13–15, 24, 45, 46, 49, 92, 96, 97, 109, 135–137, 146 Optimal control problem regularized, 45 Optimal control problem singular, 7, 14, 15, 45, 46, 97, 136 Optimal trajectory, 11 Optimal value of functional, 11, 92, 96, 97, 100, 135, 137, 139

P Partial cheap control, 79, 120, 146, 153, 173, 179, 200 Performance level, 2, 61, 63, 150, 172, 173, 176, 195–198, 200 Positive definite square root, 84, 124, 157, 182 Pursuer, 37–42, 48, 103, 105, 107, 108, 110 Pursuer control, 38–42, 48, 105, 107 Pursuer objective, 38, 40, 41, 48, 103, 105, 108 Pursuit-evasion engagement, 7, 37–41, 48, 102, 104

Q Quadratic control cost, 79, 120 Quadratic equation, 144, 198 Quadratic state cost, 46, 71

R Real-life examples, 3 Regular problem, 3, 51 Regularization method, 3, 9, 15, 22, 34, 45– 47, 73, 109, 110, 113, 146, 149, 172, 173, 175, 199 Relative velocity of vehicles, 38, 39, 41 Riccati differential equation, 35, 53, 54, 103, 172, 173 Riccati matrix algebraic equation, 113, 175 Riccati matrix differential equation, 73, 149 Robust, 1, 7, 43, 44, 49

S Saddle point, 7, 18—24, 26, 31, 33–35, 37, 39–41, 44, 46, 47, 53, 54, 57, 58, 70, 71, 73, 75, 77–80, 84, 89–91, 102, 104, 107, 109, 110, 113, 115, 117,

Index 121, 122, 125, 130–132, 141, 143, 146, 147 Saddle point of function, 7, 18, 46 Saddle-point equilibrium, 53, 73, 75, 77, 109, 110, 113, 115, 121, 146 Saddle-point equilibrium sequence of game, 73, 75, 77, 109, 110, 113, 115, 117, 121, 146 Saddle-point equilibrium solution of game, 53 Saddle-point problem, 19–23, 46 Saddle-point sequence of game, 33, 34, 37, 75, 77, 78, 117, 122 Saddle-point solution of game, 24, 31, 33– 35, 47, 53, 54, 57, 58, 71, 79, 80, 84, 89, 110, 121, 122, 125, 130, 147 Saddle-point value, 19, 21, 23, 24, 46 Sequence of functions, 14 Sequence of numbers, 9, 10, 16, 23, 37, 44, 46, 91, 132 Sequence of pairs, 21, 47 Sequence of pairs of functions, 34 Singular extremal problem, 3, 7, 44 Singular minimization problem, 9 Singular optimal control problem, 7, 14, 15, 45, 46, 97, 136 Singular saddle-point problem, 22, 46 Small positive parameter, 3, 45–47, 73, 113, 149, 175 Solution, 1–3, 7–17, 21–26, 30, 32–35, 37, 44–47, 52–64, 66, 67, 70, 71, 73– 76, 79–81, 83–85, 87, 88, 90–92, 94, 95, 97–99, 101, 103, 104, 106, 108– 110, 113–116, 118, 120–125, 127– 147, 149, 151–162, 164–169, 172, 173, 175, 177–187, 189–192, 194, 195, 198–200 Solvability conditions, 1–3, 34, 35, 47, 51, 52, 70, 71, 73, 113, 149, 164, 173, 175, 200 State transformation linear, 3, 51, 66, 71 State variable, 11, 23, 45, 46, 89, 102, 130, 160, 185 State vector, 52, 56, 61, 63, 66, 74, 105, 114, 142, 150, 176, 177, 197

T Terminal-value problem, 35, 53, 61, 79, 81– 86, 88, 91, 94, 95, 97, 99, 101, 103, 106, 108, 154–158, 160, 163–167, 171 Time constant, 39

Index Time realization of control, 54–56, 59, 60 Time realization of controller, 64 Time realization of function, 54, 55, 59, 60 Trajectory of system, 38, 40, 41, 142, 162– 165, 197 Transposition of matrix, 51, 73, 113, 149, 175 Transposition of vector, 51, 113, 149, 175

205 V Vector-valued control, 42 Vector-valued function, 42, 67, 68, 169, 170 Velocity vector, 38

W Well posed game, 34, 47 Well posed problem, 9, 15, 22, 45, 46