Lectures on mathematical control theory
The book is written on the basis of lectures read by the author at the Faculty of Mechanics and Mathematics of the Al-Fa
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7KHRUHP Let the matrix W t t ! . Then the functional (1.63) under the conditions (1.64), (1.65) is continuously Frechet differentiable, the gradient of the functional J c v u x x J vc J uc J xc J cx L I R m u L I R m u R n u R n H at any point T v u x x X is calculated by the formula J vc T >v t P t x P t x N t z t v u t @ B t \ t L I R m J uc T >v t P t x P t x N t z t v u t @ L I R m
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7KHRUHPLet the matrix W t t
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w t X t O t x x
(2.16)
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function z t X t I – solution of a differential equation A t z B t X t z t X L I R k
(2.17) The solution of the differential equation (2.14), corresponding to the control w t / is determined by the formula y t z t X O t x x N t z t X X X L I R k (2.18) where z
O t x x ) t t W t t W t t x ) t t W t t W t t ) t t x N t ) t t W t t W t t ) t t .
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X L I R u L I R 7KHRUHP Let the matrix W t t ! . In order for system (2.11) to be controlled under conditions (2.12), (2.13), it is necessary and sufficient that the value J X u , where X u L I R k u L I R m – solution of optimization problem (2.19) – (2.21). 3URRI Necessity. /HW V\VWHP XQGHU FRQGLWLRQV EH PDQDJHDEOH/HWVVKRZWKDWYDOXH J X u /HW x t t x u t I ±VROXWLRQRIWKH GLIIHUHQWLDO HTXDWLRQ ZKHUH x t t x u x x t t x u x FRQWURO u u t t I WUDQVIHUVWKHWUDMHFWRU\RIWKHV\VWHP IURP x WR x 'HQRWH E\ w t f x t t x u u t t t I 1RZUHODWLRQV FDQEHZULWWHQDV x t t x u A t x t t x u B t w t t I >t t @ z
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ZKHUH f y t u t t w t t I 1HFHVVLW\LVSURYHQ Sufficiency/HWWKHYDOXH J X u /HWVVKRZWKDWWKHSURFHVVGHVFULEHGE\ WKH GLIIHUHQWLDO HTXDWLRQ XQGHU FRQGLWLRQV LV FRQWUROODEOH ,Q IDFW WKH YDOXH J X u LI DQG RQO\ LI DOPRVW HYHU\ZKHUH IROORZLQJ HTXDOLW\ KROGVSODFH X t O t x x N t z t X f y t X u t t t I ZKHUH y t X z t X O t x x N t z t X t I /HW w t X t O t x x N t z t X f y t X u t t ZKHUH y t X x y t X x 1RZUHODWLRQV FDQEHZULWWHQDV y t X A t y t X B t w t y t x y t x ,W IROORZV WKDW y t X x t t x u x t x x t x 7KLV PHDQV WKDW V\VWHP LV FRQWUROODEOH XQGHU FRQGLWLRQV 6XIILFLHQF\ LV SURYHQ 7KH WKHRUHPLVSURYHG /HPPDLet the matrix W t t ! , function f x u t , x R n u R m , t I , is continuously differentiable by variables x u R n u R m , function F q t X T t x T t x N t z t X f y u t , where O t x x T t x T t x T t B t ) t t W t t T t
y
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z C t x C t x N t z t , C t ) t t W t t W t t , C t ) t t W t t W t t ) t t .
Then partial derivatives
wF q t >X T t x T t x N t z t f z C t x C t x N t z t u t @ , wX wF q t f u y u t >X T t x T t x N t z t f y u t @ , wu wF q t f x y u t >X T t x T t x N t z t f y u t @ , wz wF q t N t N t f x y u t >X T t x T t x N t z t f y u t @ . wz t
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wF q t satisfies the Lipschitz wq
condition. Then the functional (2.19) under the conditions (2.20), (2.21) is Fréchet differentiable, the gradient J c X u J Xc X u J uc X u L I R k u L I R m
at any point X u L I R k u L I R m can be calculated by the formula
wF q t t wF q t t , B t \ t J uc X u wX wu where q t X t u t z t X z t X , z t X , t I – solution of a differential equation J Xc X u
(2.20) when X X t , and function \ t , t I – solution of the adjoint system \
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F q t _ v t T t x T t x N t z t v f z t v C t x C t x N t z t v u t t _ q v u x x z t v z t v Then partial derivatives F v q t Fu q t F z q t F z t q t are determined by
formulas (2.23) - (2.26), respectively, and partial derivatives F x q t >T t C t f y u t @>v T t x T t x N t z t x f y u t @ )RUPXODV FDQ EH REWDLQHG GLUHFWO\ E\ GLIIHUHQWLDWLQJ WKH IXQFWLRQ F q t 'HQRWHE\ F q q t Fv Fu F x F x F z F z t q t R k m n u I /HPPD Let W t t ! U t L I R m S R n S R n convex closed sets. Function f x u t continuously differentiable by x u and inequality holds place F q q t F q q t q q ! t q q R k m n Then the functional (2.67) under the conditions (2.68), (2.69) is convex. 7KHSURRIRIWKHOHPPDLVVLPLODUWRWKHSURRIRIWKHOHPPD 'HILQLWLRQ Let's say that the derivative F q q t satisfies the Lipschitz condition by a variable q in the area R N N k m n , if F x q t >T t C t f x y u t @>v T t x T t x N t z t x f y u t @
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where Li const ! i norm _ 'q _ _ 'v 'u 'x 'x 'z 'z t _ W t t ! , function f x u t is continuously 7KHRUHP Let matrix x u R n u R m and partial derivative F q q t satisfies differentiable by variables the Lipschitz condition. Then the functional (2.67) under the conditions (2.68), (2.69) is Frechet differentiable, the gradient J c v u x x J vc v u x x J uc v u x x J cx v u x x J xc v u x x
L I R k u L I R m u R n u R n
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ZKHUH l const ! FRQVWDQWRI/LSVFKLW]IURP 7KHRUHP Let the conditions of Theorem 8, the sequence ^vn ` L I R k ^un ` U ^x n ` S ^xn ` S are determined by the formula (2.76). Then: 1) numeric sequence ^J vn un x n xn ` strictly decreases; 2) __ vn vn __o __ un un __o _ x n x n _o _ xn xn _o when n o f If, in addition, inequality (2.72) holds place, the set M v u x x ^ v u x x X J v u x x d J v u x x ` is bounded: ^vn ` ^un ` ^x n ` ^xn ` are minimizing, i.e. 3) sequences OLP J vn un x n xn n of
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V
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5) the following estimate of the convergence rate is valid d J v n u n x n xn J d
m n m n
const !
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where y t z t C t x C t x N t z t v t I 7KH SDUWLDO GHULYDWLYHV FDQ EH REWDLQHG E\ GLUHFWO\ GLIIHUHQWLDWLQJ WKH LQWHJUDQGIURP HTXDOWR * q t 'HQRWHE\ * q q t *v *u * x * x * z * z t *Z q t R N u I ZKHUH N k m s n /HPPD Let matrix W t t ! , functions f x u t F x continuously differentiable by x u , U t S S : t convex closed sets and the inequality holds place * q q t * q q t q q ! t q q R N Then the functional (2.81) under conditions (2.82) - (2.84) is convex. The proof of the lemma is similar to the proof of Lemma 2 'HILQLWLRQ Let's say that the derivative * q q t satisfies the Lipschitz condition by a variable q in the area of R N N k m s n , if
_ * v q 'q t * v q t _ d L _ 'q _ _ * u q 'q t * u q t _ d L _ 'q _ _ * x q 'q t * x q t _ d L _ 'q _ _ * x q 'q t * x q t _ d L _ 'q _ _ * z q 'q t * z q t _ d L _ 'q _ _ * z t q 'q t * z t q t _ d L _ 'q _ _ *Z q 'q t *Z q t _ d L _ 'q _
where Li const ! i _ 'q _ _ 'v 'u 'x 'x 'z 'z t 'Z _ 7KHRUHP Let matrix W t t ! , functions f x u t F x continuously differentiable by variables x u , and partial derivative * q q t satisfies the Lipschitz condition. Then the functional (2.81) under the conditions (2.82)-(2.84) is differentiable in the Frechet sense, the gradient J c [ J vc [ J uc [ J xc [ J xc [ J Zc [ L I R k u L I R m u R n u R n u L I R S H
at any point [ X
L I R k uU u S u S u : H can be calculated by the formula
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t
³*
x
q t t dt
t
J xc [
t
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x
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t
where q t v t u t x x z t v z t v Z t z t v t I solution of the differential equation (2.82) when v v t , and function \ t t I is a solution of adjoint system \
t
* z q t t A t \ \ t ³ * z t q t t dt t I t
In addition, the gradient J c [ H satisfies Lipschitz condition __ J c [ J c [ __ d l __ [ [ __ [ [ X
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Zn
vn D n J nc [ n un PU >un D n J nc [ n @ PS > x D n J xc [ n @ xn PS > x D n J xc [ n @
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/LSVFKLW] FRQVWDQW IURP [ n vn un x n xn Z n X Pu >@ PS >@ PS >@ P: >@ SURMHFWLRQV RI SRLQWV RQ VHWV U S S : UHVSHFWLYHO\ $V U L I R m S R n S R n : L I R S FRQYH[ FORVHG VHWV WKHQ HDFK SRLQW KDVDXQLTXHSURMHFWLRQRQWRWKHVHVHWV 7KHRUHP Let the conditions of the theorem 11 be satisfied, U S S : convex closed sets, sequence ^[ n ` X is determined by the formula (2.91). Then: 1. numerical sequence ^J [ n ` X strictly decreases; 2. __ [ n [ n __o when n o f If, moreover, inequality (2.87) holds place, the set M [ ^[ X J [ d J [ ` is bounded then: 3. sequence ^[ n ` X is minimizing, i.e. OLP J [ n J LQI J [
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v t O t P P N t z t v
w t 6 ^w L I R k m w t v L I R
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'HILQLWLRQLet's say that the derivative S q q t S v q t S v q t S u q t S Z q t S x q t S x q t S z q t S z t q t S d q t
satisfies the Lipschitz condition by a variable q in the area of R N N k m m s n m n m , if
_ S v q 'q t S v q t _ d L _ 'q _ _ S v q 'q t S v q t _ d L _ 'q _
_ S u q 'q t S u q t _ d L _ 'q _ _ S Z q 'q t S Z q t _ d L _ 'q _ _ S x q 'q t S x q t _ d L _ 'q _ _ S x q 'q t S x q t _ d L _ 'q _ _ S d q 'q t S d q t _ d L _ 'q _ _ S z q 'q t S z q t _ d L _ 'q _ _ S z t q 'q t S z t q t _ d L _ 'q _
where Li const ! i _ 'q _ _ 'v 'v 'u 'w 'x 'x 'd 'z 'z t _ /HPPDLet matrix W t t ! , function S q t continuously differentiable by q q R N , sets U S S : convex and closed, inequality holds place S q q t S q q t q q ! R t q q R N Then the functional (2.124) under the conditions (2.125)-(2.128) is convex. 7KHSURRIRIWKHOHPPDLVVLPLODUWRWKHSURRIRIWKHOHPPD )XQFWLRQDO JUDGLHQW 7KH IROORZLQJ WKHRUHP JLYHV DQ DOJRULWKP IRU FDOFXODWLQJ WKH JUDGLHQW RI WKH IXQFWLRQDO XQGHU WKH FRQGLWLRQV 7KHRUHPLet matrix W t t ! , functions f x u t f x u x x t , F x t continuously differentiable by variables x u x x , partial derivative S q q t satisfies Lipschitz condition. Then the functional (2.124) under conditions (2.125)-(2.128) is continuously Frechet differentiable, the gradient
N
J c [
J vc [ J vc [ J uc [ J Zc [ J xc [ J xc [ J dc [ H
at any point [ X can becalculated by the formulas J vc [
wS q t t B t \ J vc [ wv
J uc [
wS q t t J Zc [ wu
J xc [
t
wS q t t B \ wv
wS q t t J xc [ wZ
wS q t t ³t wx dt J dc [
t
wS q t t dt wx t
³
t
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³
where partial derivatives are defined by the expressions above, the function z t v v t I solution of a differential equation (2.125), and function \ t t I solution of the adjoint system \
wS q t t A t \ \ t wz
t
wS q t t dt wd t
³
In addition, the gradient J c [ [ X satisfies Lipschitz condition __ J c [ J c [ __ d K __ [ [ __ [ [ X const !
where K 7KHSURRIVRIVLPLODUWKHRUHPVDUHJLYHQDERYH 8VLQJ UDWLRV ZH FRQVWUXFW D n ^v vn u n Z n xn xn d n ` X DFFRUGLQJWRWKHIROORZLQJUXOH
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ZKHUH D n H ! K ! /LSVFKLW]FRQVWDQWIURPLQHTXDOLW\ K H :H LQWURGXFH WKH VHW / ^[ X J [ d J [ ` ZKHUH [ v v u Z x x d X VWDUWLQJSRLQWIRUWKHVHTXHQFH
0LQLPL]LQJ VHTXHQFHV 7KH IROORZLQJ WKHRUHP JLYHV D QHFHVVDU\ DQG VXIILFLHQWFRQGLWLRQIRUFRQWUROODELOLW\RIV\VWHP 7KHRUHPLet the conditions of Theorem 15 be satisfied, the sequence ^[ n ` is determined by the formula (2.136), U S S : convex closed sets. Then: 1) numerical sequence ^J [ n ` strictly decreases; 2) __ [ n [ n __o when n o f 3) If, in addition, inequality (2.132) holds place, the set / is bounded then: J [ n J LQI J [ 4) sequence ^[ n ` X is minimizing, i.e. OLP no f [ X weakly
5) sequence ^[ n ` X weakly converges to the set, i.e. X X z [ n o [ when n o f 6) the following estimate of the rate of convergence is valid d J [ n J d
m n m n
const !
7) the controllability problem (2.92) - (2.96) has a solution if and only if J [
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5)
c n c const ! n
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Then the where Li functional (6.108) is differentiable in the Frechet sense, the gradient Jc X L I R at any point X L I R is calculated by the formula Jc X
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