KWUN, YOUNG-CHEL

PARK, JONG-YEOUL

RYU, JONG-WON

Kyoto University Research Information Repository

TitleMINIMUM-TIME PROBLEM OF NONLINEAR CONTROLSYSTEM ON SEPARABLE REFLEXIVE BANACHSPACES(Nonlinear Analysis and Convex Analysis)Author(s)PARK, JONG-YEOUL; RYU, JONG-WON; KWUN,YOUNG-CHELCitation数理解析研究所講究録 (1996), 939: 110-119Issue Date1996-02URL http://hdl.handle.net/2433/60084RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityMINIMUM-TIME PROBLEM OF NONLINEAR CONTROLSYSTEM ON SEPARABLE REFLEXIVE BANACH SPACES$\mathrm{J}\mathrm{o}\mathrm{N}\mathrm{G}-\mathrm{Y}\mathrm{E}\mathrm{o}\mathrm{U}\mathrm{L}$ PARK, JONG-WON RYU AND $\mathrm{Y}\mathrm{o}\mathrm{U}\mathrm{N}\mathrm{G}^{-\mathrm{C}}\mathrm{B}\mathrm{E}\mathrm{L}$ KWUN1. INTRODUCTIONWe consider the nonlinear control system $N_{0}$ given by$\{$$\dot{x}(t)=A_{0}x(t)+Fx(t)+B_{0}u(t)$ , $t\geq 0$$x(0)=x0$in the separable reflexive Banach space $X_{0}$ . Along with $N_{0}$ , the sequence $\{N_{n}\},$ $n=$$1,2,$ $\cdots$ , of perturbed equations$\{$$\dot{x}_{n}(t)=A_{n}x_{n}(t)+^{p_{x_{n}}}(t)+B_{n}u_{n}(t)$ , $t\geq 0$$x_{n}(0)=X0,n$in the separable reflexive Banach spaces $X_{n}$ is considered with the mild solution$x_{n}(t)=S_{n}(t)X0,n+ \int_{0}^{t}S_{n}(t-s)\{Fx_{n}(s)+B_{n}u_{n}(S)\}dS$for every $u_{n}(\cdot)\in \mathrm{Y}_{n}$ , where $\mathrm{Y}_{n}$ is a control space and $B_{n}\in \mathcal{L}(\mathrm{Y}_{n},X_{n})$ . The operator$A_{n}$ and the nonlinear operator $F$ are assumed to satisfy that $A_{n}$ generates a stronglycontinuous semigroup of bounded linear operator $S_{n}(t),t\geq 0$ , on $X_{n}$ and $A_{n}+F$ isstrongly dissipative. $u_{n}(\cdot)$ is a locally summable function.Linear case $(F=0)$ of above systems in Hilbert space have been treated by Carija([3]).In this paper, we consider the case where $B_{\iota},=I_{?l}$ (the identity operator in $X_{n}$ ) andwe are to prove the existence of minimal time for the nonlinear system $N_{0}$ which steersinitial value $x_{0}$ to the target $x_{1}$ and to give conditions for the convergence of the sequenceof minimal times for the nonlinear approximate system $N_{n}$ on $X_{n},$ $n=1,2,$ $\cdots$ , to theminimal time for the original system $N_{0}$ on $X_{0}$ .939 1996 110-119 1102. $\mathrm{M}\mathrm{I}\mathrm{N}\mathrm{I}\mathrm{M}\mathrm{U}\mathrm{M}-\mathrm{T}\mathrm{I}\mathrm{M}\mathrm{E}$ PROBLEMWe consider the nonlinear control systems(1) $\dot{x}_{n}(t)=A_{n}xn(t)+FX_{n}(t)+u_{n}(t)$ $t\geq 0$in the separable reflexive Banach spaces $X_{n},$ $n=0,1,2,$ $\cdots$ , with the mild solutions(2) $x_{n}(t)=sn(t)X_{0,n}+ \int_{0}^{t}S_{n}(t-S)\{Fx_{n}(t)+u_{n}(S)\}dS$ .For each $n\geq 0$ , the set $U_{ad}^{n}$ of admissible controls is defined by$U_{ad}^{n}=$ {strongly measurable function $u_{n}(\cdot);u_{n}(t)\in \mathrm{Y}_{n}$ ,$||u_{n}(t)||\leq 1,$ $\mathrm{a}.\mathrm{e}.\}$ .For $n\geq 0$ , define$R_{n}(t)=\{(x0,n’ X_{1},n)\in X_{n^{\mathrm{X}X}n};Xn(\mathrm{o})=x_{0},n’ x_{n}(t)=x1,n$ ’for some $u_{n}\in U_{ad}^{n}$}where $x_{n}(t)$ is given by (2). Define also$R_{n}= \bigcup_{t>0R()}nt$and the minimal-time function $T_{n}$ : $R_{n}arrow R^{1}$ ,$T_{n}(x0,n’ x1,n)=inf\{t:(_{X_{0,n}},x_{1},n)\in R_{n}(t)\}$ .We now list the assumptions which will be in effect throughout this paper:(A1) there exist $M>0$ and $\omega\geq 0$ , such that, for $n=0,1,2,$ $\ldots$ and $t\geq 0$ ,$||S_{n}(t)||\leq Me^{\omega t}$where $M$ and $\omega$ are independent of $n$ ,(A2) $S_{n}(t)$ is compact,(A3) $S_{n}(t)arrow S_{0}(t)$ , uniformly for $t$ in bounded intervals,(A4) $S_{n}^{*}(t)arrow S_{0}^{*}(t)$ , uniformly for $t$ in bounded intervals.(F1) the nonlinear function $F$ is Lipschitz continuous:there exists a constant $c$ , such that$||Fx_{n}-Fy_{n}||\leq c||x_{n}-y_{n}||$ , $x_{n},y_{n}\in X_{n}$ ,(F2) $F$ has a linear growth rate on $X_{n}$ ; these exists a constant $k>0$ , such that$||Fx_{n}||\leq k(1+||X_{n}||)$ .111THEOREM 1. $Ifx_{0}\in X_{0}$ and $x_{1}\in D(A_{0}),$ $S\mathrm{u}\mathrm{C}h$ that(3) $||(A_{0}+F)x_{1}||+\omega||x0-x1||<1$ for $\omega>0$holds, then there exists $u\in U_{ad}^{0}$ which steers $x_{0}$ to $x_{1}$ in a time $T_{0}$ satisfying(4) $T_{0} \leq\omega^{-1}\log\{\frac{1-||(A_{0}+F)x_{1}||}{1-||(A_{0}+F)x_{1}||-\omega||_{X}0-X1||}\}$Proof. Consider the nonlinear equation(5) $\{$$\dot{x}(t)=A_{0}x(t)+Fx(t)-sign(X(t)-X_{1})$$x(0)=x_{0}\in D(A_{0})$wheresign$(y)=y/||y||$ , $y\neq 0$ ,sign(0) $=\{Z\in x:||z||\leq 1\}$ .Thus,multiplying equation (5) with $x(t)-x_{1}\neq 0$ , using the dissipative of $A_{0}+F$ ,multiplying by $e^{-2vt}$‘ and then integrating $0$ to $t(\mathrm{s}\mathrm{e}\mathrm{e}[4])$ . We have$e^{-2\omega t}||x(t)-x_{1}||^{2}$$\leq||x_{0^{-x_{1}}}||^{2}-2\int_{0}^{\ell}e^{-}(2\omega s1-||(A0+F)X_{1}||)||X(_{S})-X_{1}||dS$ .By Gronwall’s inequality,$e^{-\{vt}||x(t)-x_{1}||$$\leq||x_{0}-X1||$ – $\int_{0}^{t}e^{-\omega s}(1-||(A_{0}+F)x_{1}||)ds$and then,$e^{-\omega t}||x(t)-x_{1}||$$\leq||x0-X1||+\omega^{-1}(1-||(A_{0}+F)X1||)e^{-\omega t}-\omega^{-1}(1-||(A_{0}+F)x_{1}||)$ .Thus$||x(t)-x_{1}||$$\leq e^{\omega t}||x_{0^{-X}}1||-\omega^{-1}(1-||(A_{0}+F)x_{1}||)e^{\omega t}+\omega^{-1}(1-||(A_{0}+F)x_{1}||)$.112Let $x(t)arrow x_{1}$ , then$e^{\omega T_{0}}\cdot\{(1-||(A_{0}+F)x_{1}||)-\omega||X_{0}-x1||\}\leq 1-||(A_{0}+F)x_{1}||)$.We also$e^{\omega T_{0}} \leq\frac{1-||(A_{0}+F)x_{1}||}{1-||(A0+F)_{X_{1}}||-\omega||x0-X_{1}||}$ .Hence$T_{0} \leq\omega^{-1}\log\{\frac{1-||(A_{0}+F)x_{1}||}{1-||(A_{0}+F)x_{1}||-\omega||X_{0}-X1||}\}$.We will assume that a mild solution exists for every $u_{n}(\cdot)\in L_{\mathrm{Y}_{n}}^{p}$ and clearly, becauseof (F1), is unique.LEMMA 1. Let conditions $(Al)-(A\mathit{4}),$ $(Fl)-(F2)$ an$d$(B1) $x_{0,n}arrow x_{0},$ $x_{1,n}arrow x_{1}$be satisfied. If(6) $(x_{0,n}, x_{1,n})\in R_{n}(t_{n})$(7) $t_{n}arrow T,$ $u_{n}arrow u$ as $narrow\infty$then $(x_{0},x_{1})\in R_{0}(T)$ .Proof. Condition (6) implies that there exists $u_{n}\in U_{ad}^{n}$ such that$x_{1,n}=S_{n}(t_{n})_{X_{0,n}}+ \int_{0}^{t_{n_{S_{n}(ts}}}n-)\{F_{X_{1}(},ns)+u_{n}(s)\}dS$By (7), there exists $T_{0}$ such that$t_{n}\leq T_{0}$ , $n\geq 1$ .For every $n\geq 1$ and every $t\in[0, T_{0}]$ , we have$||x_{1,n}-X1||$$=||S_{n}(t_{n})x0,n+ \int_{0}^{t_{n_{S}}}n(t_{n}-S)\{FX1,n(s)+u_{n}(s)\}dS$$-S_{0}(T)_{X}0- \int_{0}^{T}s_{\mathrm{o}(TS}-)\{Fx1(s)+u(s)\}ds||$$\leq||S_{n}(t_{n})x_{0},n-s\mathrm{o}(\tau)X_{0}||$$+|| \int_{0}^{t_{n_{S}}}n(t_{n}-S)u_{n}(s)d_{S}-\int_{0}^{\tau_{S\mathrm{o}(\tau)u(}}-sS)dS||$113$+|| \int_{0}^{\ell_{n}}S_{n}(tn-S)F_{X_{1}},\mathfrak{n}(s)d_{S}-\int_{0}^{T}S\mathrm{o}(T-S)FX_{1}(S)d_{S||}$$=I+II+III$.First, it is not hard to show that$S_{n}(t_{n})X_{0,n}arrow S_{0}(\tau)x0$ .$II \leq||\int_{T}^{t_{n_{S}}}n(t_{n}-S)u_{n}(_{S})d_{S}||$$+|| \int_{0}^{T}(S_{n}(tn-s)u_{n}(_{S})-S\mathrm{o}(\tau-S)un(s))ds||$$+|| \int_{0}^{T}(s\mathrm{o}(T-s)un(s)-s_{0}(T-s)u(_{S}))d_{S}||$ .The first term converges to zero by (A1). The second term converges to zero by (A4).For the moment, let us concentrate on the third term. Rom the Hahn-Banach theorem,we know that we can find $x_{n}^{*}\in B_{1}^{*}=\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}$ unit ball such that$|( \int_{0}^{\tau_{S_{0}(\tau-}}S)(un(s)-u(S))d_{S},$ $x_{n}^{*})|$$=|| \int_{0}^{T}(S\mathrm{o}(T-S)u_{n}(S)-s_{\mathrm{o}(}\tau-s)u(s))dS||$$\Rightarrow$ $| \int_{0}(u_{n}(_{S})-u(_{S})\tau, S_{0}^{*}(T-S)_{X^{*}}n)d_{S|}$$=|| \int_{0}^{T}(S\mathrm{o}(T-S)u_{n}(S)-S\mathrm{o}(T-s)u(_{S}))d_{S}||$ .Hkom Schauder’s theorem, we know that, for $T>s,$ $S_{0}^{*}(T-s)$ is compact. By Alaoglu’stheorem, we know that $B_{1}^{*}$ is $\mathrm{W}$-compact. So by passing to subsequence if necessary, wemay assume that $x_{n}^{*}arrow x^{*}\in B_{1}^{*}$ . Hence, $S_{0}^{*}(\tau-S)x_{n}^{*}arrow z^{*}(t)$ . Since $u_{n}arrow u$ ,$| \int_{0}^{t}(u_{n}(_{S)-}u(_{S}), S_{0}^{*}(\tau-S)x^{*})nd_{S|}arrow 0$$\Rightarrow$ $|| \int_{0}^{t}s_{0}(T-s)(un(S)-u.(s))ds||arrow 0$ ,as $n$ $arrow$ $\infty$ .$III \leq||\int_{T}^{t_{\mathfrak{n}}}s_{n}(tn-s)Fx1,n(S)dS||$$+|| \int_{0}^{T}(S_{n}(t_{n}-s)F_{X_{1}},n(s)-^{s}n(t_{n}-S)FX_{1}(s))ds||$$+|| \int_{0}^{T}(s_{n}(t_{n}-S)p_{X}1(S)-S0(\tau-s)Fx1(s))ds||$ .114First and third term converges to zero. Let$r_{n}(t)=I+II+$ [ $\mathrm{F}\mathrm{i}\mathrm{r}\mathrm{S}\mathrm{t}$ and third term of III] $arrow 0$ ,as $narrow\infty$ . We have$||X_{1},n(t)-x(t)||$$\leq r_{n}(t)+||\int_{0}^{T}(S_{n}(tn-s)FX_{1,n}(s)-S_{n}(t_{n}-s)F_{X_{1}}(s))d_{S}||$$\leq r_{n}(t)+MK\int_{0}^{T}e^{\omega(t-}|ns)|X1,n(S)-x_{1}(s)||d_{S}$.Using Gronwall’s inequality, we get that$||X_{1,n}(t)-x_{1}(t)||$$\leq r_{n}(t)+MK\int^{T}\mathrm{o}r_{n}(S)e\exp(\int^{T}0\omega(t_{\hslash^{-}}\theta)(t_{\hslash}-\tau)de^{\omega}\mathcal{T})ds$.But note that $\int_{0}^{T}e^{\omega(t_{\mathfrak{n}}-\tau}$ ) $d_{\mathcal{T}}\leq R$ . So we have$||X_{1,n}(t)-x_{1}(t)|| \leq r_{n}(t)+MK\exp(R)\int_{0}^{T}e^{\omega(t-S}\Gamma n(n)s)ds$ .Recall that, for all $t$ $\in$ $[0,T],$ $r_{n}(t)$ $arrow$ $0$ . So using the dominated convergencetheorem, we get that $r_{n}(\cdot)arrow 0$ . Since$\int_{0}^{T}e^{\omega(t_{\mathfrak{n}^{-S}}}r_{n}(_{S}))ds\leq M’||r_{n}||$ ,$\lim_{narrow\infty}\int_{0}^{T}e^{\omega(-S}r_{n}n)(_{S}t)dsarrow 0$ .Therefore$||x_{1,n}(t)-x_{1}(t)||arrow 0$as $narrow\infty$ . Hence $x_{1,n}(t)arrow x_{1}(t)$ , as $narrow\infty$ , for all $t\in[0,T]$ .LEMMA 2. $Ass\mathrm{u}m\mathrm{e}(Al)-(A4),$ $(B\mathit{1})$,(B2) $x_{1}\in D(A_{0})$ , $||(A_{n}+F)x_{1}||<1$ ;(B3) $x_{1,n}\in D(A_{n})$ , $A_{n^{X}1,n}arrow A_{0}x_{1}$ .115lf $(x_{0},x_{1})\in R_{0}(t)$ , then there exists a sequence $\{\gamma_{n}\}$ , convergent to zero, such that(8) $(x_{0,n}, x_{1,n})\in R_{n}(t \dagger \gamma_{n})$for $ns$ufficiently large.Proof. First of all, we prove the following assertion: if $y_{n}arrow x_{1}$ , then there exists asequence $\{\gamma_{n}\}$ convergent to zero such that(9) $(y_{n}, x_{1,n})\in R_{n}(\gamma_{n})$for $n$ sufficiently large. Indeed, since$||(A_{0}+F)x_{1}||<1$ ,there exists a positive integer $n_{1}$ such that, for $n\geq n_{1}$ , we have$||(A_{n}+F)X1,n||\leq c_{1}<1$ .Furthermore, since $y_{n}arrow x_{1}$ , we may conclude that$||(A_{n}+F)X1,n||+\omega||y_{n}-x1,n||<1,$ $n\geq n_{1}$ .So, by Theorem 1, $x_{1,n}$ can be reached from $y_{n}$ in a time $T_{n}$ which satisfies$T_{n} \leq\omega^{-1}\log\{\frac{1-||(An+F)X1,n||}{1-||(A_{n}+F)x_{1,n}||-\omega||yn-x1,n||}\}$ .Taking $\gamma_{n}$ $=$ $T_{n}$ , we obtain (9) and $\gamma_{n}$ $arrow$ $0$ as $n$ $arrow$ $\infty$ as claimed. Since$(x_{0}, x_{1})\in R_{0}(t)$ , there exists $u\in U_{ad}^{0}$ such that(10) $x_{1}=S_{0}(t)_{X}0+ \int_{0}^{t}S_{0}(t-S)\{Fx1(s)+u(_{S})\}d_{S}$.Denoting(11) $y_{n}=S_{n}(t)_{X_{0,n}}+ \int_{0}^{t}S_{n}(t-s)\{Fyn(s)+u_{n}(S)\}d_{S}$ .116$||y_{n}(t)-x_{1}(t)||$$\leq||S_{n}(t)X0,n-s0(t)x0||$$+|| \int_{0}^{\ell_{S_{n}(}}t-s)(Fyn(s)+u(s))ds-\int_{0}^{t}s_{0}(t-S)(FX1(s)+u(s))dS||$$\leq||S_{n}(t)_{X}0,n-s_{0}(t)X0||$$+ \int_{0}^{t}||S_{n}(t-S)-s\mathrm{o}(t-S)||||un(_{S})||ds$$+ \int_{0}^{t}||S_{0}(t-S)(un(S)-u(S))||dS$$+ \int_{0}^{t}||S_{n}(t-s)-S_{0}(t-S)||||FX1(S)||d_{S}$$+ \int_{0}^{t}||s_{n}(t-s)||||Fyn(S)-Fx1(s)||d_{S}$$=J1+J2+J3+J4+J5$ .$J1,$ $J2$ and $J4$ are converge to zero as $narrow\infty$ . By same method of Lemma 2, J3 isconverge to zero as $narrow\infty$ . Let $k_{n}(t)=J1+J2+J3+J4arrow 0$ as $narrow\infty$ . We have$||y_{n}(t)-x_{1}(t)||$$\leq k_{n}(t)+\int_{0}^{t}||S_{n}(t-s)||||Fy_{n}(_{S})-Fx1(s)||d_{S}$$\leq k_{n}(t)+MKl\ell|e^{\omega(S}t-)|y_{n}(s)-x1(s)||dS$.Using Gronwall’s inequality, we get that$||y_{n}(t)-x_{1}(t)||$$\leq k_{n}(t)+MK\int^{t}\mathrm{o}(k_{n}(s)e-\exp\int^{t}\omega(tS)0e^{\omega}(t-r)d\mathcal{T})d_{S}$.But note that $\int_{0^{e^{\omega}}}^{t}(t-\mathcal{T})d\tau\leq R’$ . So, we have$||y_{n}(t)-x_{1}(t)|| \leq k_{n}(t)+MK\exp(R’)\int_{0}^{t}e^{\omega(ts}-)k_{n}(S)d_{S}$ .Recall that, for any $t>0,$ $k_{n}(t)arrow 0$ . So using the dominated convergence theorem,we get that $k_{n}(\cdot)arrow 0$ . Since$\int_{0}^{t}e^{\omega(-S}kt)(nS)d_{S}\leq M’’||k_{n}||$ ,117where $M”$ is constant,$\lim_{narrow\infty}\int 0dte^{\omega}(\ell-\delta)k_{n}(s)sarrow 0$.Therefore $||yn(t)-x1(t)||$ $arrow 0$ as $narrow\infty$ . Hence $y_{n}arrow x_{1}$ , as $narrow\infty$ , for any$t>0$ . Therefore, (9) holds for $y_{n}$ defined by (11).FinaUy, by (9) and (11), we obtain (8), thereby completing our proofs.THEOREM 2. Under conditions $(Al)-(A\mathit{4}),$ $(Fl)-(F\mathit{2})$, assume that $(Bl)-(B\mathit{3})$ and$(x_{0}, x_{1})\in R_{0}$ . Then, the following results hold:(a) $(x_{0,n}, x_{1,n})\in R_{n}$ , for $n$ sufficiently large.$(b) \lim_{narrow\infty}T_{n}(X0,n’ x_{1,n})=T_{0}(x0, x_{1})$ .Proof. By Lemma 2, there exists a subsequence of $\{T_{n}(x_{0},n’ x_{1,n})\}$ , denoted by $\{T_{n}’\}$ ,which converges, say to $T’$ . Using once again Lemma 2, with $t=T_{0}(x_{0}, x_{1})$ , we obtain$T_{n’}\leq T_{0}(X_{0}, x_{1})+\gamma_{n’}$ .Hence, we obtain$T’\leq\tau_{0}(x0, x_{1})$ .Finally, using Lemma 1, we may infer that$\tau_{0}(x0, x_{1})\leq T’$ ,and thus we obtain$T_{0}(x_{0}, x_{1})=T’$ .Since the last equality can be obtained for all convergent subsequence, the proof iscomplete.References[1] N. U. Ahemed, Finite-Time Null Controllability for a Class of Linear Evolution Equa-tions on a Banach Space with Control Constraints, JOTA Vol. 47, No. 2, PP 129-57,1985.[2] O. Carja, On the Minimal Time Function for Distributed Control Systems in BanachSpaces, JOTA Vol. 44, No. 3, PP 397-406, 1984.[3] O. Carja, On Variational Perturbations of Control Problems : Mini- mum-TimeProblem and Minimum-Effort Problem, JOTA Vol.44, No. 3, PP.407-33, 1984.[4] K. Naito, Feedback Controllability of a Semilinear Control System, Control-Theoryand Advanced Technology Vol. 7, No. 2, PP. 217-36, 1991.118[5] N. S. Papageorgiou, Existence of Optimal Controls for Nonlinear Systems in BanachSpaces, JOTA Vol. 53, No.3, PP 451-9, 1987.Dept. of Math., Pusan National University, Pusan 609-735,KoreaDept. of Math., Dong-A University, Pusan 604-714, Korea119

MINIMUM-TIME PROBLEM OF NONLINEAR CONTROL SYSTEM ON SEPARABLE REFLEXIVE BANACH SPACES(Nonlinear Analysis and Convex Analysis)

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MINIMUM-TIME PROBLEM OF NONLINEAR CONTROL SYSTEM ON SEPARABLE REFLEXIVE BANACH SPACES(Nonlinear Analysis and Convex Analysis)

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