\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 194, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/194\hfil Controllability of impulsive systems] {Controllability of impulsive functional differential systems with nonlocal conditions} \author[Y. Liu, D. O'Regan \hfil EJDE-2013/194\hfilneg] {Yansheng Liu, Donal O'Regan} % in alphabetical order \address{Yansheng Liu \newline Department of Mathematics, Shandong Normal University, Jinan, 250014, China} \email{yanshliu@gmail.com} \address{Donal O'Regan \newline Department of Mathematics, National University of Ireland, Galway, Ireland} \email{donal.oregan@nuigalway.ie} \thanks{Submitted April 12, 2012. Published August 30, 2013.} \subjclass[2000]{34K10, 34K21, 34K35} \keywords{Controllability; fixed point theorem; nonlocal conditions; \hfill\break\indent impulsive functional differential equations} \begin{abstract} In this article, we study the controllability of impulsive functional differential equations with nonlocal conditions. We establish sufficient conditions for controllability, via the measure of noncompactness and M\"{o}nch fixed point theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Consider the impulsive functional differential equation $$\label{e1.1} \begin{gathered} x'(t)=A(t)x(t) + f(t, x(t), x_t) + Bu(t),\quad\text{a.e. } t\in [0, a];\\ \Delta x\big|_{t=t_i}=I_i(x(t_i)),\quad i=1, 2, \dots k;\\ x(t)=\phi(t),\quad t\in [-\tau, 0);\\ x(0)+M(x)=x_0, \end{gathered}$$ where $\Delta x |_{t=t_i}=x(t_i+0)- x(t_i-0)$, $A(t)$ is a family of linear operators which generates an evolution operator $$U: \Delta =\{(t,s)\in J\times J: 0\le s\le t\le a\}\to L(X),$$ $X$ is a Banach space, $J=[0,a]$, $L(X)$ is the space of all bounded linear operators in $X$, $M: PC(J,X)\to X$, $B$ is a bounded linear operator from a Banach space $V$ to $X$ and the control function $u(\cdot )$ is given in $L^2(J,V)$, $0=t_00: \Omega\text{ has a finite$\varepsilon$-net in }X\}$ (see \cite{bg,koz}). In this paper, the Hausdorff measure of noncompactness of a bounded set in $X$, $PC(J, X)$, and $L([-\tau, 0], X)$ are denoted by $\beta(\cdot)$, $\beta_{PC}(\cdot)$, and $\beta_{\tau}(\cdot)$, respectively. As in \cite{h}, we have the following result on the Hausdorff noncompactness measure. \begin{lemma} \label{l2.1} Suppose $E$ is a Banach space. Let $H$ be a countable set of strongly measurable function $x: J\to E$ such that there exists a $\mu\in L[J, R^+]$ with $\|x(t)\|\le \mu(t)$ a.e. $t\in J$ for all $x\in H$. Then $\beta (H(t))\in L[J, R^+]$ and $$\beta\big(\big\{\int_J x(t)dt: x\in H\Big\}\Big)\le 2\int_J\beta (H(t))dt,$$ where $\beta(\cdot)$ denotes the Hausdorff noncompactness measure, $J=[0,a]$. \end{lemma} \begin{lemma}[M\"{o}nch fixed point theorem \cite{m}] \label{l2.2} Suppose $E$ is a Banach space. Let $D$ be a closed and convex subset of $E$ and $u\in D$. Assume that the continuous operator $A: D\to D$ has the following property: $C\subset D$ countable, $C \subset \overline{co}(\{u\}\cup A(C))$ implies $C$ is relatively compact. Then $A$ has a fixed point in $D$. \end{lemma} \begin{definition} \label{d2.1} \rm A function $x\in PC(J;X)$ is said to be a mild solution of \eqref{e1.1} if $x(0) + M(x) = x_0$ and $$x(t)=U(t, 0)x(0)+\int_0^tU(t, s)\big((f(s, x(s), x_s)+Bu(s)\big)ds + \sum_{0 0 such that \|U(t, s)\| \le L_U for any (t, s)\in J \times J. More details about evolution systems can be found in \cite{p}. \section{Main results} We will use the following hypotheses: \begin{itemize} \item[(S1)] A(t) is a family of linear operators, A(t):\mathscr{D}(A)\to X, \mathscr{D}(A) not depending on t is a dense subset of X, generating an equicontinuous evolution system \{U(t, s) : (t, s)\in J\times J\}, i.e., (t, s)\to \{U(t, s)x : x\in\Omega\} is equicontinuous for t > 0 and for all bounded subsets \Omega. \item[(S2)] f: J\times X\times L([-\tau, 0], X)\to X satisfies: \begin{itemize} \item[(i)] t\to f(t,x, y) is strongly measurable for each x\in X, y\in L([-\tau, 0], X); (x, y)\to f(t, x, y) is continuous for almost all t\in J; \item[(ii)] there exist functions a_1, b_1, \mu_1\in L(J;R^+) such that$$\|f(t, x, y)\|\le a_1(t)\|x\| + b_1(t)\|y\|_{L[-\tau, 0]} + \mu_1(t), $$for all t\in J, x\in X, y\in L([-\tau, 0], X); \item[(iii)] there exist l_1, l_2\in L^1(J;R^+) such that for any bounded subsets B_1\subset X, B_2\subset L([-\tau, 0], X),$$ \beta(f(t,B_1, B_2))\le l_1(t)\beta(B_1)+l_2(t)\beta_{\tau}(B_2); $$\end{itemize} \item[(S3)] M: PC(J, X) \to X is a continuous operator and there exist nonnegative numbers a_2, b_2, l_3 such that \begin{gather*} \|M(y)\|\le a_2\|y\| +b_2,\quad \forall y\in PC(J, X);\\ \beta(M(B_1))\le l_3\beta_{PC}(B_1),\quad \text{for any bounded } B_1\subset PC(J, X); \end{gather*} \item[(S4)] the linear operator W: L^2(J,V)\to X defined by$$ Wu=\int_0^aU(a, s)Bu(s)ds $$is such that: \begin{itemize} \item[(i)] W has an invertible operator W^{-1} which take values in L^2(J,V)/kerW and there exist positive constants L_B and L_W such that \|B\|\le L_B and \|W^{-1}\|\le L_W; \item[(ii)] there is K_W\in L^1(J,R^+) such that, for any bounded set Q \subset X,$$\beta_V((W^{-1}Q)(t))\le K_W(t)\beta(Q).\end{itemize} \item[(S5)] I_i: X\to X (i = 1, \dots, k) is a continuous operator and there exist nonnegative numbers c_i, d_i, k_i\ (i=1, 2, \dots, k) such that: \begin{gather*} \|I_i(x)\|\le c_i\|x\|+d_i,\quad \forall x\in X,\;i=1, 2, \dots, k;\\ \beta(I_i(B_1))\le k_i\beta(B_1),\quad \text{for any bounded } B_1\subset X,\; i=1, 2, \dots, k. \end{gather*} \end{itemize} \begin{theorem}\label{t2} Assume that {\rm (S1)--(S5)} are satisfied. In addition, assume that \begin{gather} \label{e3.1} \begin{aligned} c&:=L_U\Big[(1+L_BL_Wa^{1/2})\Big(a_2+\int_0^a\big(a_1(s)+\tau b_1(s)\big)ds +\sum_{i=1}^kc_i\Big)\\ &\quad +L_UL_BL_Wa_2a^{1/2}\Big]<1, \end{aligned}\\ \label{ed} \begin{aligned} d&:=L_U\Big[\big(l_3 + 2\int_0^a(l_1(s)+ \tau l_2(s))ds + \sum_{i=1}^kk_i \big)\big(1+2L_BL_U\int_0^aK_W(s)ds\big)\\ &\quad +2l_3L_B\int_0^aK_W(s)ds \Big]< 1. \end{aligned} \end{gather} Then the impulsive functional differential system \eqref{e1.1} is nonlocally controllable on J. \end{theorem} \begin{proof} From (S4)(i), one can define the control: \label{e3.2} \begin{aligned} u_x(t)&= W^{-1}[x_1-M(x) -U(a, 0)(x_0-M(x))\\ &\quad -\int_0^aU(a, s)f(s, x(s), x_s)ds - \sum_{i=1}^k U(a, t_i)I_i(x(t_i)) ](t), \end{aligned} for all x\in PC(J, X). Using this control, define the following operator on PC(J, X) by \label{e3.3} \begin{aligned} (Gx)(t)=& U(t, 0)(x_0-M(x)) + \int_0^tU(t, s)\big(f(s, x(s), x_s)+Bu_x(s)\big)ds\\ &\quad + \sum_{0< t_i< t} U(t, t_i)I_i(x(t_i)),\quad \forall x\in PC(J, X). \end{aligned} Obviously, Gx\in PC(J, X). We shall show that G has a fixed point, which is then a solution of \eqref{e1.1}. Clearly, if x is a fixed point of G, then x_1=M(x) + G(x)(a), which implies that the system \eqref{e1.1} is controllable. First we show that G is continuous. To do this, suppose x_n, x \in PC(J, X) and x_n\to x as n\to +\infty. Then by (S3) and (S5) we know that \label{e3.4} \begin{aligned} &\|Gx_n - Gx\|_{PC} \\ &\leq L_U\Big(\|M(x_n)-M(x)\|+\int_0^a\| f(s, x_n(s), x_{ns})- f(s, x(s), x_s)\|ds \\ &\quad + L_B\int_0^a\|u_{x_n}(s)- u_x(s)\|ds + \sum_{i=1}^k \|I_i(x_n(t_i))- I_i(x(t_i))\|\Big)\\ &\le L_U\Big(\|M(x_n)-M(x)\|+\int_0^a\| f(s, x_n(s), x_{ns}) - f(s, x(s), x_s)\|ds \\ &\quad + L_B a^{1/2}\|u_{x_n}- u_x\|_{L^2}+ \sum_{i=1}^k \|I_i(x_n(t_i)- I_i(x(t_i))\|\Big). \end{aligned} Notice that $$\label{e3.5} \|x_{ns}-x_s\|_{L[-\tau, 0]}\leq \tau\|x_n-x\|_{PC}.$$ From \eqref{e3.2}, we have \label{e3.6} \begin{aligned} &\|u_{x_n}- u_x\|_{L^2}\\ &\le L_W\|M(x_n)-M(x)\|+L_WL_U \Big[\|M(x_n)-M(x)\|\\ &\quad +\int_0^a\| f(s, x_n(s), x_{ns})- f(s, x(s), x_s)\|ds + \sum_{i=1}^k \|I_i(x_n(t_i)- I_i(x(t_i))\|\Big]. \end{aligned} Then by \eqref{e3.4}--\eqref{e3.6}, (S2)--(S5), and the Lebesgue dominated convergence theorem, we obtain \|Gx_n - Gx\|_{PC} \to 0\quad {\rm as} \quad n\to +\infty, so G is continuous. Next, choose a positive number r satisfying \label{er} \begin{aligned} r&>\frac{L_U}{1-c}\Big[(1+L_UL_BL_Wa^{1/2})\Big(\|x_0\|+b_2 +\int_0^ab_1(s)ds\cdot \|\phi\|_{L[-\tau, 0]}\\ &\quad+ \int_0^a\mu_1(s)ds +\sum_{i=1}^kd_i\Big)+L_BL_Wa^{1/2}(\|x_1\|+b_2)\Big]. \end{aligned} We now show that $$\label{e3.7} G: B(0, r)\to B(0, r),$$ where B(0, r)=\{x\in PC(J, X): \|x\|_{PC}\le r\}. In fact, for each x\in PC(J, X), by \eqref{e3.2}, we have \begin{align*} \|u_x\|_{L^2} &= \Big(\int_0^a\|u_x(s)\|^2ds\Big)^{1/2} \\ &\leq L_W(\|x_1\|+ a_2\|x\|_{PC}+b_2) + L_WL_U \Big[\|x_0\|+ a_2\|x\|_{PC}+b_2 \\ &\quad + \int_0^a\big(a_1(s)\|x(s)\|+b_1(s)\|x_s\|_{L[-\tau, 0]}+\mu_1(s)\big)ds + \sum_{i=1}^k(c_i\|x(t_i)\|+d_i)\Big] \\ &\leq L_W(\|x_1\|+ a_2\|x\|_{PC}+b_2) + L_WL_U \Big[\|x_0\|+ a_2\|x\|_{PC}+b_2\\ &\quad + \int_0^a\big(a_1(s)\|x\|_{PC}+b_1(s)(\tau\|x\|_{PC} +\|\phi\|_{L[-\tau, 0]}) +\mu_1(s)\big)ds\\ &\quad +\sum_{i=1}^k(c_i\|x\|_{PC}+d_i)\Big]. \end{align*} This together with \eqref{e3.3} guarantees that \begin{align*} &\|Gx\|_{PC} \\ &\leq L_U\Big[\|x_0\|+ \|M(x)\| + \int_0^a \|f(s, x(s), x_s)+Bu_x(s)\|ds + \sum_{i=1}^k \|I_i(x(t_i))\|\Big] \\ &\leq L_U\Big[\|x_0\|+ a_2\|x\|_{PC}+b_2 + \int_0^a\big(a_1(s)\|x(s)\|+b_1(s)\|x_s\|_{L[-\tau, 0]}+\mu_1(s)\big)ds \\ &\quad+ L_B\int_0^a\|u_x(s)\|ds+ \sum_{i=1}^k(c_i\|x(t_i)\|+d_i)\Big]\\ &\leq L_U\Big[\|x_0\|+ a_2\|x\|_{PC}+b_2 + \int_0^a\big(a_1(s)\|x\|_{PC}+b_1(s)(\tau\|x\|_{PC}+\|\phi\|_{L[-\tau, 0]})\\ &\quad +\mu_1(s)\big)ds + L_Ba^{1/2}\|u_x\|_{L^2}+ \sum_{i=1}^k(c_i\|x\|_{PC}+d_i)\Big] \\ &\leq c\|x\|_{PC}+ L_U\Big[(1+L_UL_BL_Wa^{1/2})\Big(\|x_0\|+b_2 +\int_0^ab_1(s)ds\cdot \|\phi\|_{L[-\tau, 0]}\\ &\quad +\int_0^a\mu_1(s)ds+ \sum_{i=1}^kd_i\Big)+L_BL_Wa^{1/2}(\|x_1\|+b_2)\Big]. \end{align*} From \eqref{er} we have \|Gx\|_{PC}\le r if \|x\|_{PC}\le r; that is, \eqref{e3.7} holds. Next we prove that if D\subset B(0, r) is countable and $$\label{e3.9} D \subset \overline{co}(\{u_0\}\cup G(D)),$$ where u_0\in B(0, r), then D is relatively compact. Without loss of generality, suppose that D =\{x_n\}_{n=1}^\infty. First we show \{Gx_n\}_{n=1}^\infty is equicontinuous on each J_i, i = 0, \dots, k. If this is true then \overline{co}(\{u_0\}\cup G(D))  is also equicontinuous on each J_i. To this end, notice that for each x\in D, t', t''\in J_i, we have \label{e3.10} \begin{aligned} &\|(Gx)(t'')- (Gx)(t')\|\\ &= \|[U(t'',0)-U(t', 0)](x_0-M(x))\| +\|\sum_{j=1}^i\big(U(t'',t_j)-U(t', t_j)\big)I_j(x(t_j))\|\\ &\quad +\|\int_0^{t''}U(t'',s)\big(f(s, x(s), x_s)+Bu_x(s)\big)ds\\ &\quad -\int_0^{t'}U(t', s)\big(f(s,x(s), x_s) +Bu_x(s)\big)ds\|\\ &\leq \|[U(t'', 0)-U(t',0)](x_0-M(x))\| +\sum_{j=1}^i\|\big(U(t'', t_j)-U(t',t_j)\big)I_j(x(t_j))\|\\ &\quad +\int_0^{t'}\|U(t'', s)-U(t',s)\big(f(s, x(s), x_s)+Bu_x(s)\big)\|ds\\ &\quad +\int_{t'}^{t''}\|U(t'', s)\|\cdot\|f(s, x(s), x_s)+Bu_x(s)\|ds \end{aligned} From the equicontinuity property of U(\cdot, s) and the absolute continuity of the Lebesgue integral, we see that the right-hand side of the inequality \eqref{e3.10} tends to zero independent of x\in D as |t''- t'|\to 0, t'', t'\in J_i. Therefore, G(D) is equicontinuous on every J_i. Next notice that \|x_{ns}-x_{ms}\|_{L[-\tau, 0]}\le \tau\|x_n-x_m\|_{PC}, \quad \|x_{n}(s)-x_{m}(s)\|\le \|x_n-x_m\|_{PC}, s\in J, $$which implies$$ \beta_{\tau}(\{x_{ns}\}_{n=1}^\infty )\le \tau\beta_{PC}(\{x_n\}_{n=1}^\infty),\quad \beta(\{x_n(s) \}_{n=1}^\infty )\le \beta_{PC}(\{x_n\}_{n=1}^\infty), s\in J. Then from (S2), (S3), (S4) and (S5), for each t\in J, we have \begin{align*} &\beta_V(\{u_{x_n}(t)\}_{n=1}^\infty )\\ &\leq K_W(t)\beta\Big(\{M(x_n) +U(a, 0)(x_0-M(x_n))+\int_0^aU(a, s)f(s, x_n(s), x_{ns})ds\\ &\quad + \sum_{i=1}^k U(a, t_i)I_i(x_n(t_i)) \}_{n=1}^\infty\Big)\\ &\leq K_W(t)\Big( l_3(1+L_U)\beta_{PC}(\{x_n\}_{n=1}^\infty) + 2L_U\int_0^a\big[l_1(s)\beta(\{x_n(s)\}_{n=1}^\infty )\\ &\quad + l_2(s)\beta_{\tau}(\{x_{ns}\}_{n=1}^\infty)\big]ds +L_U\sum_{i=1}^kk_i\beta(\{x_n(t_i)\}_{n=1}^\infty) \Big)\\ &\leq K_W(t)\Big( l_3(1+L_U) + 2L_U\int_0^a\big[l_1(s)+ \tau l_2(s)\big]ds +L_U\sum_{i=1}^kk_i \Big)\beta_{PC}(\{x_n\}_{n=1}^\infty), \end{align*} and \label{e3.11} \begin{aligned} &\beta(\{(Gx_n)(t)\}_{n=1}^\infty)\\ &\leq \beta\Big(\{U(t, 0)(x_0-M(x_n)) \}_{n=1}^\infty \Big)\\ &\quad +\beta\Big(\{\int_0^tU(t, s)\big(f(s, x_n(s), x_{ns})+Bu_{x_n}(s)\big)ds \}_{n=1}^\infty \Big)\\ &\quad + \beta\Big(\{\sum_{0< t_i< t} U(t, t_i)I_i(x_n(t_i)) \}_{n=1}^\infty \Big)\\ &\leq L_U l_3\beta_{PC}(\{x_n\}_{n=1}^\infty) + 2L_U\int_0^a\big[l_1(s)\beta(\{x_n(s)\}_{n=1}^\infty )+l_2(s)\beta_{\tau}(\{x_{ns}\}_{n=1}^\infty )\big]ds\\ &\quad +2L_UL_B\int_0^a\beta_V(\{u_{x_n}(s)\}_{n=1}^\infty )ds+L_U\sum_{i=1}^kk_i\beta(\{x_n(t_i)\}_{n=1}^\infty)\\ &\leq L_U\Big[l_3 + 2\int_0^a\big[l_1(s)+ \tau l_2(s)\big]ds + 2L_B\Big( l_3(1+L_U) + 2L_U\int_0^a\big[l_1(s)+\tau l_2(s)\big]ds\\ &\quad +L_U\sum_{i=1}^kk_i \Big)\int_0^aK_W(s)ds + \sum_{i=1}^kk_i \Big]\beta_{PC}(\{x_n\}_{n=1}^\infty)\\ &\leq L_U\Big[\big(l_3 + 2\int_0^a(l_1(s)+ \tau l_2(s))ds + \sum_{i=1}^kk_i \big)\big(1+2L_BL_U\int_0^aK_W(s)ds\big)\\ &\quad +2l_3L_B\int_0^aK_W(s)ds \Big]\beta_{PC}(\{x_n\}_{n=1}^\infty)\\ &= d\cdot\beta_{PC}(\{x_n\}_{n=1}^\infty). \end{aligned} Note since \{Gx_n\}_{n=1}^\infty is equicontinuous on each J_i, i = 0, \dots, k we have (from a well known result on measures of noncompactness) \beta_{PC}(\{Gx_n\}_{n=1}^\infty)= \sup_{ 0 \leq i \leq k}\,\sup_{t\in J_i}\beta(\{(Gx_n)(t)\}_{n=1}^\infty). $$This together with \eqref{ed}, \eqref{e3.9} and \eqref{e3.11} guarantees that$$ \beta_{PC}(\{x_n\}_{n=1}^\infty) \le \beta_{PC}(\{Gx_n\}_{n=1}^\infty) \le d \cdot\beta_{PC}(\{x_n\}_{n=1}^\infty),  which implies that $D =\{x_n\}_{n=1}^\infty$ is relatively compact. From M\"{o}nch's fixed point theorem, $G$ has a fixed point in $B(0, r)$ and immediately the system \eqref{e1.1} is nonlocally controllable on $J$. \end{proof} \begin{remark}\label{rmk3} \rm Note that \eqref{e1.1} with no effect of time delay was considered in \cite{jlw}. The assumptions on $f$, $M$, and $I_i$ in \cite{jlw} are relaxed in this paper. For example $M$ is not necessarily compact here, and the the assumptions (S2), (S3), and (S5) in our paper are weaker than assumptions (H2), (H3), and (H5) in \cite{jlw}. \end{remark} \subsection*{Acknowledgements} The authors wish to thank the anonymous referees for their valuable suggestions. This research was supported by grants 11171192 from the NNSF of China, and BS2010SF025 from the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province. \begin{thebibliography}{00} \bibitem{bg} J. Banas, K. Goebel; \emph{Measure of Noncompactness in Banach Spaces}, Marcel Dekker, New York, 1980. \bibitem{Bysz} L. 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