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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 80, pp. 1--11.\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-2013/80\hfil Optimal control problems]
{Optimal control problems for impulsive systems with integral boundary
conditions}
\author[A. Ashyralyev, Y. A. Sharifov \hfil EJDE-2013/80\hfilneg]
{Allaberen Ashyralyev, Yagub A. Sharifov} % in alphabetical order
\address{Allaberen Ashyralyev \newline
Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul,
Turkey. \newline
ITTU, Ashgabat, Turkmenistan}
\email{aashyr@fatih.edu.tr}
\address{Yagub A. Sharifov \newline
Baku State University, Institute of Cybernetics of ANAS, Baku,
Azerbaijan}
\email{sharifov22@rambler.ru}
\thanks{Submitted February 2, 2013. Published March 22, 2013.}
\subjclass[2000]{34B10, 34A37, 34H05}
\keywords{Nonlocal boundary conditions; impulsive systems;
\hfill\break\indent optimal control problem}
\begin{abstract}
In this article, the optimal control problem is considered when the
state of the system is described by the impulsive differential equations
with integral boundary conditions. Applying the Banach contraction principle
the existence and uniqueness of the solution is proved for the corresponding
boundary problem by the fixed admissible control. The first and second
variation of the functional is calculated. Various necessary conditions of
optimality of the first and second order are obtained by the help of the
variation of the controls.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks
\section{Introduction}
Impulsive differential equations have become important in recent years as
mathematical models of phenomena in both the physical and social sciences.
There has been a significant development in impulsive theory especially in the
area of impulsive differential equations with fixed moments; see for
instance the monographs \cite{b1,b2,l1,s1} and the references therein.
Many of the physical systems can better be described by integral
boundary conditions. Integral boundary conditions are encountered in various
applications such as population dynamics, blood flow models, chemical
engineering and cellular systems. Moreover, boundary value problems with
integral conditions constitute a very interesting and important class of
problems. They include two point, three point, multi-point and nonlocal
boundary value problems as special cases; see \cite{a1,b3,b5}.
For boundary-value problems with nonlocal boundary conditions and
comments on their importance,
we refer the reader to the papers \cite{b4,c1,k1} and the references therein.
The optimal control problems with boundary conditions have been investigated
by several authors (see,e.g., \cite{m2,s2,v1,v2,v3}).
Note that optimal control problems
with integral boundary condition are considered in \cite{m3,m4}
and the first-order necessary conditions are obtained.
In certain cases the first order
optimality conditions are ``left degenerate'';
i.e., they are fulfilled trivially on a series of admissible controls. In
this case it is necessary to obtain the second order optimality conditions.
In the present paper, we investigate an optimal control problem in which the
state of the system is described by differential equations with integral
boundary conditions. Note that this problem is a natural generalization of
the Cauchy problem. The matters of existence and uniqueness of solutions of
the boundary value problem are investigated, first and second variations of
the functional are calculated. Using the variations of the controls, various
optimality conditions of the second order are obtained.
Consider the following impulsive system of differential equations with
integral boundary condition
\begin{gather}
\frac{dx}{dt}=f(t,x,u(t)),\quad 00$ is a rather small number and $\delta u(t)$ is
some piecewise continuous function. Then the increment of the functional
$\Delta J(u)=J(\tilde{u})-J(u)$ for
the fixed functions $u(t),\Delta u(t)$ is the
function of the parameter $\varepsilon $. If the representation
\begin{equation}
\Delta J(u)=\varepsilon \delta J(u)+\frac{1}{2}
\varepsilon ^2\delta ^2J(u)+o(\varepsilon ^2)
\label{18}
\end{equation}
is valid, then $\delta J(u)$ is called the first variation of
the functional and $\delta ^2J(u)$ is called the second
variation of the functional. Further, we obtain an obvious expression for
the first and second variations. To achieve the object we have to select in $
\Delta x(t)$ the principal term with respect to $\varepsilon $.
Assume that
\begin{equation}
\Delta x(t)=\varepsilon \delta x(t)+o(
\varepsilon ,t), \label{19}
\end{equation}
where $\delta x(t)$ is the variation of the trajectory. Such a
representation exists and for the function $\delta x(t)$ one
can obtain an equation in variations. Indeed, by definition of $\Delta
x(t)$, we have
\begin{equation}
\begin{aligned}
\Delta x(t)
&=(E+B)^{-1}C+\int_0^TK(t,\tau )\Delta f(\tau,x(\tau ),u(\tau ))d\tau
\\
&\quad +\sum_{00$, $v$ is some $r$-dimensional vector.
By \eqref{25} the corresponding variation of the trajectory is
\begin{equation}
\delta x(t)=a(t)\varepsilon +o(\varepsilon ,t),\quad t\in (0,T), \label{36}
\end{equation}
where $a(t)$ is a continuous bounded function.
Substitute variation \eqref{35} in to \eqref{33} and select the
principal term with respect to $\varepsilon $. Then
\begin{align*}
\delta ^2J(u)
&=-\int_{\theta }^{\theta +\varepsilon }v'\frac{
\partial ^2H(t,\psi (t),x(t),u(t))}{\partial u^2}vdt+o(\varepsilon )
\\
&=-\varepsilon v'\frac{\partial ^2H(\theta ,\psi (\theta
),x(\theta ),u(\theta ))}{\partial u^2}v+o_1(\varepsilon ).
\end{align*}
Thus, considering the second condition of \eqref{28}, we obtain the
Legandre-Klebsh criterion \eqref{34}. The proof is complete.
\end{proof}
The condition \eqref{34} is the second-order optimality condition. It is
obvious that when the right-hand side of system \eqref{e1} and function
$F(t,x,u)$ are linear with respect to control parameters,
condition \eqref{34} also degenerates; i.e., it is fulfilled trivially.
Following \cite[p. 27]{g1} and \cite[p. 40]{m1},
if for all $\theta \in (0,T)$, $\nu \in\mathbb{R}^{r}$,
\[
\frac{\partial H(\theta ,\psi (\theta ),x(\theta ),u(\theta ))}{\partial u}
=0,\quad
\nu '\frac{\partial ^2H(\theta ,\psi (\theta
),x(\theta ),u(\theta ))}{\partial u^2}\nu =0,
\]
then the admissible control $u(t)$ is said be a singular
control in the classical sense.
\begin{theorem} \label{thm5.3}
For optimality of the singular control $u(t)$ in the classical sense,
\begin{equation}
\begin{aligned}
&\nu '\Big\{ \int_0^T\int_0^T\langle \frac{\partial
f(t,x,u)}{\partial u}R(t,s),\frac{\partial f(s,x,u)}{\partial u}
\rangle \,dt\,ds
\\
&+ 2\int_0^T\int_0^T\langle \frac{\partial ^2H(t,\psi
,\,x,u)}{\partial x\partial u}G(t,s),\frac{\partial f(s,x,u)}{
\partial u}\rangle \,dt\,ds\Big\} \nu \leq 0
\end{aligned} \label{37}
\end{equation}
is satisfied for all $\nu \in\mathbb{R}^{r}$.
\end{theorem}
Condition \eqref{37} is a necessary condition of optimality of an
integral type for singular controls in the classical sense. Choosing special
variation in different way in formula \eqref{36}, we can get various
necessary optimality conditions.
\subsection*{Conclusion}
In this work, the optimal control problem is considered when the state of
the system is described by the impulsive differential equations with
integral boundary conditions. Applying the Banach contraction principle the
existence and uniqueness of the solution is proved for the corresponding
boundary problem by the fixed admissible control. The first and second
variation of the functional is calculated. Various necessary conditions of
optimality of the first and second order are obtained by the help of the
variation of the controls. These statements are formulated in
\cite{a2} without proof. Of course, such type of existence and uniqueness
results hold under the same sufficient conditions on nonlinear terms for
the system of nonlinear impulsive differential equations \eqref{e1},
subject to multi-point nonlocal and integral boundary conditions
\begin{equation}
Ex(0)+\int_0^Tm(t)x(t)
dt+\sum_{j=1}^{J}B_{j}x(\lambda _{j})
=\int_0^Tg(s,x(s))ds, \label{uu}
\end{equation}
and impulsive conditions
\begin{equation}
x(t_i^{+})-x(t_i)
=I_i(x(t_i)),\quad i=1,2,\dots ,p,\; 0