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\def\dx{\dot x}  \def\hx{\hat x}  \def\hu{\hat u}
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\begin{document}

\title{Necessary conditions in optimal control problems 
with integral equations \\ of Volterra type}
\author{A.V. Dmitruk}
\institute{Russian Academy of Sciences, CEMI}
\email{vraimax@mail.ru}
\coauthor{N.P. Osmolovskii}
%\institute{University of Technology and Humanities, Radom, Poland}
%\email{osmolovski@uph.edu.pl} 
\mymaketitle

\section{Abstract}

On a fixed time interval $[0,T]$ consider the control system:
$$
x(t) = x(0) + \int_0^t f(t,s,x(s),u(s))ds,
$$
where $x \in C^n[0,T],\;\, u\in L^r_\infty[0,T].$
Among its solutions satisfying endpoint constraints $\eta(p)=0,\;\, \f(p)\le 0$ 
with $p= (x(0), x(T)),$ state and mixed constraints
$\;\Phi(t,x(t)) \leqslant 0, \q  F(t,x,u) \leqslant 0, \q G(t,x,u)=0,$ 
of dimensions  $d(\eta),\, d(\f),\, d(\Phi),\, d(F),\, d(G),$ respectively,
one should minimize an endpoint cost functional: $\; J= \f_0(p)\to\min.$
\ssk

The data functions are assumed to be smooth. We assume that the endpoints of 
the reference state $\hx(t)$ do not lie on the boundary of state constraints, i.e.,
$\Phi_k(0,\hx(0)) < 0,\q \Phi_k(T,\hx(T)) < 0,$ for all $k=1,\,\ldots,\, d(\Phi).$
Moreover, we assume that the mixed constraints are  {\it regular}, i.e.
at any point $(t,x,u),$ the gradients with respect to $u$ of all active inequality 
constraints and all equality constraints are positively--linearly independent. \ms

{\bf Theorem 1.} %(First order necessary conditions).}
Let a process $(\hx(t),\hu(t))$ provide a weak minimum, i.e., a local minimum 
in the norm $||x||_C+||u||_\infty\,.\,$ Then $\exists\, (\a_0,\ldots,\a_{d(\f)})\ge 0,$ $(\b_1,\ldots, \b_{d(\eta)}),$ measurable bounded functions 
$h_i(t)\ge 0,\; m_j(t),$ nondecreasing functions $\mu_k(t)$ continuous at $0$ and $T,$
and an $n-$vector function of bounded variation $\psi(t)$ also continuous at $0$ and $T,$
not all of them identically vanish, such that the following conditions hold: \\
endpoint complementary slackness: 
$\q \a_i\, \f_i(\hat p)=0,\qq i=1,\ldots,{d(\f)},$ \\[4pt] 
pointwise complementary slackness: $\;d\mu_k(t)\,\Phi_k(t,{\hat x}(t)) = 0$ \\[4pt]
and $\;h_i(t)\,F_i(t,{\hat x}(t), {\hat u}(t)) =0$ \qq for all $k,\,i;$ \\[4pt] 
transversality conditions $\q \psi(0) =\; l_{x(0)}\,, \quad\;  \psi(T) =\; -l_{x(T)}\,,$
where \\[4pt]
$ l(x_0, x_T) = 
\left(\sum \alpha_i\varphi_i + \sum \beta_j \eta_j\right)(x_0,x_T)$
is the endpoint Lagrange function,
\ms

\noindent
stationarity with respect to the control
$$
\;\overline H_u(s,{\hat x}(s),{\hat u}(s)) =0\;
$$
and the adjoint (costate) equation
$$
\; \dot\psi(s) =\; - \overline H_x (s,{\hat x}(s),{\hat u}(s)).\;
$$ \ssk 

\noindent
Here $\;\; f_x(t,s) = f_x(t,s,{\hat x}(s),{\hat u}(s)),\;$
$\; F_{iu}(s)= F_{iu}(s,\hat x(s),\hat u(s)),\;$ etc,  \\
$\dot\psi$ denotes the generalized derivative, 
we use the convenient {\it modified \\ Pontryagin function} 
\vadjust{\kern-7pt} 
$$
H(s,x,u) =\; \psi(s) f(s,s,x,u) + \int_s^T \psi(t)\,f_t(t,s,x,u)\,dt,
$$
and the {\it extended modified Pontryagin function} $\; \overline H(s,x,u) =\;$
$$
 H(s,x,u)-\sum_i h_i(s)\, F_i(s,x,u) -
\sum_j m_j(s)\,G_j(s,x,u) - \sum_k \dot\mu_k(s)\,\Phi(s,x).
$$ \ssk 

To prove this, we consider our problem as a particular case of an abstract 
optimization problem in a Banach space with smooth equality constraints and 
possibly nonsmooth inequality constraints, we use the Lagrange multipliers 
rule for this problem (which can be obtained by the Dubovitskii--Milyutin 
approach \cite{MDO}), and then apply the general optimality conditions to our 
specific problem. In this last step we use a theorem on the absence of singular 
components in the Lagrange multipliers at the mixed constraints \cite{VNIISI90, MDO}. 


\begin{thebibliography}{9}
\bibitem{VNIISI90}
{\sc A.V. Dmitruk},\, {\em Maximum principle for a general optimal control problem with
state and regular mixed constraints}, Computation Math. and Modeling,\,
v. 4, no. 4 (1993), p. 364--377.

\bibitem{MDO}
{\sc A.A. Milyutin, A.V. Dmitruk, and N.P. Osmolovskii}, \,
{\em Maximum principle in optimal control}, Moscow State University,
Faculty of Mechanics and Mathematics, Moscow, 2004 (in Russian), 168 p.


\end{thebibliography}


\end{document}

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