% Converted from Microsoft Word to LaTeX
% by Chikrii SoftLab Word2TeX converter (version 2.0)
% Copyright (C) 1999-2001 Kirill A. Chikrii, Anna V. Chikrii
% Copyright (C) 1999-2001 Chikrii SoftLab.
% All rights reserved.
% http://www.word2tex.com/
% mailto: info@word2tex.com, support@word2tex.com


\documentclass [12pt]{article}
\usepackage {color}

\usepackage {makeidx}
\usepackage {hyperref}
\makeindex


\hypersetup{urlbordercolor=0 0 1,pdfborder=0 0 1 [3 2]}


% Base URL
\hyperbaseurl{http://word2tex.com}


% Color definitions (RGB model)
\definecolor{mycolor1}{rgb}{0.000,0.502,0.502}
\definecolor{mycolor2}{rgb}{0.502,0.000,0.502}

\begin{document}

\title{Linear Differential Games of Pursuit with Integral Block of Control in its Dynamics}
\author{Kirill A. Chikrii, Anna V. Chikrii}
\date{\today}
\maketitle


\section{Abstract}

The game problem of bringing a trajectory of dynamic system to the terminal 
set, which has cylindrical form, is treated. Here the case is analyzed, when 
controls enter the system equation in integral form. Sufficient conditions 
for the game termination in some guaranteed time are derived on the basis of 
the \href{http://www.word2tex.com/aachik.html}{Method of Resolving 
functions}\index{Method of Resolving functions}. The result is supported by 
a model example (see section~\ref{sec:model}) and is compared 
with the game ``simple motions\index{simple motions}''.

\section{Problem Statement}

We consider the dynamical process of the form


\begin{equation}
\label{eq1}
{\frac{{dz}}{{dt}}} = A\left( {t} \right)z\left( {t} \right) + 
{\int\limits_{t_{0}} ^{t} {B\left( {t,s} \right)\varphi \left( {u\left( {s} 
\right),\upsilon \left( {s} \right)} \right)ds}} ,\,z\left( {t_{0}}  \right) 
= z_{0} ,
\end{equation}



\noindent
evolving in condition of conflict\index{conflict}. Here the phase vector $z$ 
takes its values in the finite-dimensional Euclidian space $R^{n}$; $A\left( 
{t} \right)$ is $n$ square matrix, continuously depending on $t,\;t \ge 
t_{0} ,$ and $B\left( {t,s} \right),\;t_{0} \le s \le t < \infty $ is a 
matrix function, continuous in all its variables. The block of control is 
defined by function $\varphi \left( {u,\upsilon}  \right)$, continuous on 
the direct product of compacts $U$ and $V$. The pair $\left( {t,z} \right)$ 
will be referred to as a current state of the process, and $\left( {t_{0} 
,z_{0}}  \right)$ -- as its initial state. As admissible controls the 
players employ Lebesgue-measurable functions $u\left( {s} \right)$ and 
$\upsilon \left( {s} \right)$ with values in the sets $U$ and $V$ 
respectively. By virtue of the above assumptions function $\varphi \left( 
{u,\upsilon}  \right)$ satisfies the condition on superpositional 
measurability and $\psi \left( {s} \right) = \varphi \left( {u\left( {s} 
\right),\upsilon \left( {s} \right)} \right)$ is a bounded measurable 
function.

In addition, the terminal set, having cylindrical form, is given:


\begin{equation}
\label{eq2}
M^{\ast}  = M_{0} + M,
\end{equation}



Here $M_{0} $ is a linear subspace in $R^{n}$ and $M$ is a convex compact 
from $L$, which is the orthogonal complement to $M_{0} $ in $R^{n}$.

One can easy see that in the case $B\left( {t,s} \right) = \delta \left( {t 
- s} \right)E$, where $\delta \left( {t - s} \right)$ is $\delta $-function 
and $E$ is a unit matrix, the conflict-controlled process (\ref{eq1}), (\ref{eq2}) reduces 
to ordinary differentional 
game~\cite{Krasovskii:1970,Pontryagin:1988,aachik:1997}.

We study the problem of bringing a trajectory of system (\ref{eq1}) to the terminal 
set (\ref{eq2}) in a some guaranteed time. In so doing, the first player $\left( {u} 
\right)$ employs quasi-strategies, that is, at each current instant of time 
he constructs his control in the form 


\[
u\left( {t} \right) = u\left( {t_{0} ,z_{0} ,\upsilon _{t} \left( { \cdot}  
\right)} \right),
\]



\noindent
where $\upsilon _{t} \left( { \cdot}  \right) = {\left\{ {\upsilon \left( 
{s} \right):s \in {\left[ {t_{0} ,t} \right]}} \right\}}$, for any control 
$\upsilon $ of the second player 
\cite{Krasovskii:1970,Pontryagin:1988,aachik:1997}.

\section{Lemma}
\label{sec:lemma}

For any chosen admissible controls of the players solution of system (\ref{eq1}) may 
be presented in the form


\begin{equation}
\label{eq3}
z\left( {t} \right) = \Phi \left( {t,t_{0}}  \right)z_{0} + 
{\int\limits_{t_{0}} ^{t} {C\left( {t,s} \right)\varphi \left( {u\left( {s} 
\right),\upsilon \left( {s} \right)} \right)ds,}} 
\end{equation}



\noindent
where $C\left( {t,s} \right) = {\int\limits_{s}^{t} {\Phi \left( {t,\tau}  
\right)B\left( {\tau ,s} \right)d\tau ,}} $ and $\Phi \left( {t,t_{0}}  
\right)$ is the fundamental matrix of homogeneous system (\ref{eq1}).

{\color{mycolor2} \textit{Proof}}. From formula Cauchy\index{Formula!Cauchy} 
~\cite{Krasovskii:1970} as applied to system (\ref{eq1}) it follows


\[
z\left( {t} \right) = \Phi \left( {t,t_{0}}  \right)z_{0} + 
{\int\limits_{t_{0}} ^{t} {\Phi \left( {t,\tau}  \right){\int\limits_{t_{0} 
}^{\tau}  {B\left( {\tau ,s} \right)\varphi \left( {u\left( {s} 
\right),\upsilon \left( {s} \right)} \right)ds\,d\tau} } }} .
\]



Then, using Fubini theorem~\index{theorem!Fubini}\cite{Natanson:1974} we 
have


\[
z\left( {t} \right) = \Phi \left( {t,t_{0}}  \right)z_{0} + 
{\int\limits_{t_{0}} ^{t} {\left( {{\int\limits_{s}^{t} {\Phi \left( {t,\tau 
} \right)}} B\left( {\tau ,s} \right)d\tau}  \right)}} \varphi \left( 
{u\left( {s} \right),\upsilon \left( {s} \right)} \right)ds,
\]



\noindent
whence follows formula (\ref{eq3}).

Denote by $\pi $ the orthoprojector, acting from $R^{n}$ onto $L$. Let us 
study the set-valued mappings


\[
W\left( {t,s,\upsilon}  \right) = \pi C\left( {t,s} \right)\varphi \left( 
{U,\upsilon}  \right),
\]




\[
W\left( {t,s} \right) = {\bigcap\limits_{\upsilon \in V} {W\left( 
{t,s,\upsilon}  \right),}} 
\]



\noindent
where $\varphi \left( {U,\upsilon}  \right) = {\left\{ {\varphi \left( {u,v} 
\right):u \in U} \right\}},\;t \ge s \ge t_{0} ,\;\upsilon \in V.$

In the sequel, Pontryagin's condition is assumed to hold


\begin{equation}
\label{eq4}
W\left( {t,s} \right) \ne \emptyset \quad \forall \left( {t,s} \right) \in 
\Delta = {\left\{ {\left( {t,s} \right):t_{0} \le s \le t < \infty}  
\right\}}.
\end{equation}



By virtue of the assumptions on parameters of process (\ref{eq1}), (\ref{eq2}) and condition 
(\ref{eq4}) the mapping $W\left( {t,s} \right)$ has at least a single measurable 
selection $\gamma \left( {t,s} \right)$~\cite{Ioffe:1974}. Fix it and set


\[
\xi \left( {t,z,\gamma \left( {t, \cdot}  \right)} \right) = \pi \Phi \left( 
{t,t_{0}}  \right)z + {\int\limits_{t_{0}} ^{t} {\gamma \left( {t,s} 
\right)ds.}} 
\]



Let us introduce the resolving function\index{resolving function} by the 
formula


\begin{equation}
\label{eq5}
\begin{array}{l}
 \alpha \left( {t,s,z,\upsilon ,\gamma \left( {t, \cdot}  \right)} \right) = 
\sup {\left\{ {\alpha \ge 0:{\left[ {W\left( {t,s,\upsilon}  \right) - 
\gamma \left( {t,s} \right)} \right]} \cap}  \right.} \\ 
 {\left. { \cap \alpha {\left[ {M - \xi \left( {t,z,\gamma \left( {t, \cdot 
} \right)} \right)} \right]} \ne \emptyset}  \right\}} , \\ 
 \end{array}
\end{equation}



Define the set-valued mapping


\begin{equation}
\label{eq6}
T\left( {t_{0} ,z,\gamma \left( { \cdot , \cdot}  \right)} \right) = 
{\left\{ {t \ge t_{0} :{\int\limits_{t_{0}} ^{t} {{\mathop {\inf 
}\limits_{\upsilon \in V}} \alpha \left( {t,s,z,\upsilon ,\gamma \left( {t, 
\cdot}  \right)} \right)ds \ge 1}} } \right\}}.
\end{equation}



The properties of similar functions are thoroughly studied 
in~\cite{aachik:1997}. We only note that $T\left( {t_{0} ,z,\gamma 
\left( { \cdot , \cdot}  \right)} \right) = \emptyset $, if inequality in 
(\ref{eq6}) fails for finite $t \ge t_{0} $.

\section{Theorem}

Let for the game problem (\ref{eq1}), (\ref{eq2}) Pontryagin's 
condition\index{Condition!Pontryagin's} hold.

Then, if a measurable selection\index{measurable selection} $\gamma \left( 
{t,s} \right) \in W\left( {t,s} \right),\;\left( {t,s} \right) \in \Delta $ 
exists such that $T \in T\left( {t_{0} ,z_{0} ,\gamma \left( { \cdot , \cdot 
} \right)} \right) \ne \emptyset $ then a trajectory of the 
process\index{trajectory of process} (\ref{eq1}) may be brought in a finite time 
from the initial state $\left( {t_{0} ,z_{0}}  \right)$ to set (\ref{eq2}). In so 
doing the first player employs quasi-strategies\index{quasi-strategies}.

{\color{mycolor2} \textit{The proof}} is conducted by the scheme, presented 
in~\cite{aachik:1997}. 

By way of illustration below is given a simple example.



\section{Model Example}
\label{sec:model}

Let $A\left( {t} \right) \equiv 0$, $B\left( {t,s} \right) \equiv E$, 
$\varphi \left( {u,\upsilon}  \right) = u - \upsilon $, $M^{\ast}  = 
{\left\{ {0} \right\}}$, $U = aS$, $a > 1$, $V = S$, where $S$ is a square 
ball in $R^{n}$, centered at the origin. Set $t_{0} = 0$.

Thus, a trajectory of the process


\[
{\frac{{dz}}{{dt}}} = {\int\limits_{0}^{t} {\left( {u\left( {s} \right) - 
\upsilon \left( {s} \right)} \right)ds,\;u \in aS,\;\upsilon \in S,}} 
\]



\noindent
should be brought in a finite time into the origin.

In our case $M_{0} = {\left\{ {0} \right\}}$ and $M = {\left\{ {0} 
\right\}}$ therefore $L = R^{n}$ and the orthoprojector $\pi $ is an 
operator of identical transform and defined by the unit matrix. In the turn, 
since $A\left( {t} \right) \equiv 0$ then $\Phi \left( {t,0} \right) = E$.

Then $C\left( {t,s} \right) = \left( {t - s} \right)E$ and the following 
presentation for the set-valued mapping is true


\[
W\left( {t,s,\upsilon}  \right) = \left( {t - s} \right)\left( {aS - 
\upsilon}  \right),
\]




\[
W\left( {t,s} \right) = \left( {t - s} \right)\left( {a - 1} \right)S.
\]



Therefore Pontryagin's condition holds if $a \ge 1$ and $\left( {t,s} 
\right) \in \Delta $. Since $0 \in W\left( {t,s} \right)$, $\left( {t,s} 
\right) \in \Delta $ then we can pick $\gamma \left( {t,s} \right) \equiv 
0$. From formula (\ref{eq5}) we deduce that function $\alpha \left( {t,s,z,\upsilon 
,0} \right)$ appears as the greatest root of the quadratio equation


\[
{\left\| {\left( {t - s} \right)\upsilon - \alpha z} \right\|} = \left( {t - 
s} \right)a
\]



\noindent
and has the form


\[
\alpha \left( {t,s,z,\upsilon ,0} \right) = \left( {t - s} \right)\alpha 
\left( {z,\upsilon}  \right),
\]



\noindent
where


\[
\alpha {\color{red} \left( {\color{red} 
{{\color{red} z}{\color{red} 
,}{\color{red} \upsilon} }} \right)}{\color{black} = 
}{\color{black} {\frac{{\color{black} 
{{\color{red} \left( {\color{red} 
{{\color{red} z}{\color{red} 
,}{\color{red} \upsilon} }} \right)}{\color{black} + 
}{\color{black} \sqrt {\color{black} 
{{\color{red} \left( {\color{red} 
{{\color{red} z}{\color{red} 
,}{\color{red} \upsilon} }} \right)}{\color{black} 
^{\color{black} {{\color{black} 
2}}}}{\color{black} +} {\color{blue} {\left\| 
{\color{blue} {{\color{blue} z}}} 
\right\|}}{\color{black} \left( {\color{black} 
{{\color{black} a}{\color{black} 
^{\color{black} {{\color{black} 
2}}}}{\color{black} -} {\color{black} {\left\| 
{\color{black} {{\color{black} \upsilon} }} 
\right\|}}{\color{black} ^{\color{black} 
{{\color{black} 2}}}}}} \right)}}}} }}}{{\color{black} 
{{\color{blue} {\left\| {\color{blue} 
{{\color{blue} z}}} \right\|}}{\color{black} 
^{\color{black} {{\color{black} 
2}}}}}}}}}{\color{black} .}
\]



Minimum of function $\alpha {\color{red} \left( 
{\color{red} {{\color{red} 
z}{\color{red} ,}{\color{red} \upsilon} }} 
\right)}$ in $\upsilon $ is furnished by the element $\upsilon = - 
{\frac{{z}}{{{\color{blue} {\left\| {\color{blue} 
{{\color{blue} z}}} \right\|}}}}}$. Then


\[
{\mathop {\min} \limits_{\upsilon \le 1}} \alpha \left( {t,s,z,\upsilon ,0} 
\right) = \left( {t - s} \right){\frac{{a - 1}}{{{\left\| {z} \right\|}}}}
\]



\noindent
and therefore


\[
t_{\ast}  = \min {\left\{ {t \ge 0:t \in T\left( {0,z,0} \right)} \right\}}
\]



\noindent
is a root of the equation


\[
{\int\limits_{0}^{t} {\left( {t - s} \right){\frac{{a - 1}}{{{\left\| {z} 
\right\|}}}}ds = 1.}} 
\]



Thus,


\[
t_{\ast}  = \left( {{\frac{{2{\left\| {z} \right\|}}}{{a - 1}}}} 
\right)^{{\frac{{1}}{{2}}}}.
\]



Note~\cite{aachik:1997} that in the case of simple motions


\[
{\frac{{dz}}{{dt}}} = u - \upsilon ,\;u \in aS,\;\upsilon \in S,
\]



\noindent
the time of hitting the origin is given by the expression


\[
t^{\ast}  = {\frac{{{\left\| {z} \right\|}}}{{a - 1}}}.
\]



One can easy see that times $t_{\ast}  $ and $t^{\ast} $ differ essentially.



\section{Copyright Notice}

\textsf{Copyright \copyright 2000 Kirill A. Chikrii, Anna V. Chikrii.} 

\textsf{All rights reserved.}

\href{http://www.word2tex.com}{http://www.word2tex.com}





\section{References and Notes}





\begin{thebibliography}{3}
\bibitem{Krasovskii:1970} N. N. Krasovskii. {\color{mycolor1} \textit{Game Problems on Motions Encounter}}, Moscow, Nauka, 1970.
\bibitem{Pontryagin:1988} L. S. Pontryagin. {\color{mycolor1} \textit{Selected Scientific Works}}, Vol. 2, Moscow, Nauka, 1988.
\bibitem{aachik:1997} \href{http://www.word2tex.com/aachik.html}{A. A. Chikrii}. {\color{mycolor1} \textit{Conflict-Controlled Processes}}, Dordrecht/Boston/London, Kluwer Academic Publishers, 1997.
\bibitem{Natanson:1974} I. P. Natanson. {\color{mycolor1} \textit{Theory of Functions of Real Variable}}, Moscow, Nauka, 1974.
\bibitem{Ioffe:1974} A. D. Ioffe, V. M. Tikhomirov. {\color{mycolor1} \textit{Theory of Extremal Problems}}, Moscow, Nauka, 1974.
\end{thebibliography}



\end{document}