Journal of the
Korean Mathematical Society JKMS

We consider a linear differential game described by the delay-differential equation in a Hilbert space $H$;
$$ \begin{aligned}
\frac{d}{dt} x(t)= &\ A_{0}x(t) + A_{1} x(t-h) + \int_{-h}^{0} a(s) A_{2} x(t+s)ds \\
&\ +B(t)u(t) + C(t)v(t)~~~a.e.~~ t >0 \\
x(0)=&\ g^{0},~~x(s)=g^{1}(s)~~ \in [-h,0),
\end{aligned} \tag {*} $$ where $g=(g^{0},g^{1})\in M_{2}=H \times L_{2}([-h,0);Y)$, $u\in L_{2}^{loc}(R^{+};U), v \in L_{2}^{loc}(R^{+};V)$, $U$ and $V$ are Hilbert spaces, and $B(t)$ and $C(t)$ are families of bounded operators on $U$ and $V$ to $H$, respectively. $A_{0}$ generates an analytic semigroup $T(t)=e^{tA_{0}}$ in $H$. \par The control variables $g, u$ and $v$ are supposed to be restricted in the norm bounded sets $\{ g:||g||_{M_{2}} \leq \rho \}$, $\{u:||u||_{L_{2}([0,t];U)} \leq \delta \}$ and $\{v:||v||_{L_{2}([0,t];V)} \leq \gamma \}$ $(\rho, \delta, \gamma \geq 0)$. For given $x^{0} \in H $ and a given time $t>0$, we study $\epsilon$- approximate controllability to determine $x(\cdot )$ for a given $g$ and $v( \cdot )$ such that the corresponding solution $x(t)$ satisfies $||x(t)- x^{0}|| \leq \epsilon $ ($ \epsilon >0 $ :a given error).

Keywords: max-min controllability, evader's control, minimal time

MSC numbers: 49J25, 49J35, 49K25, 49N05