Summary: We consider a linear differential game described by the delay- differential equation in a Banach space X: \[ \frac{dx(t)}{dt}=A_ 0x(t)+\int^{0}_{-h}d\eta (s)x(t+s)+B(t)u(t)+C(t)v(t)\quad a.e.\quad t>0, \] x(0)\(=g^ 0\), \(x(s)=g^ 1(s)\) a.e. \(s\in [-h,0)\), where \(g=(g^ 0,g^ 1)\in M_ p=X\times L_ p([-h,0]\); X), \(u\in L_ q^{loc}({\mathbb{R}}^+;V)\), \(v\in L_ r^{loc}({\mathbb{R}}^+;U)\), U and V are Banach spaces, p,q,r\(\in (1,\infty)\), B(t) and C(t) are families of bounded operators on U and V to X, respectively, and \(A_ 0\) generates a \(C_ 0\)-semigroup, \(\eta\) is a Stieltjes measure.
The control variables g, u and v are supposed to be restricted in the norm bounded sets \(\{\) g; \(\| g\|_{M_ p}\leq \rho \}\), \(\{\) u; \(\| u\|_{L_ q([0,t];U)}\leq \delta \}\) and \(\{\) v; \(\| v\|_{L_ r([0,t];V)}\leq \gamma \}\) (\(\rho\),\(\delta\),\(\gamma\geq 0)\). For given \(x^ 0\in X\) and a given time \(t>0\) we study \(\epsilon\)- approximate controllability to determine u(\(\cdot)\) 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).