A RELAXATION THEOREM FOR A BANACH-SPACE INTEGRAL-INCLUSION WITH DELAYS AND SHIFTS

We consider the following integral-inclusion x(t) = integral(a)(t)f(t, s, u(s), u(t - d(1)), ..., u(t - d(k))) ds + g Q(t) u(t) is an elemen t of F(t, xi (x) (t)) a.e. in T in Banach space which, in particular, includes a control system defined by partial differential equations wi th delayed or shifted controls and an ordinary differential inclusion as special cases. We prove a relaxation theorem for this integral-incl usion and discuss some properties of its relaxed solution set which in clude conditions under which the set of relaxed solution is closed or compact as well as continuous dependence of the set of relaxed solutio ns on various parameters. (C) 1994 Academic Press, Inc.