Baire's category and the Bang-Bang property for evolution differential inclusions

The Bang-Bang property is investigated for evolution differential inclusions, under a locally Lipschitz assumption on the right hand side. Our approach is an adaptation of the Baire category method, which was introduced in 1982 by De Blasi and Pianigiani in order to establish existence results results for non-convex differential inclusions in Banach spaces. In the present paper we consider an evolution differential inclusion with closed bounded and convex values and then, under appropriate assumptions, we show that most (in the sense of the Baire category) elements of its solution set, say M, are actually Bang-Bang solutions. As consequence it follows that the Bang-Bang solutions are dense in M.