Weak and generic bang-bang properties for continuous evolution inclusions and Baire’s method

Abstract

We consider evolution inclusions, in a separable and reflexive Banach space \({\mathbb{E}}\) , of the form \({(\ast) x'(t) \in Ax(t) + F(t, x(t)), x(t_0) = c}\) and \({(**) x'(t) \in Ax(t) + {\rm ext} F(t,x(t)), x(t_0) = c}\) , where A is the infinitesimal generator of a C0-semigroup, F is a continuous and bounded multifunction defined on \({[t_0, t_1] \times \mathbb{E}}\) with values F(t, x) in the space of all closed convex and bounded subsets of \({\mathbb{E}}\) with nonempty interior, and ext F(t, x(t)) denotes the set of the extreme points of F(t, x(t)). For (*) and (**) we prove a weak form of the bang-bang property, namely, the closure of the set of the mild solutions of (**) contains the set of all internal solutions of (*). The proof is based on the Baire category method. This result is used to prove the following generic bang-bang property, that is, if A is the infinitesimal generator of a compact C0-semigroup then for most (in the sense of the Baire categories) continuous and bounded multifunctions, with closed convex and bounded values \({F(t, x) \subset \mathbb{E}}\) , the bang-bang property is actually valid, that is, the closure of the the set of the mild solutions of (**) is equal to the set of the mild solutions of (*).