Two nonlinear feedback control problems on proximate retracts of Hilbert spaces

Given a nonempty, closed set K in a Hilbert space H, we consider the following two control problems involving the set K. 1. For a nonlinear control problem with state x ∈ H we study the existence of a state feedback control, taking values in a Hilbert space H1, which "stabilizes" the set K in a precise sense, which is related to the viability of K. 2. For a nonlinear control problem affected by deterministic uncertainty we look for a state feedback control, taking values in H1, such that K is invariant under any system dynamics f, which is represented by a selection of the uncertain controlled dynamics. In this paper, by using methods from multivalued analysis, we will solve these problems for a suitable class of sets K ⊂ H. that of proximate retracts. Indeed, this class will provide us the necessary continuity property for the external contingent Bouligand cone to K and for the metric projection on K, which permit to use recent results on the existence of selections for multivalued maps. These selections will represent the required feedback control laws.