Existence of viable solutions for nonconvex differential inclusions

Abstract

We show the existence result of viable solutions to the differential inclusion ẋ(t) ∈ G(x(t)) + F(t, x(t)) x(t) ∈ S on [0, T], where F : [0, T] x H → H (T > 0) is a continuous set-valued mapping, G : H → H is a Hausdorff upper semi-continuous set-valued mapping such that G(x) ⊂ ∂g(x), where g : H → ℝ is a regular and locally Lipschitz function and S is a ball, compact subset in a separate Hilbert space H.