Tangent sets, viability for differential inclusions and applications

Abstract

Let X be a real Banach space, let A : D ( A ) ⊆ X ↝ X be a given operator, K a nonempty and possible non-open subset in D ( A ) ¯ , F : K ↝ X a given multi-function. In this lecture, we consider the differential inclusion, u ′ ( t ) ∈ A u ( t ) + F ( u ( t ) ) , with (a) A = 0 , (b) A linear and (c) A nonlinear and, in each one of these cases, we give a short survey of the most important and very recent necessary and sufficient conditions for viability expressed in terms of tangent sets and A - quasi-tangent sets to K at a given point ξ ∈ K , concepts recently introduced by the authors. From a rather long list of applications, we confined ourselves only to: solutions in moving sets, a comparison result for a reaction–diffusion system, a comparison result for a nonlinear diffusion inclusion and a sufficient condition for null controllability.