Differential inclusions with state constraints

Abstract

In “Viability Theory”, we select trajectories which are viable in the sense that they always satisfy a given constraint. Since the fundamental work of Nagumo [26], we know that in order to guarantee existence of viable trajectories, we need to satisfy certain tangential conditions. In the case of differential inclusions and using the modern terminology and notation of tangent cones, this condition takes the formF(t,x) ∩TK#φ, whereF(.,.) is the orientor field involved in the differential inclusion,Kis the viability (constraint) set andTK(x) is the tangent cone toKatx. Results on the existence of viable solutions for differential inclusions can be found in Aubin–Cellina [2] and Papageorgiou [30,32].