A Relaxation Result for State-Constrained Delay Differential Inclusions

Abstract

In this paper, we consider a delay differential inclusionẋ(t) ∈ F (t, xt), where xt denotes the history function of x(t) along an interval of time. We extend the celebrated Filippov's theorem to this case. Then, we further generalize this theorem to the case when the state variable x is constrained to the closure of an open subset K ⊂ R n. Under a new "inward pointing condition", we give a relaxation result stating that the set of trajectories lying in the interior of the state constraint is dense in the set of constrained trajectories of the convexified inclusioṅ x(t) ∈ co F (t, xt).

Optimization of boundary value problems for higher order differential inclusions and duality

“…Over the past decades, great progress has been made in various areas of optimal control problems described by ordinary [1,6,8,10,11,13,14,16,19,20,24,[28][29][30][31][36][37][38][39] and partial differential equations/inclusions [4,5,15,[21][22][23]26]. In the paper [12] the averaging method is used to study singularly perturbed differential inclusions in an evolutionary triplet.…”