Abstract
In this paper we consider a state constrained differential inclusion ˙ x ∈ Ax + F (t, x), with A generator of a strongly continuous semigroup in an infinite dimensional separable Banach space. Under an inward pointing condition we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion ˙ x ∈ Ax + coF (t, x). Some applications to control problems involving PDEs are given.
Highlights
Under an “inward pointing condition” we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion x ∈ Ax + coF (t, x)
We study a class of infinite dimensional differential inclusions subject to state constraints
Differential inclusions find a natural application in a research area of great development, the control theory, and the infinite dimensional setting allows to apply our results to control problems involving PDEs
Summary
We study a class of infinite dimensional differential inclusions subject to state constraints. In the proof of Theorem 3.1, we need to approximate relaxed trajectories by relaxed trajectories lying in the interior of K This is the reason why the inward pointing condition (6) required in this case involves the set-valued map coF. The proof of Proposition 4 provided in Section 6 implies that it is still valid if (29) is replaced by the following less restrictive assumption: for x ∈ ∂K define N (x) :=Limsupz→x, z∈Z {nz} (the Kuratowski upper limit) and assume that for all x ∈ ∂K the set N (x) is compact and for every ε > 0 there exists δ > 0 such that nz ∈ N (x) + εB ∀ z ∈ Z ∩ B(x, δ). From (14), for [t, 1], fε′ (t) ∈ F (t, y(t)) ⊂ F (t, yε′ (t)) + kR(t) yε′ (t) − y(t) X B
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