A Relaxation Result for State-Constrained Delay Differential Inclusions

In this paper, we consider a delay differential inclusion x
^·
(t) ∈ F (t, x
_t
), where x
_t
denotes the history function of x(·) 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 C 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 inclusion x
^·
(t) ∈ co F (t, x
_t
).