Weak epigraphical solutions to Hamilton-Jacobi-Bellman equations on infinite horizon

Control problems on infinite horizon subject to time-dependent pure state constraints

“…(1) Proposition 4.1 extends classical viability results under restricted conditions on the regularity of the tube E (we refer the interested reader to the bibliography therein [3]). Furthermore, it is straightforward to see that Lipschitz continuity for set-valued maps imply the locally bounded variations property.…”

“…(3) For all (t, x) ∈ R + × R n the set {(f (t, x, u), L(t, x, u)) : u ∈ U (t)} is closed. 3 We recall that for a function q ∈ L 1 loc ([t0, +∞[; R) the integral ∞ t 0 q(t) dt := lim T →∞ T t 0 q(t) dt, provided this limit exists.…”

“…The value function, when differentiable, solves the HJB equation in the classical sense. However, it is well known that such a kind of notion turns out to be quite unsatisfactory for HJB equations arising in control theory and the calculus of variations (we refer the interested reader to the pioneer works [13,14] and [3] for further discussions). Indeed, the value function loses the differentiability property whenever there are multiple optimal solutions at the same initial condition or additional state constraints are present.…”

“…Sections 3-4 below). More specifically, by employing recent viability results that were investigated in [3], we establish Lipschitz regularity of the value function and viability of the system. We also demonstrate that the value function vanishes at infinity on the feasible set for all sufficiently large discount factors.…”

“…Existence of Viable Trajectories,[3]). Let E : R + R d be continuous1 , of locally bounded variations in the sense of Definition 4.1, and consider Φ : R + × R d R d a set-valued map with nonempty convex closed values such that:…”