Infinite horizon problems on stratifiable state-constraints sets

Abstract

This paper deals with a state-constrained control problem. It is well known that, unless some compatibility condition between constraints and dynamics holds, the value function has not enough regularity, or can fail to be the unique constrained viscosity solution of a HamiltonJacobi-Bellman (HJB) equation. Here, we consider the case of a set of constraints having a stratified structure. Under this circumstance, the interior of this set may be empty or disconnected, and the admissible trajectories may have the only option to stay on the boundary without possible approximation in the interior of the constraints. In such situations, the classical pointing qualification hypothesis are not relevant. The discontinuous value function is then characterized by means of a system of HJB equations on each stratum that composes the state constraints. This result is obtained under a local controllability assumption which is required only on the strata where some chattering phenomena could occur.

State-Constrained Stochastic Optimal Control Problems via Reachability Approach

“…A rich literature has been developed for dealing with state constrained optimal control problems and the associated HJB equations. In the deterministic setting, we refer to [34,35,19,22,14,21] where the HJB equation for the value function is discussed and several conditions are investigated in order to guarantee the characterization of ϑ. In the stochastic case, the problem has also attracted a great attention, see for example [24,26,23,10].…”

Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions

“…Proof. The implication (iii) ⇒ (i) is customary and well known, in particular see [20], Proposition 5.1 in the constrained framework and [18], [19], [15], [28] for the unconstrained framework. Now we prove that (i)⇒ (ii).…”

“…The first proposition states the existence of smooth trajectories for a given initial data, namely, initial point and initial velocity. The proof is analogous to the proof of Proposition 4.1 of [20] and we omit it. Lemma B.1.…”

“…Finally, we would like to mention some recent work on networks [1,24,23,10] which share the same kind of difficulty as this subject. A different framework is considered in [20], where an infinite horizon state constrained control problem with a constraint set having a stratified structure is studied. We refer also to [21], where the same approach has been used to study the minimum time problem and the Mayer problem for stratified state constraints.…”

Hamilton–Jacobi equations for optimal control on multidimensional junctions with entry costs

“…We do not know whether these results apply to the ramified sets for which interfaces of dimension d − 2 cross each other (see Figure 1b). Recent results on optimal control and HJ equations on ramified sets include Bressan & Hong [11], Camilli, Schieborn & Marchi [13], Nakayasu [24] and Hermosilla & Zidani [17] and the book of Barles & Chasseigne [9].…”