Abstract
We revisit the pioneering work of Bressan \& Hong on deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of R N . By using slightly different methods, involving more partial differential equations arguments, we (i) slightly improve the assumptions on the dynamic and the cost; (ii) obtain a comparison result for general semi-continuous sub and supersolutions (without any continuity assumptions on the value function nor on the sub/supersolutions); (iii) provide a general framework in which a stability result holds.
Highlights
In a well-known pioneering work, Bressan & Hong [14] provide a rather complete study of deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of RN. They show that the value-function satisfies some suitable Hamilton-Jacobi-Bellman (HJB) inequalities and were able to prove that, under certain conditions, one has a comparison result between sub and supersolutions of these HJB equations, ensuring that the value function is the unique solution of these equations
The article is organized as follows: in Section 2, we describe the control problem in a full generality; this gives us the opportunity to provide all the notations and recall well-known general results which are useful in the sequel
Proof — We provide the proof in the case of an (AFS), the general case resulting from a simple change of variable
Summary
In a well-known pioneering work, Bressan & Hong [14] provide a rather complete study of deterministic control problems in stratified domains, i.e. control problems for which the dynamic and the cost may have discontinuities on submanifolds of RN. We consider control problems in RN where the dynamics and costs may be discontinuous on the collection of submanifolds Mk for k < N In this first part, we describe the approach using differential inclusions and we recall all the properties of the value-function which are always true, i.e. results where the structure of the stratification does not play any role. We point out that both Theorem 2.1 and 2.2 hold in a complete general setting, independently of the stratification we may have in mind We conclude this first part by a converse result showing that supersolutions always satisfy a superdynamic programming principle: again we remark that this result is independent of the possible discontinuities for the dynamic or cost. We first do it in the case of a flat stratification; the non-flat case is reduced to the flat one by suitable local charts
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