A Bellman approach for two-domains optimal control problems in ℝ^N

Abstract

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space R N . As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

Hamilton–Jacobi equations for optimal control on junctions and networks

“…Assumption [H3] is a strong controllability condition at the vertex (wich implies the coercivity of the Hamiltonian). It has already been widely used in the framework of networks (for instance, the same assumption is made in [1,3], and the coercivity of the Hamiltonian is assumed [13]). We will see that [H3] yields the continuity of the value function.…”

“…In this part, we study sub and supersolutions of (2.1), transposing ideas coming from Barles-Briani-Chasseigne [3,4] to the present context. …”

“…It is interesting to note that in [3] and [4], a condition similar to (3.7) is introduced to characterize a particular viscosity solution of the transmission problem studied there among all the possible solutions in the sense of Ishii, (this condition is not satisfied by all subsolutions). In the present context, the fact that (3.7) is automatically satisfied by subsolutions seems to be linked to the richness of the space R(G): for any Lipschitz function u defined in a neighborhood of O, there exists ϕ ∈ R(G) such that u − ϕ has a maximum at O.…”

“…Related problems were also recently addressed by Rao, Siconolfi and Zidani [19,20]. The aim of the present paper is to focus on optimal control problems with independent dynamics and running costs in the edges, and to show that the arguments in [3] can be adapted to yield a simple proof of a comparison result.…”

“…In [13], the proof of the comparison result is rather involved and only uses arguments from the theory of partial differential equations: in the most simple case where all the Hamiltonians related to the edges are strictly convex and reach their minima at p = 0, the idea consists of doubling the variables and using a suitable test-function; then, in the general case, perturbation arguments are used for applying the results proved in the former case. In coincidence with these research efforts about networks, Barles, Briani and Chasseigne, see [3,4], have recently studied control problems with discontinuous dynamics and costs, obtaining comparison results for some Bellman equations arising in this context, with original and elegant arguments. Related problems were also recently addressed by Rao, Siconolfi and Zidani [19,20].…”