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

By Guy Barles, Ariela Briani, Emmanuel Chasseigne. 2011

73 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/451df176283e9c2204623186f7035c2e219389d0

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Well posedness for multi-dimensional junction problems with Kirchoff-type conditions (2017, methodology): “Among the long list of references on this topic with convex Hamiltonians we refer to Achdou and Tchou [1], Barles, Briani and Chasseigne[2, 3], Barles, Briani, Chasseigne and Imbert [4], Barles and Chasseigne [5], Bressan and Hong [6], Imbert and Monneu [7] and Imbert and Nguen [8].”
Viscosity solutions for junctions: well posedness and stability (2016, methodology): “Among the long list of references on this topic with convex Hamiltonians, in addition to [5], we refer to Barles and Briani and Chasseigne[1, 2], Barles and Chasseigne [3], Bressan and Hong [4] and Imbert and Nguen [6].”
Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks (2013, background): “Contributions to such problems are [9, 10, 38, 37, 2].”