A Bellman Approach for Regional Optimal Control Problems in $\mathbb{R}^N$

Barles, Guy; Briani, Ariela; Chasseigne, Emmanuel

doi:10.1137/130922288

Cited by 43 publications

(128 citation statements)

References 25 publications

Supporting

Mentioning

124

Contrasting

Order By: Relevance

“…More recently, there is an increasingly interest in control problems in stratified domains, see [36,7,8,37]. In those papers, the control problem is formulated in the whole space R N with a given stratification, and under a strong controllability assumption that guarantees the continuity of the value function and provides an appropriate framework for analysing the transmission conditions.…”

Section: Discussion and Commentsmentioning

confidence: 99%

Infinite horizon problems on stratifiable state-constraints sets

Hermosilla

Zidani

2015

Journal of Differential Equations

View full text Add to dashboard Cite

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.

show abstract

Section: Discussion and Commentsmentioning

confidence: 99%

Infinite horizon problems on stratifiable state-constraints sets

Hermosilla

Zidani

2015

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…The first one, given below, is inspired by Lions & Souganidis [21,22] by using arguments from the theory of PDE. The second one (displayed in the appendix) is inspired by the works of Achdou, Oudet & Tchou [3] and Barles, Briani & Chasseigne [7,8] by using arguments from the theory of optimal control and PDE techniques. Both proofs make use of the following important properties of viscosity subsolutions displayed in the next lemma.…”

Section: Theorem 316 (Comparison Principle) Under Assumptions [A] Amentioning

confidence: 99%

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

Dao

Djehiche

2020

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton-Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is done under the usual strong controllability assumption and also under a weaker condition, coined 'moderate controllability assumption'.

show abstract

“…Measurable Hamiltonians have been considered in [7][8][9]11]. The notion of viscosity solution has been also adapted in a recent paper by Barles et al [3] to study Bellman equations related to deterministic control problems in which dynamics and costs are different in complementary domains, and consequently discontinuities may arise at the boundary of these domains. However, to the best of our knowledge, this is the first time that viscosity solutions to discontinuous Hamilton-Jacobi equations are applied to the study of Mather measures.…”

Section: L(x(t)ẋ(t)) Dtmentioning

confidence: 99%

The Mather problem for lower semicontinuous Lagrangians

Gomes

Terrone²

2013

Nonlinear Differ. Equ. Appl.

View full text Add to dashboard Cite

Abstract. In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler-Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity. Mathematics Subject Classification (2010). List of symbolsThe canonical basis of R . , ∂ xN ψ) x(t),ẍ(t)The first and second derivative of a functionThe tangent space of a smooth manifold M at the pointThe Dirac mass concentrated at x 0 μ X (x)The projection on X of a measure μ(x, y) onThe jump of the function r : I ⊂ R → R at t 0 , that is lim t→t

show abstract

A Bellman Approach for Regional Optimal Control Problems in $\mathbb{R}^N$

Cited by 43 publications

References 25 publications

Infinite horizon problems on stratifiable state-constraints sets

Infinite horizon problems on stratifiable state-constraints sets

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

The Mather problem for lower semicontinuous Lagrangians

Contact Info

Product

Resources

About