Hamilton–Jacobi equations for optimal control on junctions and networks

Abstract

Abstract. We consider continuous-state and continuous-time control problems where the admissible trajectories of the system are constrained to remain on a network. A notion of viscosity solution of Hamilton-Jacobi equations on the network has been proposed in earlier articles. Here, we propose a simple proof of a comparison principle based on arguments from the theory of optimal control. We also discuss stability of viscosity solutions.Résumé. On considère des problèmes de contrôle optimal pour lesquels l'état est contraintà rester sur un réseau. Une notion de solution de viscosité deséquations de Hamilton-Jacobi associées aété proposée dans des travaux antérieurs. Ici, on propose une preuve simple d'un principe de comparaison. Cette preuve est basée sur des arguments de contrôle optimal. La stabilité des solutions de viscosité est aussiétudiée.1991 Mathematics Subject Classification. 34H05, 49J15.The dates will be set by the publisher.

Infinite horizon problems on stratifiable state-constraints sets

“…On another hand, several papers have been devoted to the case of control problems on networks [25,2,26], where the framework is also different from the one considered in Corrollary 2.1. Indeed, in the above cited papers, the dynamics is defined only on each branch and is not Lipschitz continuous in the whole network.…”

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

“…Then u ε converges uniformly to u in L ∞ (P i , R) and there exists a function m : (0, +∞) → (0, +∞) such that lim ε→0 m(ε) = 0 and the function u ε −m(ε) is a viscosity subsolution of (3.27) on a neighborhood of Γ. [3] is adopted to prove Lemma 6.2 and finally, the existence of a function m : (0, +∞) → (0, +∞) with m(0 + ) = 0 such that u ε i − m(ε) is a subsolution of (3.27) on a neighborhood of Γ was proved by Lions [20] or Barles & Jakobsen [10].…”

“…In Section 3, we study the control problem under the strong controllability condition, where we derive the system of HJ equations associated with the optimal control problem, propose a comparison principle, which leads to the well-posedness of (1.1)-(1. 3), and prove that the value function of the optimal control problem is the unique discontinuous solution of the HJ system. We suggest two different proofs of the comparison principle.…”

Hybrid Thermostatic Approximations of Junctions for Some Optimal Control Problems on Networks

“…Proof. We are going to use the notations in (2). We also recall that the stateconstraint problem in branch i is the optimal control problem restricted to the branch i and such that, at the point x = 0 we can only use controls that make us to not leave the branch.…”