Effective transmission conditions for Hamilton–Jacobi equations defined on two domains separated by an oscillatory interface

Abstract

We consider a family of optimal control problems in the plane with dynamics and running costs possibly discontinuous across an oscillatory interface Γ ε . The oscillations of the interface have small period and amplitude, both of the order of ε, and the interfaces Γ ε tend to a straight line Γ. We study the asymptotic behavior as ε → 0. We prove that the value function tends to the solution of Hamilton-Jacobi equations in the two half-planes limited by Γ, with an effective transmission condition on Γ keeping track of the oscillations of Γ ε .

Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case

“…The reader can observe that this strong uniqueness result is very similar to the comparison principle obtained in the present paper; the two works were independent and achieved approximately at the same time. A two-domain Hamilton-Jacobi equation of the form (1.3) also appears naturally in the singular perturbation problem studied in [3].…”

Effective nonlinear Neumann boundary conditions for 1D nonconvex Hamilton–Jacobi equations

“…In [21], the author studied the case of linear Neumann boundary condition. For first-order Hamilton-Jacobi equations, Barles and Lions prove a comparison principle result in [7] under a nondegeneracy condition on the boundary nonlinearity (see (1) below). The second-order case was treated by Ishii and Barles in [19,6,8].…”

“…This error estimate has been improved in [14] to order ∆x 1 2 . There are also applications in optimal control, for example in [1] where the authors study problem related to flux-limited functions.…”

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

“…The optimal control problem on networks is related to ndimensional optimal control problems on multi-domains, where the dynamics and costs incur in discontinuities when crossing some fixed hypersurfaces. These problems, started with Bressan-Hong [15], [16], have been studied, in connection with HJB, in Barles-Briani-Chasseigne [10], Barnard-Wolenski [14], Rao-Zidani [30], Barles-Briani-Chasseigne [11], Rao-Siconolfi-Zidani [29], Barles-Chasseigne [13], Achdou-Oudet-Tchou [3], Barles-Briani-Chasseigne-Imbert [12], Imbert-Monneau [25]. In this paper we study a possible approximation of an optimal control problem on a network with a junction.…”