On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations

Abstract

We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong bracket generation condition on the vector fields. We first prove the existence of a critical value of the Hamiltonian by means of the ergodic approximation. Next we prove the existence of a possibly discontinuous viscosity solution to the critical equation. We show that the long-time behaviour of solutions to the evolutive Hamilton-Jacobi equation is described in terms of the critical constant and a critical solution. As in the classical weak KAM theory we find a fixed point of the Hamilton-Jacobi-Lax-Oleinik semigroup, although possibly discontinuous. Finally we apply the existence and properties of the critical value to the periodic homogenization of stationary and evolutive H-J equations.

A Dijkstra-Type Algorithm for Dynamic Games

“…with δ, l > 0 and Ω a rectangle, say Ω = (0, c) × (0, d) (see [9] for other results on such equation). An associated differential game can be chosen with dynamics The locally uniform convergence of the solution W of the discrete Isaacs equation (11) to the lower value function v − as h and the mesh size k of the grid X tend to zero was proved in [5] in the case v − is continuous, T has a Lipschitz boundary, and k/ h → 0, see also [17], the survey by Bardi et al [6], and the recent book by Falcone and Ferretti [19].…”

Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension

“…When the coercivity condition of H in p is dropped, one loses control on the derivatives of the solutions of equation (HJ ε ) and of the associated "cell" problem, which are no longer Lipschitz continuous in general. As a consequence, homogenization of (HJ ε ) is known to fail even in the periodic case, regardless to the fact that the Hamiltonian is or is not convex in p, see for instance the introductions in [19,20] and some examples in [15]. In this generality, supplementary conditions need to be assumed to compensate the lack of coercivity of the Hamiltonian.…”

On cell problems for Hamilton–Jacobi equations with non-coercive Hamiltonians and their application to homogenization problems

“…We also refer the reader to [2] as a related work to [8]. Homogenization results with non-coercive Hamiltonians can be seen in [4,5,6,9,10,18,22]. Hamiltonians with some partial coercivity is studied in [4], and [5] treats equations with u/ε-term.…”

“…Homogenization results with non-coercive Hamiltonians can be seen in [4,5,6,9,10,18,22]. Hamiltonians with some partial coercivity is studied in [4], and [5] treats equations with u/ε-term. The papers [6,22,18] are concerned with homogenization on spaces with a (sub-Riemannian) geometrical condition.…”