Multiscale Homogenizations for first-order Hamilton-Jacobi-Bellman Equations

TL;DR

This paper investigates the homogenization of first-order Hamilton-Jacobi-Bellman equations in quasi-periodic and almost periodic settings, establishing ergodicity and limits for these classes of equations.

Contribution

It provides new results on quasi-periodic and almost periodic homogenization of Hamilton-Jacobi-Bellman equations, extending previous work in the field.

Findings

Established ergodicity under non-resonance conditions.

Solved almost periodic homogenization as a limit of quasi-periodic cases.

Extended the theoretical framework for homogenization in these settings.

Abstract

The quasi-periodic homogenization for some classes of first-order Hamilton-Jaconi-Bellman equation is studied in this paper. The cell problem of the quasi-periodic homogenization satisfies the non-resonance condition, under which the corresponding deterministic system is ergodic. The almost periodic homogenization for the same classes of equations is also solved, as a limit of a sequence of quasi-periodic homogenizations. Here, the almost periodicity is in the sense of H. Bohr. This result has been cited by some authors, for example: by H. Ishii, "Almost periodic homogenization of Hamilton-Jacobi equations", in Int. Conf. on Diff. Eqs., vol.1, Berlin, 1999, World Scientific, River Edge, NJ 2000, pp. 600-605; and by P.-L. Lions, and P.E. Souganidis, "Correctors for the Homogenizations of Hamilton-Jacobi Equations in the stationary ergodic setting", Comm. Pure Appl. Math. LVI, (2003), pp. 1501-1524.