On ergodic control problem for viscous Hamilton--Jacobi equations for weakly coupled elliptic systems

Abstract

In this article we study ergodic problems in the whole space ${\mathbb{R}}^N$ for weakly coupled systems of viscous Hamilton-Jacobi equations with coercive right-hand sides. The Hamiltonians are assumed to have a fairly general structure and the switching rates need not be constant. We prove the existence of a critical value ${\lambda }^{\ast }$ such that the ergodic eigenvalue problem has a solution for every $\lambda \le {\lambda }^{\ast }$ and no solution for $\lambda >{\lambda }^{\ast }$. Moreover, the existence and uniqueness of non-negative solutions corresponding to the value ${\lambda }^{\ast }$ are also established. We also exhibit the implication of these results to the ergodic optimal control problems of controlled switching diffusions.

25 pages

References in corpus (1)

• On unbounded solutions of ergodic problems in r^m for viscous hamilton-jacobi equations