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

In this article, we study ergodic problems in the whole space RN 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* such that the ergodic eigenvalue problem has a solution for every lambda < lambda* and no solution for lambda > lambda*. Moreover, the existence and uniqueness of non-negative solutions corresponding to the value lambda* are also established. We also exhibit the implication of these results to the ergodic optimal control problems of controlled switching diffusions. (c) 2022 Elsevier Inc. All rights reserved.