On the Solvability of Some Ergodic Control Problems in $\mathbb{R}^d$

Abstract

This paper is concerned with stochastic ergodic control problems where both the state and the control space are $\mathbb{R}^d$. We are interested in giving conditions on the fixed drift, the cost function, and the Lagrangian function that are sufficient for synthesizing an optimal control of feedback type. In order to obtain such conditions, we propose an approach that combines the Lyapunov method and the approximation of the problem on bounded sets with reflection of the diffusions at the boundary. We first develop a general framework, and then study particular cases which show how Lyapunov functions can be constructed from the solutions of the approximating problems.

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