Vanishing discount limit and nonexpansive optimal control and differential games

A classical problem in ergodic control consists of studying the limit behavior of the optimal value $V_\lambda$ of a discounted cost functional with infinite horizon as the discount factor $lambda$ tends to zero. In the literature, this problem has been addressed under various conditions ensuring that the rescaled value function $\lambda V_\lambda$ converges uniformly to a constant limit. The main goal of this paper is to study this problem without such conditions, so that the aforementioned limit need not be constant. So, under a nonexpansivity assumption, we derive Lipschitz bounds which yield compactness of $\{\lambda V_\lambda\}$ for both control systems and differential games. Then, we study the convergence of solutions to Hamilton–Jacobi equations under the hypothesis that the Hamiltonian is radially nondecreasing, hence allowing for the existence noncoercivity directions. Using PDE methods, we show that the convergence is monotone and we characterize the limit as the maximal subsolution of a certain Hamilton–Jacobi equation. Finally, we return to optimal control problems and differential games for nonexpansive dynamics obtaining explicit representation formulas for the uniform limit of $\lambda V_\lambda$ as $lambda\to 0$.