Optimal Control and Vanishing Discount Limit in Continuous and Discrete Time

This thesis investigates the vanishing discount limit within infinite-horizon optimal control, exploring its asymptotic behavior across both continuous and discrete time settings. After reviewing and establishing the fundamental theory of finite- and infinite-horizon optimal control, we use the infinite-horizon framework to characterize the vanishing discount limit, namely the limit of the rescaled value function \(λ V_λ\) as the discount factor \(λ\) tends to zero. This problem has been addressed in earlier works under controllability and ergodicity assumptions ensuring that the rescaled value function converges uniformly to a constant limit. In contrast, we do not impose such conditions. When a uniform limit exists, it is in general a function that can be characterized as the supremum of the family of viscosity subsolutions to the corresponding system of Hamilton--Jacobi equations. When the subsolution is itself a viscosity solution of the corresponding system, we obtain not only the convergence of the vanishing discount limit, but also a specific rate of convergence. An analogous analysis is carried out in discrete time via Shapley and Bellman operators. Exploiting the additional structure of Bellman operators, we show that the uniform limit of the rescaled discounted value function \(α v_α\) as \(α\) tends to zero can be characterized as the supremum of the directing vectors of sub-invariant half-lines.