On Representation Formulas for Long Run Averaging Optimal Control Problem

We investigate an optimal control problem with an averaging cost. The asymptotic behavior of the values is a classical problem in ergodic control. To study the long run averaging we consider both Cesàro and Abel means. A main result of the paper says that there is at most one possible accumulation point in the uniform convergence topology of the values, when the time horizon of the Cesàro means converges to infinity or the discount factor of the Abel means converges to zero. This unique accumulation point is explicitly described by representation formulas involving probability measures on the state and control spaces. As a byproduct we obtain the existence of a limit value whenever the Cesàro or Abel values are equicontinuous. Our approach allows to generalise several results in ergodic control, and in particular it allows to cope with cases where the limit value is not constant with respect to the initial condition. Introduction Let us consider the following control system x(t) = f(x(t), α(t)), α(t) ∈ A, t ∈ R, (1) with its unique solution x(t, x0, α) such that x(0) = x0. Here f : R ×A 7→ R is supposed to be bounded and Lipschitz, and A is some compact metric space. We will denote by A the set of all measurable controls α : R 7→ A. Given a bounded cost function l : R × A 7→ R, we consider two ways of averaging the cost along the trajectory x(t, x0, α), t ≥ 0, (Cesàro Mean) 1 T ∫ T 0 l(x(s, x0, , α), α(s))ds, This work has been partially supported by the French National Research Agency ANR-10-BLAN 0112.