Asymptotic Behaviour of Optimal Control Systems with Low Discount Rates

Let x be the path and u the control that maximizes the cumulative return discounted at rate δ. The purpose is to analyze the dependence of x and u on the rate δ. Under certain conditions, as δ tends to 0, the corresponding solution (x, u) tends to the solution for rate 0. We use results from geometric control theory and the theory of semiflows.