Stability Analysis of Discrete-Time Infinite-Horizon Optimal Control With Discounted Cost

We analyze the stability of general nonlinear discrete-time systems controlled by an optimal sequence of inputs that minimizes an infinite-horizon discounted cost. First, assumptions related to the controllability of the system and its detectability with respect to the stage cost are made. Uniform semiglobal and practical stability of the closed-loop system is then established, where the adjustable parameter is the discount factor. Stronger stability properties are thereupon guaranteed by gradually strengthening the assumptions. Next, we show that the Lyapunov function used to prove stability is continuous under additional conditions, implying that stability has a certain amount of nominal robustness. The presented approach is flexible and we show that robust stability can still be guaranteed when the sequence of inputs applied to the system is no longer optimal but near-optimal. We also analyze stability for cost functions in which the importance of the stage cost increases with time, opposite to discounting. Finally, we exploit stability to derive new relationships between the optimal value functions of the discounted and undiscounted problems, when the latter is well-defined.