Finite-Horizon Discounted Optimal Control: Stability and Performance

Motivated by (approximate) dynamic programming and model predictive control problems, we analyse the stability of deterministic nonlinear discrete-time systems whose inputs minimize a discounted finite-horizon cost. We assume that system satisfies stabilizability detectability properties with respect to stage Then, Lyapunov function for closed-loop is constructed uniform semiglobal property ensured, where adjustable parameters are both discount factor horizon length, which corresponds number iterations algorithms like value iteration. Stronger such as global exponential also provided strengthening initial assumptions. give bounds on length under holds provide conditions these less conservative than literature infinite-horizon cost undiscounted costs, respectively. In addition, new relationships between optimal functions discounted, undiscounted, costs respectively, very different from those available in approximate literature. These rely assumptions more likely be satisfied context. Finally, investigate when only near-optimal sequence available, covering iteration particular case.