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

中文翻译：

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有限视野折扣最优控制：稳定性和性能


在（近似）动态规划和模型预测控制问题的推动下，我们分析了确定性非线性离散时间系统的稳定性，该系统的输入使贴现的有限范围成本最小化。我们假设系统满足阶段成本的稳定性和可检测性特性。然后，构造闭环系统的Lyapunov函数并确保均匀的半全局稳定性，其中可调参数既是折扣因子又是层位长度，对应于值迭代等动态规划算法的迭代次数。通过加强初始假设还提供了更强的稳定性特性，例如全局指数稳定性。我们给出了贴现因子和保持稳定性的范围长度的界限，并提供了这些条件，在这些条件下，这些条件分别比贴现无限范围成本和未贴现有限范围成本的文献界限保守。此外，我们分别提供了贴现成本、未贴现成本、无限范围成本和有限范围成本的最优价值函数之间的新关系，这与近似动态规划文献中提供的关系非常不同。这些关系依赖于在控制环境中更有可能得到满足的假设。最后，我们研究了当只有折扣有限范围成本的近乎最优输入序列可用时的稳定性，涵盖近似值迭代作为特定情况。