在弱ergodic性假设下平均成本最优控制：相对价值迭代
Average cost optimal control under weak ergodicity hypotheses: Relative value iterations
AI总结：
本文研究了在弱ergodic性假设下，通过相对价值迭代算法求解平均成本最优控制问题，探讨了Bellman方程的正确性以及滚动时间策略的稳定性与渐近最优性。
摘要：
我们研究了具有波兰状态和动作空间的马尔可夫决策过程。动作空间是状态依赖的，并不一定是紧致的。我们首先仅使用接近单调性假设在运行成本上建立最优ergodic占用测度的存在性。然后在额外假设存在小集的情况下研究Bellman方程的正确性，即通常所知的平均成本最优性方程的正确性。我们偏离通常基于消失折扣方法的常规方法，而是将问题映射到一个受控分裂链等价的问题上。我们利用泊松方程的随机表示来推导Bellman方程。接下来，在适当假设下，我们建立了计算Bellman方程解的'相对价值迭代'算法的收敛性结果。此外，我们还展示了与相关滚动时间策略相关的稳定性和渐近最优性结果。
英文摘要：
We study Markov decision processes with Polish state and action spaces. The action space is state dependent and is not necessarily compact. We first establish the existence of an optimal ergodic occupation measure using only a near-monotone hypothesis on the running cost. Then we study the well-posedness of Bellman equation, or what is commonly known as the average cost optimality equation, under the additional hypothesis of the existence of a small set. We deviate from the usual approach which is based on the vanishing discount method and instead map the problem to an equivalent one for a controlled split chain. We employ a stochastic representation of the Poisson equation to derive the Bellman equation. Next, under suitable assumptions, we establish convergence results for the 'relative value iteration' algorithm which computes the solution of the Bellman equation recursively. In addition, we present some results concerning the stability and asymptotic optimality of the associated rolling horizon policies.
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