带有$L^\infty$和积分成本泛函的最优控制
Optimal Control with $L^\infty$ and Integral Cost Functionals
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中文总结 AI 辅助
本文研究结合$L^\infty$与积分成本泛函的最优控制问题，通过松弛框架证明最优控制存在，引入光滑逼近方法求解，并以排队问题验证峰值与累积性能的权衡。
中文摘要 AI 辅助
许多控制问题传统上采用积分成本来建模，以捕捉系统的累积性能；另一类控制问题则采用上确界或$L^\infty$成本来捕捉峰值或最坏情况行为。当累积性能和峰值性能都重要时，将两种成本结合起来构造目标函数是很自然的。尽管每种准则都已得到充分研究，但它们的组合尚未得到广泛探索，本文正是针对这类控制问题展开研究。为了证明最优解的存在性，我们首先考虑松弛框架，其中控制通过概率分布来表示。利用这类控制空间众所周知的紧性和凸性性质，我们证明了最优松弛控制的存在性，最终还证明了对于任意$\epsilon > 0$，存在$\epsilon$-最优的纯（或经典）控制。尽管有这些存在性结果，由于上确界项引入的非光滑性，计算最优策略仍然具有挑战性，且对于我们的组合问题，动态规划原理是否成立尚不明确。为解决这一问题，我们引入一族光滑逼近，生成具有明确定义的最优（纯）解的标准控制问题。利用最大值定理，我们证明了纯控制下光滑问题的解对于原问题是$\epsilon$-最优的，且当光滑参数收敛到零时，$\epsilon$趋于零。最后，利用本文提出的方法，我们研究了一个排队问题，以说明（除其他外）可以实现峰值拥塞水平与累积性能之间所需的权衡。
英文摘要
Many control problems are classically formulated using integral costs that capture the cumulative performance of a system. Peak or worst-case behavior is captured via supremum or $L^\infty$-costs in another variety of control problems. When both cumulative and peak performance are important, it is natural to consider objective functions that combine the two costs. Although each criterion is well studied, their combination has not been explored extensively and we precisely work on such control problems. Towards establishing the existence, we first consider the relaxed framework, where control is considered using probability distributions. Using the well-known compactness and convexity properties of such control spaces, we establish the existence of an optimal relaxed control---we eventually establish the existence of an $ε$-optimal pure (or classical) control, for every $ε> 0.$ Despite these existence results, computing optimal policies remains challenging due to the non-smoothness introduced by the supremum term, and it is not clear whether the dynamic programming principle holds for our combined problem. To address this, we introduce a family of smooth approximations that yield standard control problems with well-defined optimal (pure) solutions. Using Maximum Theorem, we establish that the solutions of the smooth problems among pure controls form $ε$-optimal for the original problem, with $ε$ tending to zero as the smoothing parameter converges to zero. Finally using the methods proposed in this paper, we study a queueing problem to illustrate (among others) that the required trade-off between peak congestion levels and cumulative performance can be achieved.
发表机构
- IIT Bombay（印度理工学院孟买分校）
机构由 AI 辅助整理，请以论文原文为准。
