AI 中文总结
本文针对带噪声的线性抛物型PDE，提出极小极大约束密度控制的数值最优控制框架，将问题转化为凸半无限规划，构建高效算法求解并通过数值示例验证。
AI 中文摘要
本文针对一类带噪声的线性抛物型偏微分方程（PDE），尤其是带噪声的热方程，提出了一种用于极小极大约束密度控制的数值最优控制框架。研究目标是在固定时间范围内，将初始密度迁移至目标密度，同时相对于控制动作最小化指定代价、相对于扰动最大化该代价，且需满足给定的凸路径约束。为解决该问题，采用有限差分近似离散PDE中的空间导数，将原问题转化为时间域内的常微分方程组。随后对控制与扰动轨迹的容许空间进行有限参数化，并在适度假设下将所得最优控制问题构建为凸半无限规划（SIP）。通过利用凸SIP理论的新数值工具，本文建立了精确解的保证，该保证考虑了无限组扰动实现下的约束满足情况，还构建了一种基于优化的计算高效算法以求解这些精确解。文中包含全面的数值示例，用于展示和验证本文的研究结果。
英文摘要
This article introduces a numerical optimal control framework for minmax constrained density control for a class of noisy linear parabolic partial differential equations (PDEs), in particular the noisy heat equation. The goal is to transport an initial density to a target density while minimizing a specified cost with respect to control actions and maximizing it with respect to disturbances, all within a fixed time horizon while satisfying given convex path constraints. To address this, the spatial derivatives in the PDE are discretized using finite-difference approximations, transforming the problem into a system of ordinary differential equations in time. The admissible space of control and disturbance trajectories is then finitely parametrized, and the resulting optimal control problem is formulated as a convex semi-infinite program (SIP) under mild assumptions. By leveraging new numerical tools from convex SIP theory, we establish guarantees for exact solutions that account for constraint satisfaction under an infinite family of disturbance realizations, and we establish an optimization-based computationally efficient algorithm to recover these solutions. Comprehensive numerical examples to demonstrate and validate our findings are included.
CommentsRevised version of a journal submission; 15 pages (2-col)
