This paper describes an optimization framework to control a distributed parameter system (DPS) using a team of mobile actuators. The framework simultaneously seeks optimal control of the DPS and optimal guidance of the mobile actuators such that a cost function associated with both the DPS and the mobile actuators is minimized subject to the dynamics of each. The cost incurred from controlling the DPS is linear–quadratic, which is transformed into an equivalent form as a quadratic term associated with an operator-valued Riccati equation. This equivalent form reduces the problem to seeking only for guidance because the optimal control can be recovered once the optimal guidance is obtained. We establish conditions for the existence of a solution to the proposed problem. Since computing an optimal solution requires approximation, we also establish the conditions for convergence to the exact optimal solution of the approximate optimal solution. That is, when evaluating these two solutions by the original cost function, the difference becomes arbitrarily small as the approximation gets finer. Two numerical examples demonstrate the performance of the optimal control and guidance obtained from the proposed approach.

本文描述了使用一组移动执行器来控制分布式参数系统 (DPS) 的优化框架。该框架同时寻求 DPS 的最佳控制和移动执行器的最佳引导，使得与 DPS 和移动执行器相关的成本函数在每个人的动态下最小化。控制 DPS 所产生的成本是线性-二次的，转化为二次项的等价形式与算子值 Riccati 方程相关联。这种等价形式将问题简化为仅寻求指导，因为一旦获得最佳指导，就可以恢复最佳控制。我们为所提出问题的解决方案的存在建立条件。由于计算最优解需要近似，我们还建立了收敛到近似最优解的精确最优解的条件。也就是说，当通过原始成本函数评估这两个解决方案时，随着近似变得更精细，差异变得任意小。两个数值例子证明了从所提出的方法中获得的最优控制和引导的性能。