Optimal control of a 2D diffusion-advection process with a team of mobile actuators under jointly optimal guidance

TL;DR

This paper develops an optimization framework for controlling a 2D diffusion-advection process with mobile actuators, jointly optimizing control and guidance to minimize a combined cost, with proven convergence and demonstrated effectiveness.

Contribution

It introduces a novel approach that transforms the control problem into guidance optimization, with convergence guarantees and practical numerical demonstrations.

Findings

The framework effectively minimizes combined control and guidance costs.

Convergence conditions ensure approximate solutions approach the optimal.

Numerical examples validate the proposed method's performance.

Abstract

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 for guidance only 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.