Bilinear Control of Convection-Cooling: From Open-Loop to Closed-Loop

TL;DR

This paper develops and compares open-loop and closed-loop control strategies for enhancing convection-cooling using bilinear control of an incompressible velocity field, with theoretical analysis and numerical validation.

Contribution

It introduces a systematic approach to design and analyze both open-loop and feedback control laws for convection-cooling problems, including optimality conditions and numerical schemes.

Findings

Closed-loop feedback control is easier to implement than open-loop control.

Numerical experiments validate the effectiveness of the proposed control strategies.

The feedback laws derived improve convection-cooling efficiency.

Abstract

This paper is concerned with a bilinear control problem for enhancing convection-cooling via an incompressible velocity field. Both optimal open-loop control and closed-loop feedback control designs are addressed. First and second order optimality conditions for characterizing the optimal solution are discussed. In particular, the method of instantaneous control is applied to establish the feedback laws. Moreover, the construction of feedback laws is also investigated by directly utilizing the optimality system with appropriate numerical discretization schemes. Computationally, it is much easier to implement the closed-loop feedback control than the optimal open-loop control, as the latter requires to solve the state equations forward in time, coupled with the adjoint equations backward in time together with a nonlinear optimality condition. Rigorous analysis and numerical experiments are presented to demonstrate our ideas and validate the efficacy of the control designs.