Optimal Control of Convection-Cooling and Numerical Implementation

TL;DR

This paper develops a rigorous mathematical framework and numerical methods for optimally controlling convection-cooling in fluid systems, demonstrating effectiveness through computational experiments.

Contribution

It provides the first rigorous proof of existence, optimality conditions, and numerical implementation for bilinear control of convection-cooling.

Findings

Existence of optimal control proven.

First order optimality conditions derived.

Numerical experiments validate control effectiveness.

Abstract

This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion-convection model. This essentially leads to a bilinear optimal control problem. A rigorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.