A duality-based approach for solving linear parabolic control constrained optimal control problems

TL;DR

This paper develops a duality-based approach for solving linear parabolic optimal control problems with control constraints, providing explicit dual formulations, proving solution properties, and designing efficient numerical methods.

Contribution

It introduces a Fenchel duality scheme for constrained parabolic control problems, deriving explicit dual problems and proposing conjugate gradient and semismooth Newton methods.

Findings

Dual problem explicitly derived and analyzed.

Proposed methods are efficient and accurate.

Numerical examples validate the approach.

Abstract

This paper is concerned with the optimal control problem governed by a linear parabolic equation and subjected to box constraints on control variables. This type of problem has important applications in heating and cooling systems. By applying the scheme of Fenchel duality, we derive the dual problem explicitly where the control constraints in primal problem are embedded in the dual problem's objective functional. The existence and uniqueness of the solution to the dual problem are proved and the first-order optimality conditions are also derived. In addition, we discuss the saddle point property between solution of the primal problem and the dual problem. The solution of primal problem can be readily obtained by the solution of the dual problem. To solve the dual problem numerically, we design two implementable method: conjugate gradient method and semismooth Newton method. Three example problems are solved, numerical results show that the proposed method is efficient and accurate.