Optimal control of parabolic equations -- a spectral calculus based approach

TL;DR

This paper introduces a spectral calculus-based method for solving constrained parabolic optimal control problems, providing closed-form solutions, sensitivity analysis, and efficient numerical approximations using rational Krylov techniques.

Contribution

It offers a novel geometric analysis and a discretization-independent numerical approach for parabolic control problems using spectral calculus and rational Krylov approximations.

Findings

Closed-form solution for the control problem

Efficient rational Krylov approximation technique

Robust numerical method with exponential convergence

Abstract

In this paper we consider a constrained parabolic optimal control problem. The cost functional is quadratic and it combines the distance of the trajectory of the system from the desired evolution profile together with the cost of a control. The constraint is given by a term measuring the distance between the final state and the desired state towards which the solution should be steered. The control enters the system through the initial condition. We present a geometric analysis of this problem and provide a closed-form expression for the solution. This approach allows us to present the sensitivity analysis of this problem based on the resolvent estimates for the generator of the system. The numerical implementation is performed by exploring efficient rational Krylov approximation techniques that allow us to approximate a complex function of an operator by a series of linear problems. Our method does not depend on the actual choice of discretization. The main approximation task is to construct an efficient rational approximation of a generalized exponential function. It is well known that this class of functions allows exponentially convergent rational approximations, which, combined with the sensitivity analysis of the closed form solution, allows us to present a robust numerical method. Several case studies are presented to illustrate our results.