Final-state, Open-loop Control of Parabolic PDEs with Dirichlet Boundary Conditions

TL;DR

This paper develops a method for controlling second-order parabolic PDEs with Dirichlet boundary conditions using open-loop, final-state control, and demonstrates convergence of approximations with explicit solutions for unconstrained cases.

Contribution

It introduces a semi-discrete Galerkin approximation approach for optimal control of parabolic PDEs, with explicit solutions for unconstrained problems and convergence analysis.

Findings

Convergence of approximate solutions to the original control problem.

Explicit characterization of solutions for unconstrained control problems.

Numerical examples demonstrating the effectiveness of the method.

Abstract

In this paper, a quadratic optimal control problem is considered for second-order parabolic PDEs with homogeneous Dirichlet boundary conditions, in which the "point" control function (depending only on time) constitutes a source term. These problems involve choosing a control function (with or without "peak-value" constraints) to approximately steer the solution of the PDE in question to a desired function at the end of a prescribed (finite) time-interval. To compute approximations to the desired optimal control functions, semi-discrete, Galerkin approximations to the equation involved are introduced and the corresponding (approximating) control problems are tackled. It is shown that the sequences of solutions to both the constrained and unconstrained approximating (finite-dimensional) control problems converge, respectively, to the optimal solutions of the control problems involving the original initial/boundary value problem. The solution to the unconstrained approximating problem can be quite explicitly characterized, with the main numerical step for its computation requiring only the solution of a Lyapunov equation. Whereas approximate solutions to the constrained control problems can be obtained on the basis of Lagrangian duality and piecewise linear multipliers. These points are worked out in detail and illustrated by numerical examples involving the heat equation (HEq).