Departmentof Mathematics

Introduction

Here we present a simple distributed optimal control problem of the heat equation. Problems of this type are examined in detail within [Tröltzsch, 2010, Chapter 3]. The problem was derived as a test for the paper Güttel and Pearson [2017], which required optimal states and controls that are not polynomial in spatial or time variables. The problem is generically usable in dimensions 1 through 3 and maintains a parameter dependence for the regularization parameter $\beta$ to serve as a test case for the $\beta$ dependence of solvers.

The 2d version of this problem and analytical solution appear in [Güttel and Pearson, 2017, Section 6.1], where computations for $\beta =0.05$ and final times $T=1$ were conducted. The implementation is provided for $d=2$ as well.

Variables & Notation

Unknowns

$$\begin{array}{ccccc}\hfill u& \in L^2\left(\Omega \times I\right)\hfill & \hfill & \text{control variable}\hfill & \hfill \\ \hfill y& \in W\left(0,T\right):=L^2\left(I;H_0^1\left(\Omega \right)\right)\cap H^1\left(I;H^{-1}\left(\Omega \right)\right)\hfill & \hfill & \text{state variable}\hfill \end{array}$$

Given Data

$$\begin{array}{ccccc}\hfill d& \in \left\{1,2,3\right\}\hfill & \hfill & \text{dimension of the problem}\hfill & \hfill \\ \hfill T& >0\hfill & \hfill & \text{length of time interval}\hfill \\ \hfill \Omega & ={\left(-1,1\right)}^d\hfill & \hfill & \text{spatial domain}\hfill \\ \hfill I& =\left(0,T\right)\hfill & \hfill & \text{time interval}\hfill \\ \hfill Q& ={\left(-1,1\right)}^d\times \left(0,T\right)\hfill & \hfill & \text{space-time domain}\hfill \\ \hfill \Sigma & =\partial \Omega \times \left(0,T\right)\hfill & \hfill & \text{lateral boundary of }Q\hfill \\ \hfill \beta & >0\hfill & \hfill & \text{regularization parameter}\hfill \\ \hfill y_{d,T}& =\left[\frac{d{\pi }^2}{4}+\frac{4}{d{\pi }^2\beta }\right]e^T+\left[1-\frac{d{\pi }^2}{4}-\frac{4}{\left(4+d{\pi }^2\right)\beta }\right]e^t\hfill & \hfill & \text{aux. function for desired state}\hfill \\ \hfill y_d& =y_{d,T}\left(t\right)\prod _{k=1}^d\cos <!--FUNCTION\; APPLICATION-->\left(\frac{\pi x_k}{2}\right)\hfill & \hfill & \text{desired state}\hfill \\ \hfill y_0& =\left(\frac{4}{d{\pi }^2\beta }{e}^T-\frac{4}{\left(4+d{\pi }^2\right)\beta }\right)\prod _{k=1}^d\cos <!--FUNCTION\; APPLICATION-->\left(\frac{\pi x_k}{2}\right)\hfill & \hfill & \text{initial state}\hfill \end{array}$$

Problem Description

Optimality System

The following optimality system for the control $u\in L^2\left(\Omega \times I\right)$, the state $y\in W\left(0,T\right)$, and the adjoint state $p\in W\left(0,T\right)$, given in the strong form, characterizes the unique minimizer.

$$\begin{array}{ccccc}\hfill y_t-△y& =u\hfill & \hfill & \text{in }Q\hfill & \hfill \\ \hfill y& =0\hfill & \hfill & \text{on }\Sigma \hfill \\ \hfill y& =y_0\hfill & \hfill & \text{at }t=0\hfill \\ \hfill -p_t-△p& =y-y_d\hfill & \hfill & \text{in }Q\hfill \\ \hfill p& =0\hfill & \hfill & \text{on }\Sigma \hfill \\ \hfill p& =0\hfill & \hfill & \text{at }t=T\hfill \\ \hfill u& =-\frac{1}{\beta }p\hfill & \hfill & \text{in }Q\hfill \end{array}$$

Supplementary Material

The optimal state, adjoint state, and control are known analytically:

$$\begin{array}{ccc}\hfill y& =\left(\frac{4}{d{\pi }^2\beta }{e}^T-\frac{4}{\left(4+d{\pi }^2\right)\beta }{e}^t\right)\prod _{k=1}^d\cos <!--FUNCTION\; APPLICATION-->\left(\frac{\pi x_k}{2}\right)\hfill & \hfill \\ \hfill p& =-\left(e^T-e^t\right)\prod _{k=1}^d\cos <!--FUNCTION\; APPLICATION-->\left(\frac{\pi x_k}{2}\right)\hfill \\ \hfill u& =\frac{1}{\beta }\left(e^T-e^t\right)\prod _{k=1}^d\cos <!--FUNCTION\; APPLICATION-->\left(\frac{\pi x_k}{2}\right)\hfill \end{array}$$

Notice that the sign of $p$ is reversed in [Güttel and Pearson, 2017, Section 6.1]. Consequently, the control law reads $u=\frac{1}{\beta }p$ in [Güttel and Pearson, 2017, Section 6.1].

References