## mpdist2 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Heat:
• linear parabolic operator of order 2.
• Defined on a 1-3-dim domain in 1-3-dim space
• Time dependent.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2017-06-02 by John Pearson. Published on 2017-12-11

## mpdist2 description:

### Introduction

Here we present a simple distributed optimal control problem of the heat equation. Problems of this type are examined in detail within [Tröltzsch2010, Chapter 3]. The problem was derived as a test for the paper Güttel and Pearson , 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 Pearson2017, 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.

### Optimality System

The following optimality system for the control $u\in {L}^{2}\left(\Omega ×I\right)$, 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.

### Supplementary Material

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

 $\begin{array}{ccc}\hfill y& =\left(\frac{4}{d\phantom{\rule{0.3em}{0ex}}{\pi }^{2}\phantom{\rule{0.3em}{0ex}}\beta }\phantom{\rule{0.3em}{0ex}}{e}^{T}-\frac{4}{\left(4+d\phantom{\rule{0.3em}{0ex}}{\pi }^{2}\right)\phantom{\rule{0.3em}{0ex}}\beta }\phantom{\rule{0.3em}{0ex}}{e}^{t}\right)\prod _{k=1}^{d}cos\left(\frac{\pi \phantom{\rule{0.3em}{0ex}}{x}_{k}}{2}\right)\hfill & \hfill \\ \hfill p& =-\left({e}^{T}-{e}^{t}\right)\prod _{k=1}^{d}cos\left(\frac{\pi \phantom{\rule{0.3em}{0ex}}{x}_{k}}{2}\right)\hfill \\ \hfill u& =\frac{1}{\beta }\phantom{\rule{0.3em}{0ex}}\left({e}^{T}-{e}^{t}\right)\prod _{k=1}^{d}cos\left(\frac{\pi \phantom{\rule{0.3em}{0ex}}{x}_{k}}{2}\right)\hfill \end{array}$

Notice that the sign of $p$ is reversed in [Güttel and Pearson2017, Section 6.1]. Consequently, the control law reads $u=\frac{1}{\beta }\phantom{\rule{0.3em}{0ex}}p$ in [Güttel and Pearson2017, Section 6.1].

### References

S. Güttel and J. W. Pearson. A rational deferred correction approach to parabolic optimal control problems. IMA Journal of Numerical Analysis, online-ﬁrst, 2017. doi: 10.1093/imanum/drx046. URL https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drx046/4372128.

F. Tröltzsch. Optimal Control of Partial Diﬀerential Equations, volume 112 of Graduate Studies in Mathematics. American Mathematical Society, Providence, 2010.