## ccparfin1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

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

Design constraints:

• box of order 0

State constraints:

• none

Mixed constraints:

• none

Submitted on 2013-05-27 by Fredi Tröltzsch. Published on 2013-06-21

## ccparfin1 description:

### Introduction

This example is an optimal control problem for a semilinear heat equation with cubic nonlinearity in a two dimensional domain. There are four time-dependent control functions restricted by box constraints. The example is constructed such that a locally optimal solution is explicitly known. It was used in the context of model reduction by POD to test an a posteriori error estimator for optimality, and appears in [Kammann et al.2013, Section 4.2].

### Variables & Notation

#### Given Data

The graphs of the functions ${w}_{1},\dots ,{w}_{4}$ are shown in Figure 0.1. Figure 0.1: Weight functions ${w}_{1},\dots ,{w}_{4}$ for the controls ${u}_{i}$.

### Optimality System

The following optimality system for the state $y$, the control $u$, and the adjoint state $p$, given in the strong form, represents ﬁrst-order necessary optimality conditions.

### Supplementary Material

A set of locally optimal controls ${u}_{i}$ with associated state $y$ and adjoint state $p$ are known analytically.

The controls are shown in Figure 0.2.

To show the local optimality of this solution, we verify that $\left(ȳ,{ū}_{1},\dots ,{ū}_{4}\right)$ obeys the standard second-order suﬃcient optimality conditions.1 1This analysis is not presented in Kammann et al.  but it was carried out by the authors in 2014 (unpublished). To this end, we introduce the Lagrangian; writing $u:=\left({u}_{1},\dots ,{u}_{4}\right)$, we deﬁne

 $\mathsc{ℒ}\left(y,u,p\right):=J\left(y,u\right)-{\int }_{Q}\left\{\frac{\partial y}{\partial t}-△y+{y}^{3}+d-\sum _{i=1}^{4}{w}_{i}\phantom{\rule{0.3em}{0ex}}{u}_{i}\right\}p\phantom{\rule{3.0235pt}{0ex}}\mathrm{\text{d}}x\phantom{\rule{0.3em}{0ex}}\mathrm{\text{d}}t.$

The second-order derivative of $\mathsc{ℒ}$ with respect to $\left(y,u\right)$ at $\left(ȳ,ū,\stackrel{̄}{p}\right)$ is

Notice that $|ȳ\left(x,t\right)|\le 1$ and $|\stackrel{̄}{p}\left(x,t\right)|\le 0.01$ is satisﬁed. Invoking [Tröltzsch2010, Theorem 5.17], we obtain that $ū$ is locally optimal with respect to the topology of ${L}^{\infty }{\left(0,T\right)}^{4}$. (The quoted theorem is formulated for a control function $u:Q\to ℝ$, but it obviously extends to the case $u:Q\to {ℝ}^{4}$.)  Figure 0.2: Optimal controls ${ū}_{1},{ū}_{2}$. By symmetry, ${ū}_{1}={ū}_{3}$ and ${ū}_{2}={ū}_{4}$ holds.

### Revision History

• 2016–06–10: added comment on second-order suﬃcient conditions
• 2014–10–30: formulas for the control weights ${w}_{1},\dots ,{w}_{4}$ corrected to match the calculations and ﬁgures in [Kammann et al.2013, Section 4.2]
• 2013–05–27: problem added to the collection

### References

E. Kammann, F. Tröltzsch, and S. Volkwein. A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD. ESAIM. Mathematical Modelling and Numerical Analysis, 47(2):555–581, 2013. ISSN 0764-583X. doi: 10.1051/m2an/2012037.

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