## 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.

### 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. [2013] but it was carried out by the authors in 2014 (unpublished).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 $u:Q\to {ℝ}^{4}$.)

### 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.