## rddist2 details:

Keywords:

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Schlögel or Nagumo :
• semi-linear parabolic operator of order 2.
• Defined on a 1-dim domain in 1-dim space
• Time dependent.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2013-05-24 by Fredi Tröltzsch. Published on 2013-05-21

## rddist2 description:

### Introduction

This is a distributed optimal control problem for a semilinear 1D parabolic reaction-diﬀusion equation, where traveling wave fronts occur. The state equation is known as Schlögl model in physics and as Nagumo equation in neurobiology. In this context, various goals of optimization are of interest, for instance the stopping, acceleration, or extinction of a traveling wave. Here, we discuss the problem of re-directing a wave front after a certain time. The control is acting only on two subdomains near the boundary, which occupy the portion $q$ of the entire spatial domain. This problem appears in [Buchholz et al.2013, Section 5.2]. In the same paper, additional examples can be found which cover the other optimization goals mentioned above.

### Variables & Notation

#### Given Data

The natural uncontrolled state ${u}_{\text{nat}}$ is shown in Figure 0.1. In the ﬁgure, the horizontal axis shows the spatial variable $x$ while the vertical one displays the time $t$. The speed $c$ of the uncontrolled wave front was determined numerically.  Figure 0.1: Initial state ${u}_{0}$ (left) and natural uncontrolled state ${u}_{\text{nat}}$ (right).

### Problem Description

Notice that the PDE has a non-monotone nonlinearity. The associated homogeneous elliptic (stationary) equation admits three diﬀerent solutions; namely, the functions ${u}_{1}\left(x\right)\equiv -\sqrt{3}$, ${u}_{2}\left(x\right)\equiv 0$, and ${u}_{3}\left(x\right)\equiv \sqrt{3}$.

### Optimality System

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

### Supplementary Material

Figure 0.2 displays the desired state, as well as the optimal state and optimal control.   Figure 0.2: Re-directing a wave front with control support of size $q=0.6$; desired state ${u}_{Q}$ (top left), optimal state $u$ (top right) and optimal control $f$ (bottom).

### References

R. Buchholz, H. Engel, E. Kammann, and F. Tröltzsch. On the optimal control of the Schlögl model. Computational Optimization and Applications, 56(1):153–185, 2013. doi: 10.1007/s10589-013-9550-y.