## rddist1 details:

Keywords: analytic solution

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 2013-05-23 by Fredi Tröltzsch. Published on 2013-06-21

## rddist1 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 stopping a wave front at a certain time and keeping it ﬁxed afterwards. This problem appears in [Buchholz et al.2013, Section 5.4]. In the same paper, additional examples can be found which cover the other optimization goals mentioned above. It has the explicitly known optimal control (forcing) ${f}_{\text{stop}}$ deﬁned below and displayed in Figure 0.2.

### 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 $t$. An analytical expression for ${u}_{\text{nat}}$ is not known.  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

The optimal state and the optimal control are given by:

These functions are shown in Figure 0.2.  Figure 0.2: Control ${f}_{\text{stop}}$ (left) and desired state ${u}_{Q}$ (right).

### 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.