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.