rddist1 details:

Keywords: analytic solution

Global classification: nonlinear-quadratic

Functional: convex quadratic

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

Design constraints:

State constraints:

Mixed constraints:



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) fstop defined below and displayed in Figure 0.2.

Variables & Notation

Unknowns

f L2(Q) control variable (forcing) u L2(0,T;H1(Ω)) H1(0,T;H1(Ω)) L(Q)state variable

Given Data

Ω = (0,L) spatial domain L = 20 side length of domain Q = Ω × (0,T) computational domain T = 5 terminal time u0(x) = 1.23,x [9,11] 0,  elsewhere,  initial condition λ = 106 Tikhonov regularization parameter uQ(,t) = unat(,t), t [0,2.5] unat(,2.5),t (2.5,T] desired state unat solution of the PDE (0.1) for f 0.

The natural uncontrolled state unat 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 unat is not known.


PICPIC

Figure 0.1: Initial state u0 (left) and natural uncontrolled state unat (right).

Problem Description

Minimize 1 2Q(u(x,t) uQ(x,t))2dxdt + λ 2Qf2(x,t)dxdt s.t. u t (x,t) 2u x2(x,t) + 1 3u3(x,t) u(x,t) = f(x,t)in Q u(x,0) = u0(x) in Ω u x(0,t) = u x(L,t) = 0 in (0,T). (0.1) Notice that the PDE has a non-monotone nonlinearity. The associated homogeneous elliptic (stationary) equation admits three different solutions; namely, the functions u1(x) 3, u2(x) 0, and u3(x) 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.

u t (x,t) 2u x2(x,t) + 1 3u3(x,t) u(x,t) = f(x,t) in Q u(x,0) = u0(x) in Ω u x(0,t) = u x(L,t) = 0 in (0,T), p t (x,t) 2p x2(x,t) + u2(x,t)p(x,t) p(x,t) = u(x,t) u Q(x,t)in Q p(x,T) = 0 in Ω p x(0,t) = p x(L,t) = 0 in (0,T), f(x,t) = 1 λp(x,t) in Q.

Supplementary Material

The optimal state and the optimal control are given by:

u(x,t) = uQ(x,t), fstop(x,t) = 0  for t 2.5, 1 3unat3(x,2.5) unat(x,2.5) 2 x2unat(x,2.5), for t > 2.5.

These functions are shown in Figure 0.2.


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Figure 0.2: Control fstop (left) and desired state uQ (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.