# Bang-bang approximation of a traveling wave

Bang-bang approximation of a traveling wave
State dimension: 2
Differential states: 1
Discrete control functions: 1
Path constraints: 2

The following problem is an academic example of a PDE constrained optimal control problem with integer control constraints and was introduced in [Hante2009]Address: Berlin, Heidelberg
Author: Hante, Falk M.; Leugering, G\"unter
Booktitle: HSCC '09: Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control
Pages: 209--222
Publisher: Springer-Verlag
Title: Optimal Boundary Control of Convention-Reaction Transport Systems with Binary Control Functions
Year: 2009
.

The control task consists of choosing the boundary value of a transport equation from the extremal values of a traveling wave such that the $L^2$-distance between the traveling wave and the resulting flow is minimized.

## Mathematical formulation

$\begin{array}{ll} \displaystyle \min_{x, q} & \displaystyle \int_0^1\int_0^1 |x(t,s)-x_d(t,s)|^2\,ds\,dt + c \bigvee_0^1 q(t)\,dt \\[1.5ex] \mbox{s.t.} & \displaystyle \frac{\partial}{\partial t}x(t,s)+\frac{\partial}{\partial s}x(t,s) = 0,\quad 0

where

$x_d(t,s)=\frac12\sin(5\pi(t-s))+1,\quad 0\leq t \leq 1,~0\leq s\leq 1$

is the traveling wave (oscillating between 0 and 1), $c>0$ is a (small) regularization parameter and $\bigvee_0^1 q(t)\,dt$ denotes the variation of $q(\cdot)$ over the interval $[0,1]$. Thereby, the solution of the transport equation has to be understood in the usual weak sense defined by the characteristic equations. Systems biology

## Reference solution

For $c=0.0075$ the best known solution is given by

$q^*(t)=\chi_{[0,0.2]}(t)+\chi_{[0.4,0.6]}(t)+\chi_{[0.8,1]}(t),\quad 0\leq t\leq 1$

where $\chi_{[a,b]}(t)$ denotes the indicator function of the interval $[a,b]$.

## References

 [Hante2009] Hante, Falk M.; Leugering, G\"unter (2009): Optimal Boundary Control of Convention-Reaction Transport Systems with Binary Control Functions. Springer-Verlag, HSCC '09: Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control