# Bang-bang approximation of a traveling wave

Bang-bang approximation of a traveling wave | |
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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*

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

## Mathematical formulation

where

is the traveling wave (oscillating between 0 and 1), is a (small) regularization parameter and denotes the variation of over the interval . 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 the best known solution is given by

where denotes the indicator function of the interval .

## 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 |