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Sparse optimal control of the Schlögl and FitzHugh-Nagumo system

Christopher Ryll:

 

We investigate optimal control problems for two reaction-di ffusion problems, namely the so called Schlögl or Nagumo model and the FitzHugh-Nagumo equations. Under appropriate initial conditions, the uncontrolled solutions of these systems behave like a traveling wave or a spiral wave. It is a natural task to control such wave type solutions in an optimal way. Often, it is desired to apply controls only in small parts of the spatial domain, since it is not always realistic to apply distributed controls in the whole spatial domain. This is a typical situation, where the theory of sparse optimal control can be applied. Therefore, the objective functional is extended by the weighted L1-norm of the control. The topic of sparse controls itself has recently been studied actively for elliptic partial diff erential equations and the needed techniques for dealing with the not-diff erentiable part of the objective functional can be easily applied to our case, namely semilinear parabolic equations or systems.
Since they are of non-monotone type, the theory of existence, uniqueness and regularity of associated solutions is more delicate than for equations of monotone type. We prove existence and uniqueness of a solution of the FitzHugh-Nagumo equations that works in Lipschitz domains of spatial dimension one, two or three. By an L- approach, we show the second-order Fréchet-diff erentiability of the control-to-state mapping. Based on this foundation, we derive first order necessary optimality conditions for sparse optimal controls.
Further, we study various numerical examples in spatial dimension one and two. We control traveling wave fronts as solutions to the Schlögl-equation. In the case of the FitzHugh-Nagumo system, spiral waves occur. Controlling such patterns is geometrically impressive but numerically fairly demanding. To our best knowledge, sparse optimal controls for such equations were not yet discussed in literature. However, there is a rich literature on feedback control problems in the community of Physics.

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