Sparse Optimal Control of the Schlogl and FitzHugh-Nagumo Systems

作者：Casas Eduardo^*; Ryll Christopher; Troeltzsch Fredi

来源：Computational Methods in Applied Mathematics, 2013, 13(4): 415-442.

DOI：10.1515/cmam-2013-0016

• Optimal Control
• Reaction-Diffusion System
• Schlogl Model
• FitzHugh-Nagumo System
• Sparse Control
• Traveling Wave Fronts
• Spiral Waves

We investigate the problem of sparse optimal controls for the so-called Schlogl model and the FitzHugh-Nagumo system. In these reaction-diffusion equations, traveling wave fronts occur that can be controlled in different ways. The L 1 norm of the distributed control is included in the objective functional so that optimal controls exhibit effects of sparsity. We prove the differentiability of the control-to-state mapping for both dynamical systems, show the well-posedness of the optimal control problems and derive first-order necessary optimality conditions. Based on them, the sparsity of optimal controls is shown. The theory is illustrated by various numerical examples, where wave fronts or spiral waves are controlled in a desired way.

• 出版日期2013-10