Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems

Karl Kunisch; Sérgio S. Rodrigues

doi:10.3934/dcds.2019276

Article Contents

2019, Volume 39, Issue 11: 6355-6389. Doi: 10.3934/dcds.2019276

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Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems

Karl Kunisch^1, and
Sérgio S. Rodrigues^2, ,

1.
Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Heinrichstr. 36, 8010 Graz, Austria
2.
Johann Radon Institute for Computational and Applied Mathematics, ÖAW, Altenbergerstraße 69, A-4040 Linz, Austria

^* Corresponding author
^* Corresponding author

Received: October 31, 2018

Revised: February 28, 2019

Published: August 2019

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Abstract

Global feedback stabilizability results are derived for nonautonomous coupled systems arising from the linearization around a given time-dependent trajectory of FitzHugh-CNagumo type systems. The feedback is explicit and is based on suitable oblique (nonorthogonal) projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. Subsequently, local feedback stabilizability to time-dependent trajectories results are derived for nonlinear coupled parabolic-ODE systems of the FitzHugh-CNagumo type. Simulations are presented showing the stabilizing performance of the feedback control.

Keywords:

Mathematics Subject Classification: 93D15, 93B52, 93C05, 93C20.

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Figure 1. The domain $ \Omega $, and the (computed) first Neumann Laplacian eigenvalues

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Figure 2. Norm of solution. Linearization around the steady state

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Figure 3. Norm of solution. Linearization around the time-dependent trajectory

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Figure 4. Norm of solution. Around the time-dependent trajectory

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Figure 5. Norm of solution. Linearization around the steady state

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Figure 6. Norm of solution. Linearization around the time-dependent trajectory

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Figure 7. Norm of solution. Around the time-dependent trajectory

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Figure 8. Norm of solution. Linearization around the steady state

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Figure 9. Norm of solution. Linearization around the time-dependent trajectory

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Figure 10. Norm of solution. Around the time-dependent trajectory

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Figure 11. Norm of solution. Around the time-dependent trajectory

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Figure 12. Norm of solution. Linearization around the time-dependent trajectory

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Figure 13. Norm of solution. Around the time-dependent trajectory

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Figure 14. Norm of solution. Around the time-dependent trajectory

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Figure 15. Norm of solution. Linearization around the time-dependent trajectory

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Figure 16. PDE only. With $ d = d_2 = \gamma = \rho = 0 $ (uncoupled system) and $ z_0 = 0 $. Solution norm $ \left|{(y,z)}\right|_{{H\times H}}^2 = \left|{y}\right|_{{H}}^2 $. Linearization around the time-dependent trajectory

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