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November  2019, 39(11): 6355-6389. doi: 10.3934/dcds.2019276

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

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

Received  November 2018 Revised  March 2019 Published  August 2019

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.

Citation: Karl Kunisch, Sérgio S. Rodrigues. Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6355-6389. doi: 10.3934/dcds.2019276
References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965, URL https://bookstore.ams.org/chel-369-h/. Google Scholar

[2]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos, Solitons & Fractals, 7 (1996), 293-301. doi: 10.1016/0960-0779(95)00089-5. Google Scholar

[3]

V. BarbuS. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478. doi: 10.1137/100785739. Google Scholar

[4]

V. Barbu and G. Wang, Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems, Indiana Univ. Math. J., 54 (2005), 1521-1546. doi: 10.1512/iumj.2005.54.2663. Google Scholar

[5]

T. BreitenK. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh-Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713. doi: 10.1137/15M1038165. Google Scholar

[6]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6. Google Scholar

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[8]

M. Hieber, N. Kajiwara, K. Kress and P. Tolksdorf, Strong time periodic solutions to the bidomain equations with Fitzhugh–Nagumo type nonlinearities, Analysis of PDEs, (2017), URL https://arXiv.org/abs/1708.05304.Google Scholar

[9]

B. Hu, Blow-Up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, 2018. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4. Google Scholar

[10]

V. Y. Ivrii, Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary, Funct. Anal. Appl., 14 (1980), 98-106. doi: 10.1007/BF01086550. Google Scholar

[11]

J. Keener and J. Sneyd, Mathematical Physiology. Vol. II: Systems Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/Ⅱ. Springer, New York, 2009. Google Scholar

[12]

P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in euclidean space, J. Funct. Anal., 106 (1992), 353-357. doi: 10.1016/0022-1236(92)90052-K. Google Scholar

[13]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., (2018). doi: 10.1051/cocv/2018054. Google Scholar

[14]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form ${P}u_t = -{A}u+{\mathcal F}(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386. doi: 10.1007/BF00263041. Google Scholar

[15]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod et Gauthier-Villars, Paris, 1969. Google Scholar

[16]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren Math. Wiss. Einzeldarstellungen, vol. Ⅰ. Springer-Verlag, 1972. doi: 10.1007/978-3-642-65161-8. Google Scholar

[17]

A. Lunardi, Stabilizability of time-periodic parabolic equations, SIAM J. Control Optim., 29 (1991), 810-828. doi: 10.1137/0329044. Google Scholar

[18]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar

[19]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Syst., 30 (2018), Art. 11, 50 pp. doi: 10.1007/s00498-018-0218-0. Google Scholar

[20]

S. S. Rodrigues and K. Sturm, On the explicit feedback stabilisation of 1D linear nonautonomous parabolic equations via oblique projections, IMA J. Math. Control Inform., (2018). doi: 10.1093/imamci/dny045. Google Scholar

[21]

J. M. Rogers and A. D. McCulloch, A collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng, 41 (1994), 743-757. doi: 10.1109/10.310090. Google Scholar

[22]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050. Google Scholar

[23]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001, http://www.ams.org/bookstore-getitem/item=CHEL-343-H. doi: 10.1090/chel/343. Google Scholar

[24]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479. doi: 10.1007/BF01456804. Google Scholar

[25]

W. H. Young, On classes of summable functions and their Fourier series, Proc. R. Soc. Lond., 87 (1912), 225-229. doi: 10.1098/rspa.1912.0076. Google Scholar

show all references

References:
[1]

S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965, URL https://bookstore.ams.org/chel-369-h/. Google Scholar

[2]

R. R. Aliev and A. V. Panfilov, A simple two-variable model of cardiac excitation, Chaos, Solitons & Fractals, 7 (1996), 293-301. doi: 10.1016/0960-0779(95)00089-5. Google Scholar

[3]

V. BarbuS. S. Rodrigues and A. Shirikyan, Internal exponential stabilization to a nonstationary solution for 3D Navier-Stokes equations, SIAM J. Control Optim., 49 (2011), 1454-1478. doi: 10.1137/100785739. Google Scholar

[4]

V. Barbu and G. Wang, Feedback stabilization of periodic solutions to nonlinear parabolic-like evolution systems, Indiana Univ. Math. J., 54 (2005), 1521-1546. doi: 10.1512/iumj.2005.54.2663. Google Scholar

[5]

T. BreitenK. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh-Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713. doi: 10.1137/15M1038165. Google Scholar

[6]

F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, EDP Sciences, Les Ulis, 2012. doi: 10.1007/978-1-4471-2807-6. Google Scholar

[7]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical J., 1 (1961), 445-466. doi: 10.1016/S0006-3495(61)86902-6. Google Scholar

[8]

M. Hieber, N. Kajiwara, K. Kress and P. Tolksdorf, Strong time periodic solutions to the bidomain equations with Fitzhugh–Nagumo type nonlinearities, Analysis of PDEs, (2017), URL https://arXiv.org/abs/1708.05304.Google Scholar

[9]

B. Hu, Blow-Up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, 2018. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18460-4. Google Scholar

[10]

V. Y. Ivrii, Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary, Funct. Anal. Appl., 14 (1980), 98-106. doi: 10.1007/BF01086550. Google Scholar

[11]

J. Keener and J. Sneyd, Mathematical Physiology. Vol. II: Systems Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/Ⅱ. Springer, New York, 2009. Google Scholar

[12]

P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in euclidean space, J. Funct. Anal., 106 (1992), 353-357. doi: 10.1016/0022-1236(92)90052-K. Google Scholar

[13]

K. Kunisch and S. S. Rodrigues, Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators, ESAIM Control Optim. Calc. Var., (2018). doi: 10.1051/cocv/2018054. Google Scholar

[14]

H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form ${P}u_t = -{A}u+{\mathcal F}(u)$, Arch. Ration. Mech. Anal., 51 (1973), 371-386. doi: 10.1007/BF00263041. Google Scholar

[15]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod et Gauthier-Villars, Paris, 1969. Google Scholar

[16]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren Math. Wiss. Einzeldarstellungen, vol. Ⅰ. Springer-Verlag, 1972. doi: 10.1007/978-3-642-65161-8. Google Scholar

[17]

A. Lunardi, Stabilizability of time-periodic parabolic equations, SIAM J. Control Optim., 29 (1991), 810-828. doi: 10.1137/0329044. Google Scholar

[18]

J. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061-2070. doi: 10.1109/JRPROC.1962.288235. Google Scholar

[19]

D. Phan and S. S. Rodrigues, Stabilization to trajectories for parabolic equations, Math. Control Signals Syst., 30 (2018), Art. 11, 50 pp. doi: 10.1007/s00498-018-0218-0. Google Scholar

[20]

S. S. Rodrigues and K. Sturm, On the explicit feedback stabilisation of 1D linear nonautonomous parabolic equations via oblique projections, IMA J. Math. Control Inform., (2018). doi: 10.1093/imamci/dny045. Google Scholar

[21]

J. M. Rogers and A. D. McCulloch, A collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Trans. Biomed. Eng, 41 (1994), 743-757. doi: 10.1109/10.310090. Google Scholar

[22]

R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 66. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. doi: 10.1137/1.9781611970050. Google Scholar

[23]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI, 2001, http://www.ams.org/bookstore-getitem/item=CHEL-343-H. doi: 10.1090/chel/343. Google Scholar

[24]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71 (1912), 441-479. doi: 10.1007/BF01456804. Google Scholar

[25]

W. H. Young, On classes of summable functions and their Fourier series, Proc. R. Soc. Lond., 87 (1912), 225-229. doi: 10.1098/rspa.1912.0076. Google Scholar

Figure 1.  The domain $ \Omega $, and the (computed) first Neumann Laplacian eigenvalues
Figure 2.  Norm of solution. Linearization around the steady state
Figure 3.  Norm of solution. Linearization around the time-dependent trajectory
Figure 4.  Norm of solution. Around the time-dependent trajectory
Figure 5.  Norm of solution. Linearization around the steady state
Figure 6.  Norm of solution. Linearization around the time-dependent trajectory
Figure 7.  Norm of solution. Around the time-dependent trajectory
Figure 8.  Norm of solution. Linearization around the steady state
Figure 9.  Norm of solution. Linearization around the time-dependent trajectory
Figure 10.  Norm of solution. Around the time-dependent trajectory
Figure 11.  Norm of solution. Around the time-dependent trajectory
Figure 12.  Norm of solution. Linearization around the time-dependent trajectory
Figure 13.  Norm of solution. Around the time-dependent trajectory
Figure 14.  Norm of solution. Around the time-dependent trajectory
Figure 15.  Norm of solution. Linearization around the time-dependent trajectory
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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