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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 |
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:
Exponential stabilization,
explicit feedback,
monodomain equations,
finite-dimensional controller,
oblique projections.
Mathematics Subject Classification:
93D15, 93B52, 93C05, 93C20.
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/. |
| [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. |
| [3] |
V. Barbu, S. 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. |
| [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. |
| [5] |
T. Breiten, K. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
J. Keener and J. Sneyd, Mathematical Physiology. Vol. II: Systems Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/Ⅱ. Springer, New York, 2009. |
| [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. |
| [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. |
| [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. |
| [15] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod et Gauthier-Villars, Paris, 1969. |
| [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. |
| [17] |
A. Lunardi,
Stabilizability of time-periodic parabolic equations, SIAM J. Control Optim., 29 (1991), 810-828.
doi: 10.1137/0329044. |
| [18] |
J. Nagumo, S. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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/. |
| [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. |
| [3] |
V. Barbu, S. 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. |
| [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. |
| [5] |
T. Breiten, K. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
J. Keener and J. Sneyd, Mathematical Physiology. Vol. II: Systems Physiology, Second edition, Interdisciplinary Applied Mathematics, 8/Ⅱ. Springer, New York, 2009. |
| [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. |
| [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. |
| [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. |
| [15] |
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod et Gauthier-Villars, Paris, 1969. |
| [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. |
| [17] |
A. Lunardi,
Stabilizability of time-periodic parabolic equations, SIAM J. Control Optim., 29 (1991), 810-828.
doi: 10.1137/0329044. |
| [18] |
J. Nagumo, S. 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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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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