September  2017, 7(3): 369-391. doi: 10.3934/mcrf.2017013

Boundary feedback stabilization of the monodomain equations

1. 

Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstr. 36,8010 Graz, Austria

2. 

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Tobias Breiten

Received  July 2016 Revised  November 2016 Published  July 2017

Boundary feedback control for a coupled nonlinear PDE-ODE system (in the two and three dimensional cases) is studied. Particular focus is put on the monodomain equations arising in the context of cardiac electrophysiology. Neumann as well as Dirichlet based boundary control laws are obtained by an algebraic operator Riccati equation associated with the linearized system. Local exponential stability of the nonlinear closed loop system is shown by a fixed-point argument. Numerical examples are given for a finite element discretization of the two dimensional monodomain equations.

Citation: Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control and Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013
References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, Academic Press, 2003.

[2]

M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst., 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.

[3]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. application to the Navier-Stokes system, SIAM Journal on Control and Optimization, 49 (2011), 420-463.  doi: 10.1137/090778146.

[4]

V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197-206.  doi: 10.1051/cocv:2003009.

[5]

V. BarbuI. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoirs of the American Mathematical Society, 181 (2006), x+128 pp.  doi: 10.1090/memo/0852.

[6]

V. Barbu and G. Wang, Feedback stabilization of semilinear heat equations, in Abstract and Applied Analysis, Hindawi Publishing Corporation, 12 (2003), 697-714.  doi: 10.1155/S1085337503212100.

[7]

P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: A state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52.  doi: 10.1002/gamm.201310003.

[8]

A. Bensoussan, G. D. Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, vol. 1, Birkhäuser Boston, Inc. , Boston, MA, 1992.

[9]

T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the FitzHugh-Nagumo model, SIAM Journal on Control and Optimization, 52 (2014), 4057-4081.  doi: 10.1137/140964552.

[10]

T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the Fitzhugh-Nagumo model, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 241-262.  doi: 10.1051/cocv/2015047.

[11]

J. BurnsE. Sachs and L. Zietsman, Mesh independence of Kleinman-Newton iterations for Riccati equations in Hilbert space, SIAM Journal on Control and Optimization, 47 (2008), 2663-2692.  doi: 10.1137/060653962.

[12]

E. CasasC. Ryll and F. Tröltzsch, Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation, SIAM Journal on Control and Optimization, 53 (2015), 2168-2202.  doi: 10.1137/140978855.

[13]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[14]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs of the American Mathematical Society, 166 (2013), viii+114 pp. doi: 10.1090/memo/0788.

[15]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[16]

A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Korea, 1996, Lecture Notes no. 34.

[17]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, 1985.

[18]

G. Grubb and V. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, Mathematica Scandinavica, 69 (1991), 217-290.  doi: 10.7146/math.scand.a-12380.

[19]

J. Keener and J. Sneyd, Mathematical Physiology, Vol. I: Cellular Physiology, vol. 8 of Interdisciplinary Applied Mathematics, 2nd edition, Springer, New York, 2009. doi: 10.1007/978-0-387-79388-7.

[20]

D. Kleinman, On an iterative technique for Riccati equation computations, IEEE Transactions on Automatic Control, 13 (1968), 114-115.  doi: 10.1109/TAC.1968.1098829.

[21]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories, Cambridge University Press, 2000.

[22]

J. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. Ⅰ/Ⅱ, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer-Verlag, Berlin, 1972.

[23]

B. NielsenT. RuudG. Lines and A. Tveito, Optimal monodomain approximations of the bidomain equations, Applied Mathematics and Computation, 184 (2007), 276-290.  doi: 10.1016/j.amc.2006.05.158.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM Journal on Control and Optimization, 45 (2006), 790-828.  doi: 10.1137/050628726.

[26]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, Journal de Mathématiques Pures Et Appliquées, 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.

[27]

O. Savin and E. Valdinoci, Density estimates for a nonlocal variational model via the Sobolev inequality, SIAM Journal on Mathematical Analysis, 43 (2011), 2675-2687.  doi: 10.1137/110831040.

[28]

F. Schlögl, A characteristic critical quantity in nonequilibrium phase transitions, Zeitschrift für Physik B Condensed Matter, 52 (1983), 51-60. 

[29]

L. ThevenetJ.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional Burgers equation, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 929-955.  doi: 10.1051/cocv/2009028.

[30]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators North-Holland Publishing Company, 1978.

[31]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkäuser, 2009. doi: 10.1007/978-3-7643-8994-9.

show all references

References:
[1]

R. Adams and J. Fournier, Sobolev Spaces, Academic Press, 2003.

[2]

M. Badra, Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control, Discrete Contin. Dyn. Syst., 32 (2012), 1169-1208.  doi: 10.3934/dcds.2012.32.1169.

[3]

M. Badra and T. Takahashi, Stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers. application to the Navier-Stokes system, SIAM Journal on Control and Optimization, 49 (2011), 420-463.  doi: 10.1137/090778146.

[4]

V. Barbu, Feedback stabilization of Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 197-206.  doi: 10.1051/cocv:2003009.

[5]

V. BarbuI. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Memoirs of the American Mathematical Society, 181 (2006), x+128 pp.  doi: 10.1090/memo/0852.

[6]

V. Barbu and G. Wang, Feedback stabilization of semilinear heat equations, in Abstract and Applied Analysis, Hindawi Publishing Corporation, 12 (2003), 697-714.  doi: 10.1155/S1085337503212100.

[7]

P. Benner and J. Saak, Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: A state of the art survey, GAMM-Mitteilungen, 36 (2013), 32-52.  doi: 10.1002/gamm.201310003.

[8]

A. Bensoussan, G. D. Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, vol. 1, Birkhäuser Boston, Inc. , Boston, MA, 1992.

[9]

T. Breiten and K. Kunisch, Riccati-based feedback control of the monodomain equations with the FitzHugh-Nagumo model, SIAM Journal on Control and Optimization, 52 (2014), 4057-4081.  doi: 10.1137/140964552.

[10]

T. Breiten and K. Kunisch, Compensator design for the monodomain equations with the Fitzhugh-Nagumo model, ESAIM: Control, Optimisation and Calculus of Variations, 23 (2017), 241-262.  doi: 10.1051/cocv/2015047.

[11]

J. BurnsE. Sachs and L. Zietsman, Mesh independence of Kleinman-Newton iterations for Riccati equations in Hilbert space, SIAM Journal on Control and Optimization, 47 (2008), 2663-2692.  doi: 10.1137/060653962.

[12]

E. CasasC. Ryll and F. Tröltzsch, Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation, SIAM Journal on Control and Optimization, 53 (2015), 2168-2202.  doi: 10.1137/140978855.

[13]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4224-6.

[14]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs of the American Mathematical Society, 166 (2013), viii+114 pp. doi: 10.1090/memo/0788.

[15]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[16]

A. Fursikov and O. Imanuvilov, Controllability of Evolution Equations, Seoul National University, Korea, 1996, Lecture Notes no. 34.

[17]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, 1985.

[18]

G. Grubb and V. Solonnikov, Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods, Mathematica Scandinavica, 69 (1991), 217-290.  doi: 10.7146/math.scand.a-12380.

[19]

J. Keener and J. Sneyd, Mathematical Physiology, Vol. I: Cellular Physiology, vol. 8 of Interdisciplinary Applied Mathematics, 2nd edition, Springer, New York, 2009. doi: 10.1007/978-0-387-79388-7.

[20]

D. Kleinman, On an iterative technique for Riccati equation computations, IEEE Transactions on Automatic Control, 13 (1968), 114-115.  doi: 10.1109/TAC.1968.1098829.

[21]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Volume 1, Abstract Parabolic Systems: Continuous and Approximation Theories, Cambridge University Press, 2000.

[22]

J. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. Ⅰ/Ⅱ, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Springer-Verlag, Berlin, 1972.

[23]

B. NielsenT. RuudG. Lines and A. Tveito, Optimal monodomain approximations of the bidomain equations, Applied Mathematics and Computation, 184 (2007), 276-290.  doi: 10.1016/j.amc.2006.05.158.

[24]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[25]

J.-P. Raymond, Feedback boundary stabilization of the two-dimensional Navier-Stokes equations, SIAM Journal on Control and Optimization, 45 (2006), 790-828.  doi: 10.1137/050628726.

[26]

J.-P. Raymond, Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations, Journal de Mathématiques Pures Et Appliquées, 87 (2007), 627-669.  doi: 10.1016/j.matpur.2007.04.002.

[27]

O. Savin and E. Valdinoci, Density estimates for a nonlocal variational model via the Sobolev inequality, SIAM Journal on Mathematical Analysis, 43 (2011), 2675-2687.  doi: 10.1137/110831040.

[28]

F. Schlögl, A characteristic critical quantity in nonequilibrium phase transitions, Zeitschrift für Physik B Condensed Matter, 52 (1983), 51-60. 

[29]

L. ThevenetJ.-M. Buchot and J.-P. Raymond, Nonlinear feedback stabilization of a two-dimensional Burgers equation, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 929-955.  doi: 10.1051/cocv/2009028.

[30]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators North-Holland Publishing Company, 1978.

[31]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkäuser, 2009. doi: 10.1007/978-3-7643-8994-9.

Figure 1.  Control setup
Figure 2.  Stabilization of perturbed initial state
Figure 3.  Stabilization of a reentry wave
Figure 4.  Stabilization of perturbed initial state
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