American Institute of Mathematical Sciences

Advanced
September  2018, 8(3&4): 777-787. doi: 10.3934/mcrf.2018034

Feedback stabilization with one simultaneous control for systems of parabolic equations

Cătălin-George Lefter , and  Elena-Alexandra Melnig

Faculty of Mathematics, University "Al. I. Cuza" Iași, Romania, Octav Mayer Institute of Mathematics, Romanian Academy, Iași Branch

* Corresponding author: Cătălin-George Lefter (catalin.lefter@uaic.ro)

Dedicated to Professor Jiongmin Yong on the occasion of his 60th anniversary

Received  November 2017 Revised  March 2018 Published  September 2018

Fund Project: The second author was supported by a grant of the Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-Ⅲ-P4-ID-PCE-2016-0011.

In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction-diffusion systems are discussed.

Keywords: Reaction-diffusion equations, unique continuation, feedback stabilization, Lyapunov equation.
Mathematics Subject Classification: 35K40, 35K57, 93D15, 93B52, 93B18.
Citation: Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034
References:
[1]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., 42 (2003), 1661-1680.  doi: 10.1137/S0363012902417826.  Google Scholar

[2]

V. Barbu and G. Wang, Feedback stabilization of semilinear heat equations, Abstr. Appl. Anal., 12 (2003), 697-714.  doi: 10.1155/S1085337503212100.  Google Scholar

[3]

V. Barbu and G. Wang, Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407.  doi: 10.1016/S0022-247X(03)00405-0.  Google Scholar

[4]

V. Barbu, Partial Differential Equations and Boundary Value Problems, Dordrecht: Kluwer Academic Publishers, 1998. doi: 10.1007/978-94-015-9117-1.  Google Scholar

[5]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems. Volume I. Boston: Birkhäuser, 1992.  Google Scholar

[6]

J.-M. Coron, Controllability and nonlinearity, ESAIM, Proc., 22 (2008), 21-39.  doi: 10.1051/proc:072203.  Google Scholar

[7]

J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term, SIAM J. Control Optim., 48 (2010), 5629-5653.  doi: 10.1137/100784539.  Google Scholar

[8]

J.-M. Coron and J.-P. Guilleron, Control of three heat equations coupled with two cubic nonlinearities, SIAM J. Control Optim., 55 (2017), 989-1019.  doi: 10.1137/15M1041201.  Google Scholar

[9]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Seoul: Seoul National Univ., 1996.  Google Scholar

[10]

C. Lefter, Feedback stabilization of 2D Navier-Stokes equations with Navier slip boundary conditions, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 70 (2009), 553-562.  doi: 10.1016/j.na.2007.12.026.  Google Scholar

[11]

C.-G. Lefter, Feedback stabilization of magnetohydrodynamic equations, SIAM J. Control Optim., 49 (2011), 963-983.  doi: 10.1137/070697124.  Google Scholar

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

show all references

References:
[1]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and I. Kostin, Controllability to the trajectories of phase-field models by one control force, SIAM J. Control Optim., 42 (2003), 1661-1680.  doi: 10.1137/S0363012902417826.  Google Scholar

[2]

V. Barbu and G. Wang, Feedback stabilization of semilinear heat equations, Abstr. Appl. Anal., 12 (2003), 697-714.  doi: 10.1155/S1085337503212100.  Google Scholar

[3]

V. Barbu and G. Wang, Internal stabilization of semilinear parabolic systems, J. Math. Anal. Appl., 285 (2003), 387-407.  doi: 10.1016/S0022-247X(03)00405-0.  Google Scholar

[4]

V. Barbu, Partial Differential Equations and Boundary Value Problems, Dordrecht: Kluwer Academic Publishers, 1998. doi: 10.1007/978-94-015-9117-1.  Google Scholar

[5]

A. Bensoussan, G. Da Prato, M. C. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems. Volume I. Boston: Birkhäuser, 1992.  Google Scholar

[6]

J.-M. Coron, Controllability and nonlinearity, ESAIM, Proc., 22 (2008), 21-39.  doi: 10.1051/proc:072203.  Google Scholar

[7]

J.-M. Coron, S. Guerrero and L. Rosier, Null controllability of a parabolic system with a cubic coupling term, SIAM J. Control Optim., 48 (2010), 5629-5653.  doi: 10.1137/100784539.  Google Scholar

[8]

J.-M. Coron and J.-P. Guilleron, Control of three heat equations coupled with two cubic nonlinearities, SIAM J. Control Optim., 55 (2017), 989-1019.  doi: 10.1137/15M1041201.  Google Scholar

[9]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Seoul: Seoul National Univ., 1996.  Google Scholar

[10]

C. Lefter, Feedback stabilization of 2D Navier-Stokes equations with Navier slip boundary conditions, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 70 (2009), 553-562.  doi: 10.1016/j.na.2007.12.026.  Google Scholar

[11]

C.-G. Lefter, Feedback stabilization of magnetohydrodynamic equations, SIAM J. Control Optim., 49 (2011), 963-983.  doi: 10.1137/070697124.  Google Scholar

[12]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[1]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079
[2]

Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673
[3]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[4]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
[5]

Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047
[6]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382
[7]

Nick Bessonov, Gennady Bocharov, Tarik Mohammed Touaoula, Sergei Trofimchuk, Vitaly Volpert. Delay reaction-diffusion equation for infection dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2073-2091. doi: 10.3934/dcdsb.2019085
[8]

Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control & Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165
[9]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
[10]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43
[11]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281
[12]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083
[13]

Wilhelm Stannat, Lukas Wessels. Deterministic control of stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020087
[14]

Tobias Breiten, Karl Kunisch. Feedback stabilization of the three-dimensional Navier-Stokes equations using generalized Lyapunov equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4197-4229. doi: 10.3934/dcds.2020178
[15]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[16]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
[17]

Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225
[18]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907
[19]

Perla El Kettani, Danielle Hilhorst, Kai Lee. A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5615-5648. doi: 10.3934/dcds.2018246
[20]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (86)
  • HTML views (331)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences