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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

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