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June  2021, 10(2): 333-351. doi: 10.3934/eect.2020069

Internal feedback stabilization for parabolic systems coupled in zero or first order terms

Elena-Alexandra Melnig 1,2,

1. 

Faculty of Mathematics, University "Al. I. Cuza" Iaşi, Romania
2. 

Octav Mayer Institute of Mathematics, Romanian Academy, Iaşi Branch, Romania

Received  January 2020 Revised  March 2020 Published  June 2020

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

We consider systems of $ n $ parabolic equations coupled in zero or first order terms with $ m $ scalar controls acting through a control matrix $ B $. We are interested in stabilization with a control in feedback form. Our approach relies on the approximate controllability of the linearized system, which in turn is related to unique continuation property for the adjoint system. For the unique continuation we establish algebraic Kalman type conditions.

Keywords: Parabolic systems, Feedback stabilization, Unique continuation property, Kalman rank condition.
Mathematics Subject Classification: Primary: 35K40, 93D15, 93B52; Secondary: 35K57, 93B18.
Citation: Elena-Alexandra Melnig. Internal feedback stabilization for parabolic systems coupled in zero or first order terms. Evolution Equations & Control Theory, 2021, 10 (2) : 333-351. doi: 10.3934/eect.2020069
References:
[1]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[2]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parbolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.  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, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and their Applications Vol. 90, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-76666-9.  Google Scholar

[5]

V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006) 2704–2746. doi: 10.1016/j.na.2005.09.012.  Google Scholar

[6]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), 128 pp. doi: 10.1090/memo/0852.  Google Scholar

[7]

V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d = 2, 3$, via feedback stabilization of its linearization, in Control of Coupled Partial Differential Equations, Internat. Ser. Numer. Math., Vol. 155, Birkhäuser, Basel, 2007. doi: 10.1007/978-3-7643-7721-2_2.  Google Scholar

[8]

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.  Google Scholar

[9]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar

[10]

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

[11]

M. Duprez and P. Lissy, Positive and negative results on the internal controllability of parabolic equations coupled by zero- and first-order terms, J. Evol. Equ., 18 (2018), 659-680.  doi: 10.1007/s00028-017-0415-1.  Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[13]

M. González-Burgos and L. de Teresa., Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[14]

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

[15]

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

[16]

C. Lefter, Internal feedback stabilization of nonstationary solutions to semilinear parabolic systems, J. Optim. Theory Appl., 170 (2016), 960-976.  doi: 10.1007/s10957-016-0964-4.  Google Scholar

[17]

P. Lissy and E. Zuazua, Internal observability for coupled systems of linear partial differential equations, SIAM J. Control Optim., 57 (2019), 832-853.  doi: 10.1137/17M1119160.  Google Scholar

[18]

A. Lunardi, Interpolation theory, third edition, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), Vol. 16, Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[19]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[20]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[21]

R. Seeley, Norms and domains of the complex powers $A_{B}z$, Amer. J. Math., 93 (1971), 299-309.  doi: 10.2307/2373377.  Google Scholar

show all references

References:
[1]

F. Ammar Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.  Google Scholar

[2]

F. Ammar-Khodja, A. Benabdallah, C. Dupaix and M. González-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parbolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.  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, Controllability and Stabilization of Parabolic Equations, Progress in Nonlinear Differential Equations and their Applications Vol. 90, Birkhäuser/Springer, Cham, 2018. doi: 10.1007/978-3-319-76666-9.  Google Scholar

[5]

V. Barbu, I. Lasiecka and R. Triggiani, Abstract settings for tangential boundary stabilization of Navier-Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal., 64 (2006) 2704–2746. doi: 10.1016/j.na.2005.09.012.  Google Scholar

[6]

V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations, Mem. Amer. Math. Soc., 181 (2006), 128 pp. doi: 10.1090/memo/0852.  Google Scholar

[7]

V. Barbu, I. Lasiecka and R. Triggiani, Local exponential stabilization strategies of the Navier-Stokes equations, $d = 2, 3$, via feedback stabilization of its linearization, in Control of Coupled Partial Differential Equations, Internat. Ser. Numer. Math., Vol. 155, Birkhäuser, Basel, 2007. doi: 10.1007/978-3-7643-7721-2_2.  Google Scholar

[8]

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.  Google Scholar

[9]

V. Barbu and R. Triggiani, Internal stabilization of Navier-Stokes equations with finite-dimensional controllers, Indiana Univ. Math. J., 53 (2004), 1443-1494.  doi: 10.1512/iumj.2004.53.2445.  Google Scholar

[10]

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

[11]

M. Duprez and P. Lissy, Positive and negative results on the internal controllability of parabolic equations coupled by zero- and first-order terms, J. Evol. Equ., 18 (2018), 659-680.  doi: 10.1007/s00028-017-0415-1.  Google Scholar

[12]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of evolution equations, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.  Google Scholar

[13]

M. González-Burgos and L. de Teresa., Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Port. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.  Google Scholar

[14]

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

[15]

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

[16]

C. Lefter, Internal feedback stabilization of nonstationary solutions to semilinear parabolic systems, J. Optim. Theory Appl., 170 (2016), 960-976.  doi: 10.1007/s10957-016-0964-4.  Google Scholar

[17]

P. Lissy and E. Zuazua, Internal observability for coupled systems of linear partial differential equations, SIAM J. Control Optim., 57 (2019), 832-853.  doi: 10.1137/17M1119160.  Google Scholar

[18]

A. Lunardi, Interpolation theory, third edition, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), Vol. 16, Edizioni della Normale, Pisa, 2018. doi: 10.1007/978-88-7642-638-4.  Google Scholar

[19]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[20]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[21]

R. Seeley, Norms and domains of the complex powers $A_{B}z$, Amer. J. Math., 93 (1971), 299-309.  doi: 10.2307/2373377.  Google Scholar

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