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

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

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 2021 Early access  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.

Citation: Elena-Alexandra Melnig. Internal feedback stabilization for parabolic systems coupled in zero or first order terms. Evolution Equations and Control Theory, 2021, 10 (2) : 333-351. doi: 10.3934/eect.2020069
References:
[1]

F. Ammar KhodjaA. BenabdallahC. 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.

[2]

F. Ammar-KhodjaA. BenabdallahC. 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.

[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.

[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.

[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.

[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.

[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.

[8]

V. BarbuS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

show all references

References:
[1]

F. Ammar KhodjaA. BenabdallahC. 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.

[2]

F. Ammar-KhodjaA. BenabdallahC. 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.

[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.

[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.

[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.

[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.

[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.

[8]

V. BarbuS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

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

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