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doi: 10.3934/eect.2021052
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Exponential stabilization of a linear Korteweg-de Vries equation with input saturation

1. 

Laboratory LIMA, Faculty of sciences, Chouaib Doukkali University, El-Jadida, Morocco

2. 

LIMA, Faculty of sciences, Chouaib Doukkali University, El-Jadida, Morocco

* Corresponding author: Ahmat Mahamat Taboye

Dedicated to Professor Mohammed Elarbi Achhab on the occasion of his retirement

Received  July 2021 Revised  August 2021 Early access September 2021

This article deals with the issue of the exponential stability of a linear Korteweg-de Vries equation with input saturation. It is proved that the system is well-posed and the origin is exponentially stable for the closed loop system, by using the classical argument used in this kind of problems.

Citation: Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations and Control Theory, doi: 10.3934/eect.2021052
References:
[1]

E. Cerpa, Control of a korteweg-de Vries equation: A tutorial, Math. Control Relat. Fields, 4 (2014), 45-99.  doi: 10.3934/mcrf.2014.4.45.

[2]

E. Cerpa and J-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.  doi: 10.1109/TAC.2013.2241479.

[3]

J-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[4]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995.

[5]

H. Hedenmalm, On the uniqueness theorem of Holmgren, Math. Z., 281 (2015), 357-378.  doi: 10.1007/s00209-015-1488-6.

[6]

K. Ito and F. Kappel, Evolution Equations and Approximations, World Scientific, 2002.

[7]

M. Laabissi and A. M. Taboye, Strong stabilization of non-dissipative operators in Hilbert spaces with input saturation, Math. Control Signals Systems, 33 (2021), 553-568.  doi: 10.1007/s00498-021-00291-1.

[8]

I. Lasiecka and T. I. Seidman, Strong stability of elastic control systems with dissipative saturating feedback, Systems Control Lett., 48 (2003), 243-252.  doi: 10.1016/S0167-6911(02)00269-4.

[9]

S. Marx and E. Cerpa, Output feedback control of the linear korteweg-de vries equation, IEEE Conference on Decision and Control, (2014), 2083–2087.

[10]

S. Marx, E. Cerpa, C. Prieur and V. Andrieu, Stabilization of a linear korteweg-de vries equation with a saturated internal control, In 2015 European Control Conference (ECC), IEEE, (2015), 867–872.

[11]

S. MarxE. CerpaC. Prieur and V. Andrieu, Global stabilization of a Korteweg–de Vries equation with saturating distributed control, SIAM J. Control Optim., 55 (2017), 1452-1480.  doi: 10.1137/16M1061837.

[12]

G. P. MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.

[13]

I. Miyadera, Nonlinear Semigroups, American Mathematical Soc., 1992.

[14]

A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486.  doi: 10.1051/cocv:2005015.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[16]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[17]

L. Rosier and B-Y. Zhang, Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956.  doi: 10.1137/050631409.

[18]

T. I. Seidman and H. Li, A note on stabilization with saturating feedback, Discrete Contin. Dynam. Systems, 7 (2001), 319-328.  doi: 10.3934/dcds.2001.7.319.

[19]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Soc., Providence, RI, 1997.

[20]

M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.

[21]

B.-Y. Zhang, Exact boundary controllability of the Korteweg–de Vries equation, SIAM J. Control Optim., 37 (1999), 543-565.  doi: 10.1137/S0363012997327501.

show all references

References:
[1]

E. Cerpa, Control of a korteweg-de Vries equation: A tutorial, Math. Control Relat. Fields, 4 (2014), 45-99.  doi: 10.3934/mcrf.2014.4.45.

[2]

E. Cerpa and J-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.  doi: 10.1109/TAC.2013.2241479.

[3]

J-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[4]

R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, 21. Springer-Verlag, New York, 1995.

[5]

H. Hedenmalm, On the uniqueness theorem of Holmgren, Math. Z., 281 (2015), 357-378.  doi: 10.1007/s00209-015-1488-6.

[6]

K. Ito and F. Kappel, Evolution Equations and Approximations, World Scientific, 2002.

[7]

M. Laabissi and A. M. Taboye, Strong stabilization of non-dissipative operators in Hilbert spaces with input saturation, Math. Control Signals Systems, 33 (2021), 553-568.  doi: 10.1007/s00498-021-00291-1.

[8]

I. Lasiecka and T. I. Seidman, Strong stability of elastic control systems with dissipative saturating feedback, Systems Control Lett., 48 (2003), 243-252.  doi: 10.1016/S0167-6911(02)00269-4.

[9]

S. Marx and E. Cerpa, Output feedback control of the linear korteweg-de vries equation, IEEE Conference on Decision and Control, (2014), 2083–2087.

[10]

S. Marx, E. Cerpa, C. Prieur and V. Andrieu, Stabilization of a linear korteweg-de vries equation with a saturated internal control, In 2015 European Control Conference (ECC), IEEE, (2015), 867–872.

[11]

S. MarxE. CerpaC. Prieur and V. Andrieu, Global stabilization of a Korteweg–de Vries equation with saturating distributed control, SIAM J. Control Optim., 55 (2017), 1452-1480.  doi: 10.1137/16M1061837.

[12]

G. P. MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.

[13]

I. Miyadera, Nonlinear Semigroups, American Mathematical Soc., 1992.

[14]

A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486.  doi: 10.1051/cocv:2005015.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.

[16]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[17]

L. Rosier and B-Y. Zhang, Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956.  doi: 10.1137/050631409.

[18]

T. I. Seidman and H. Li, A note on stabilization with saturating feedback, Discrete Contin. Dynam. Systems, 7 (2001), 319-328.  doi: 10.3934/dcds.2001.7.319.

[19]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, American Mathematical Soc., Providence, RI, 1997.

[20]

M. Slemrod, Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Systems, 2 (1989), 265-285.  doi: 10.1007/BF02551387.

[21]

B.-Y. Zhang, Exact boundary controllability of the Korteweg–de Vries equation, SIAM J. Control Optim., 37 (1999), 543-565.  doi: 10.1137/S0363012997327501.

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