Stabilization of KdV equation with boundary time-delay feedback and input saturation

Ahmat Mahamat Taboye; Toufik Ennouari

doi:10.2478/candc-2024-0018

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Stabilization of KdV equation with boundary time-delay feedback and input saturation

Control and Cybernetics

Volume 53 (2024): Issue 3 (September 2024)

By: Ahmat Mahamat Taboye and Toufik Ennouari

Open Access

|May 2025

Abstract

In this paper, we study the question of stabilization of nonlinear Korteweg-de Vries equation with boundary time-delay feedback in presence of saturated source term. Thanks to Banach fixed-point theorem the well-posedness is proved. The exponential stability result is demonstrated, using an appropriate Lyapunov functional.

References

Achhab, M. E. and Laabissi, M. (2002) Feedback stabilization of a class of distributed parameter systems with control constraints. Systems & Control Letters 45, 163–171.
Search in Google Scholar Back to article
Baudouin, L. Crépeau, E. and Valein, J. (2018) Two approaches for the stabilization of nonlinear kdv equation with boundary time-delay feedback. IEEE Transactions on Automatic Control 64, 1403–1414.
Search in Google Scholar Back to article
Capistrano-Filho, R. d. A,. Chentouf, B., de Sousa, L. S. and Gonzalez Martinez, V. H. (2023) Two stability results for the kawahara equation with a time-delayed boundary control. Zeitschrift für angewandte Mathematik und Physik 74, 16.
Search in Google Scholar Back to article
Cerpa, E. (2014) Control of a Korteweg-de Vries equation: a tutorial. Math. Control Relat. Fields 4, 45–99.
Search in Google Scholar Back to article
Chen, Y., Zuo, Z. and Wang, Y. (2018) Stabilization of wave equation with boundary saturated control. In: 2018 37th Chinese Control Conference (CCC), IEEE 726–731.
Search in Google Scholar Back to article
Coron, J.-M. (2007) Control and Nonlinearity. American Mathematical Society 136.
Search in Google Scholar Back to article
Curtain, R. and Zwart, H. (2016) Stabilization of collocated systems by nonlinear boundary control. Systems & Control Letters 96, 11–14.
Search in Google Scholar Back to article
Jacob, B., Schwenninger, F. L. and Vorberg, L. A. (2020) Remarks on input-to-state stability of collocated systems with saturated feedback. Mathematics of Control, Signals, and Systems 32, 293–307.
Search in Google Scholar Back to article
Lasiecka, I. and Seidman, T. I. (2003) Strong stability of elastic control systems with dissipative saturating feedback. Systems & Control Letters 48, 243–252.
Search in Google Scholar Back to article
Linares, F. and Ponce, G. (2014) Introduction to Nonlinear Dispersive Equations. Springer.
Search in Google Scholar Back to article
Marx, S., Andrieu, V. and Prieur, C. (2017) Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces. Mathematics of Control, Signals, and Systems 29, 1–32.
Search in Google Scholar Back to article
Marx, S., Cerpa, E., Prieur, C. and Andrieu, V. (2015) Stabilization of a linear Korteweg-de Vries equation with a saturated internal control. In: 2015 European Control Conference (ECC), IEEE, 867–872.
Search in Google Scholar Back to article
Marx, S., Cerpa, E., Prieur, C. and Andrieu, V. (2016) Global stabilization of a korteweg-de vries equation with a distributed control saturated in l2-norm. IFAC-PapersOnLine 49, 122–127.
Search in Google Scholar Back to article
Marx, S., Cerpa, E., Prieur, C. and Andrieu, V. (2017) Global stabilization of a Korteweg–de Vries equation with saturating distributed control. SIAM Journal on Control and Optimization 55, 1452–1480.
Search in Google Scholar Back to article
Marx, S., Chitour, Y. and Prieur, C. (2018) Stability results for infinite-dimensional linear control systems subject to saturations. In: 2018 European Control Conference (ECC), IEEE, 2995–3000.
Search in Google Scholar Back to article
Parada, H. (2022) Stability analysis of a Korteweg-de Vries equation with saturated boundary feedback. IFAC-PapersOnLine 55, 1–6.
Search in Google Scholar Back to article
Parada, H., Timimoun, C. and Valein, J. (2023) Stability results for the kdv equation with time-varying delay. Systems & Control Letters 177, 105547.
Search in Google Scholar Back to article
Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer Science & Business Media.
Search in Google Scholar Back to article
Perla Menzala, G., Vasconcellos, C. F. and Zuazua, E. (2002) Stabilization of the Korteweg-de Vries equation with localized damping. Quarterly of Applied Mathematics 60, 111–129.
Search in Google Scholar Back to article
Prieur, C., Tarbouriech, S. and Da Silva, J. M. G. (2016) Wave equation with cone-bounded control laws. IEEE Transactions on Automatic Control 61, 3452–3463.
Search in Google Scholar Back to article
Rosier, L. (1997) Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: Control, Optimisation and Calculus of Variations 2, 33–55.
Search in Google Scholar Back to article
Rosier, L. and Zhang, B.-Y. (2006) Global stabilization of the generalized Korteweg–de Vries equation posed on a finite domain. SIAM Journal on Control and Optimization 45, 927–956.
Search in Google Scholar Back to article
Rosier, L. and Zhang, B.-Y. (2009) Control and stabilization of the Korteweg-de Vries equation: recent progresses. Journal of Systems Science and Complexity 22, 647–682.
Search in Google Scholar Back to article
Seidman, T. I. and Li, H. (2001) A note on stabilization with saturating feedback. Discrete and Continuous Dynamical Systems 7, 319–328.
Search in Google Scholar Back to article
Sene, N. (2022) Numerical methods applied to a class of seir epidemic models described by the Caputo derivative. In: Methods of Mathematical Modeling, Elsevier, 23–40.
Search in Google Scholar Back to article
Sene, N. and Ndiaye, A. (2024) Existence and uniqueness study for partial neutral functional fractional differential equation under Caputo derivative. An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 14, 208–219.
Search in Google Scholar Back to article
Slemrod, M. (1989) Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control. Mathematics of Control, Signals and Systems 2, 265–285.
Search in Google Scholar Back to article
Taboye, A. and Ennouari, T. (2024) Stabilization of linear KdV equation with boundary time-delay feedback and internal saturation. Gulf Journal of Mathematics 17, 72–86.
Search in Google Scholar Back to article
Taboye, A. and Ennouari, T. (2025) Stability result of the Kawahara equation with localized damping and time-varying delay. Evolution Equations and Control Theory 14, 36–56.
Search in Google Scholar Back to article
Taboye, A. M. and Laabissi, M. (2022) Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations and Control Theory 11, 1519–1532.
Search in Google Scholar Back to article
Tarbouriech, S., Garcia, G., da Silva Jr, J. M. G. and Queinnec, I. (2011) Stability and Stabilization of Linear Systems with Saturating Actuators. Springer Science & Business Media.
Search in Google Scholar Back to article
Valein, J. (2022) On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control and Related Fields, 12(3), 667–694.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/candc-2024-0018 | Journal eISSN: 2720-4278 | Journal ISSN: 0324-8569

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Language: English

Page range: 409 - 428

Submitted on: May 1, 2024

Accepted on: Dec 1, 2024

Published on: May 15, 2025

Published by: Systems Research Institute Polish Academy of Sciences

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

Korteweg-de Vries equation,

time-delay,

exponential stability,

input saturated,

Lyapunov functional

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Electrical engineering,

Fundamentals of electrical engineering,

Mechanical engineering,

Fundamentals of mechanical engineering,

Mathematics,

General mathematics

© 2025 Ahmat Mahamat Taboye, Toufik Ennouari, published by Systems Research Institute Polish Academy of Sciences
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 53 (2024): Issue 3 (September 2024)