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Stabilization of the Kawahara–Kadomtsev–Petviashvili equation with time-delayed feedback

Part of: Stability Equations of mathematical physics and other areas of application Control systems

Published online by Cambridge University Press:  11 September 2023

Roberto de A. Capistrano–Filho Open the ORCID record for Roberto de A. Capistrano–Filho [Opens in a new window] ,
Victor Hugo Gonzalez Martinez  and
Juan Ricardo Muñoz
Roberto de A. Capistrano–Filho
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), 50740-545, Recife PE, Brazil (roberto.capistranofilho@ufpe.br; victor.martinez@ufpe.br; juan.ricardo@ufpe.br)
Victor Hugo Gonzalez Martinez
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), 50740-545, Recife PE, Brazil (roberto.capistranofilho@ufpe.br; victor.martinez@ufpe.br; juan.ricardo@ufpe.br)
Juan Ricardo Muñoz
Affiliation:
Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), 50740-545, Recife PE, Brazil (roberto.capistranofilho@ufpe.br; victor.martinez@ufpe.br; juan.ricardo@ufpe.br)
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Abstract

Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara–Kadomtsev–Petviashvili equation are locally and globally exponentially stable. The main novelty of this work is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.

Keywords

KP systemdelayed systemdamping mechanismstabilization

MSC classification

Primary: 35Q53: KdV-like equations (Korteweg-de Vries)
Secondary: 93D15: Stabilization of systems by feedback 93D30: Scalar and vector Lyapunov functions 93C20: Systems governed by partial differential equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 1 , February 2025 , pp. 280 - 306
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Araruna, F. D., Capistrano–Filho, R. A. and Doronin, G. G.. Energy decay for the modified Kawahara equation posed in a bounded domain. J. Math. Anal. Appl. 385 (2012), 743–756.CrossRefGoogle Scholar
Besov, O. V., Il'in, V. P. and Nikol'skii, S. M., Integral Representations of Functions and Imbedding Theorems, Vol. I (New York-Toronto, Ont.-London, 1978).Google Scholar
Bona, J. L., Lannes, D. and Saut, J.-C.. Asymptotic models for internal waves. J. Math. Pures Appl. 89 (2008), 538–566.CrossRefGoogle Scholar
Brezis, H., Functional analysis, Sobolev spaces, and partial differential equations. Universitext (Springer, New York, 2011).CrossRefGoogle Scholar
Capistrano–Filho, R. A., Cerpa, E. and Gallego, F. A.. Rapid exponential stabilization of a Boussinesq system of KdV–KdV Type. Communications in Contemporary Mathematics 25 (2023), 2150111.CrossRefGoogle Scholar
Capistrano–Filho, R. A., Chentouf, B., de Sousa, L. and Gonzalez Martinez, V. H.. Two stability results for the Kawahara equation with a time-delayed boundary control. Z. Angew. Math. Phys. 74 (2023), 1–26.CrossRefGoogle Scholar
Capistrano–Filho, R. A. and Gallego, F. A.. Asymptotic behavior of Boussinesq system of KdV–KdV type. J. Differ. Equ. 265 (2018), 2341–2374.CrossRefGoogle Scholar
Capistrano–Filho, R. A. and Gonzalez Martinez, V. H.. Stabilization results for delayed fifth-order KdV-type equation in a bounded domain. Math. Control Related Fields. doi: 10.3934/mcrf.2023004Google Scholar
Chentouf, B.. Well-posedness and exponential stability of the Kawahara equation with a time-delayed localized damping. Math. Methods. Appl. Sci. 45 (2022), 10312–10330.CrossRefGoogle Scholar
de Moura, R. P., Nascimento, A. C. and Santos, G. N.. On the stabilization for the high-order Kadomtsev–Petviashvili and the Zakharov–Kuznetsov equations with localized damping. Evol. Equ. Control Theory 11 (2022), 711–727.CrossRefGoogle Scholar
Gomes, D. A. and Panthee, M.. Exponential energy decay for the Kadomtsev–Petviashvili (KP-II) equation. São Paulo J. Math. Sci. 5 (2011), 135–148.CrossRefGoogle Scholar
Gronwall, T. H.. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919), 292–296.CrossRefGoogle Scholar
Haragus, M.. Model equations for water waves in the presence of surface tension. Eur. J. Mech. Fluids 15 (1996), 471–492.Google Scholar
Hasimoto, H.. Water waves. Kagaku 40 (1970), 401–408.Google Scholar
Kadomtsev, B. B. and Petviashvili, V. I.. On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15 (1970), 539–549.Google Scholar
Karpman, V. I.. Transverse stability of Kawahara solitons. Phys. Rev. E 47 (1993), 674–676.CrossRefGoogle ScholarPubMed
Kawahara, T.. Oscillatory solitary waves in dispersive media. J. Phys. Soc. Japan 33 (1972), 260–264.CrossRefGoogle Scholar
Komornik, V.. Exact Controllability and Stabilization. The multiplier method. RAM: Research in Applied Mathematics (Masson, Paris; John Wiley & Sons, Ltd, Chichester, 1994).Google Scholar
Lannes, D., The water waves problem. Mathematical analysis and asymptotics. Mathematical Surveys and Monographs, Vol. 188 (American Mathematical Society, Providence, RI, 2013).CrossRefGoogle Scholar
Linares, F. and Ponce, G.. Introduction to Nonlinear Dispersive Equations. 2nd ed. (Springer, New York, 2015).CrossRefGoogle Scholar
Lions, J.-L.. Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988), 1–68.CrossRefGoogle Scholar
Lions, J.-L., Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués, Vol. 22. (Masson, Paris, 1988).Google Scholar
Menzala, G., Vasconcellos, C. and Zuazua, E.. Stabilization of the Korteweg-De Vries equation with localized damping. Q. Appl. Math. 60 (2002), 111–129.CrossRefGoogle Scholar
Pazoto, A. F. and Rosier, L.. Stabilization of a Boussinesq system of KdV–KdV type. Syst. Control. Lett. 57 (2008), 595–601.CrossRefGoogle Scholar
Pazy, A.. Semigroups of Linear Operators and Applications to Partial Differential Equations (Spronger-Verlag, New York, 1983).CrossRefGoogle Scholar
Showalter, R. E.. Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Vol. 49 (American Mathematical Society, Providence, RI, 2013).CrossRefGoogle Scholar
Simon, S.. Compact sets in the space $L^p(0,\,T;B)$. Annali di Mate. Pura ed App. CXLXVI (1987), 65–96.Google Scholar
Tikhonov, A. N. and Arsenin, V. Y.. Solutions of ill-Posed Problems (Winston, New York, 1977).Google Scholar
Valein, J.. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Math. Control Related Fields 12 (2022), 667–694.CrossRefGoogle Scholar