Stabilization results for delayed fifth-order KdV-type equation in a bounded domain

Roberto de A. Capistrano-Filho; Victor Hugo Gonzalez Martinez

doi:10.3934/mcrf.2023004

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2024, Volume 14, Issue 1: 284-321. Doi: 10.3934/mcrf.2023004

This issue Previous Article Zero-sum stochastic games in continuous-time with risk-sensitive average cost criterion on a countable state space Next Article On stable solutions of the dynamical reconstruction and tracking control problems for a coupled ordinary differential equation–heat equation

Stabilization results for delayed fifth-order KdV-type equation in a bounded domain

Departamento de Matemática, Universidade Federal de Pernambuco (UFPE), 50740-545, Recife (PE), Brazil

^*Corresponding author: Victor Hugo Gonzalez Martinez
^*Corresponding author: Victor Hugo Gonzalez Martinez

Received: February 2022

Revised: January 2023

Early access: February 2023

Published: March 2024

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Abstract

Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to $ 0 $ exponentially as $ t \rightarrow \infty $. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.

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Mathematics Subject Classification: Primary: 35Q53, 93D15, 93C20; Secondary: 93D30.

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