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On the boundary stabilization of the KdV-KdV system with time-dependent delay

Roberto de A. Capistrano-Filho, Boumediène Chentouf, Victor H. Gonzalez Martinez, Juan Ricardo Muñoz

Abstract

The boundary stabilization problem of the Boussinesq KdV-KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, first, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato's approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV-KdV system decays exponentially by employing the Lyapunov method.

On the boundary stabilization of the KdV-KdV system with time-dependent delay

Abstract

The boundary stabilization problem of the Boussinesq KdV-KdV type system is investigated in this paper. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a delay term is designed. Then, first, considering time-varying delay feedback together with a smallness restriction on the length of the spatial domain and the initial data, we show that the problem under consideration is well-posed. The proof combines Kato's approach and the fixed-point argument. Last but not least, we prove that the energy of the linearized KdV-KdV system decays exponentially by employing the Lyapunov method.
Paper Structure (16 sections, 9 theorems, 105 equations, 1 figure)

This paper contains 16 sections, 9 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Boussinesq system model
  3. Problem setting
  4. Main results and paper's outline
  5. Well-posedness theory
  6. Linear problem
  7. Nonlinear problem
  8. Linear stabilization result
  9. Proof of Theorem \ref{['th:Lyapunov0']}
  10. Optimization of the decay rate
  11. Concluding discussion
  12. Further comments
  13. Open problems
  14. A time-varying delay feedback
  15. Variation of feedback-law
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $0<L<\sqrt{3}\pi$. Suppose that eq:TauCond and eq:CCond are satisfied. Then, for two positive constants $\mu_1$ and $\mu_2$ with $\mu_1 L < 1$, there exist and such that the energy $E(t)$ given by eq:En associated to the linearized system of eq:KdV-KdV around the origin satisfies

Figures (1)

  • Figure 1: Ilustration of Proposition \ref{['llllll']}

Theorems & Definitions (19)

  • Remark 1
  • Theorem 1.1
  • Theorem 2.1: Kato1970
  • Theorem 2.2
  • proof
  • Claim 1
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • ...and 9 more