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Boundary Effects on the Controllability of Coupled KdV Systems

F. A. Gallego, A. F. Pazoto, I. Rivas

TL;DR

The paper addresses exact boundary controllability for a nonlinear, coupled KdV system on a finite interval, focusing on configurations with up to four boundary controls. It combines a linear–adjoint duality framework with Paley–Wiener spectral analysis to derive observability inequalities for multiple control configurations, showing controllability for all domain lengths $L>0$ under suitable coefficient conditions. Local controllability for the nonlinear system follows by a contraction mapping argument around small initial/final data, using a decoupling transform to reduce the linear part to two independent KdV equations and a boundary-integral operator to realize controls. The work extends boundary controllability results for KdV-type systems to a nonlinear, strongly coupled setting, with implications for wave interaction models in stratified fluids and dispersive media; it also identifies open problems about global well-posedness and controllability with fewer controls, especially when $r=0$.

Abstract

We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system.

Boundary Effects on the Controllability of Coupled KdV Systems

TL;DR

The paper addresses exact boundary controllability for a nonlinear, coupled KdV system on a finite interval, focusing on configurations with up to four boundary controls. It combines a linear–adjoint duality framework with Paley–Wiener spectral analysis to derive observability inequalities for multiple control configurations, showing controllability for all domain lengths under suitable coefficient conditions. Local controllability for the nonlinear system follows by a contraction mapping argument around small initial/final data, using a decoupling transform to reduce the linear part to two independent KdV equations and a boundary-integral operator to realize controls. The work extends boundary controllability results for KdV-type systems to a nonlinear, strongly coupled setting, with implications for wave interaction models in stratified fluids and dispersive media; it also identifies open problems about global well-posedness and controllability with fewer controls, especially when .

Abstract

We study the exact boundary controllability of a nonlinear coupled system of two Korteweg-de Vries equations on a bounded interval. The model describes the interactions of two weakly nonlinear gravity waves in a stratified fluid. Due to the nature of the system, six boundary conditions are required. However, to study the controllability property, we consider a different combination of the control inputs, with a maximum of four. Firstly, the results are obtained for the linearized system through a classical duality approach and some hidden regularity properties of the boundary terms. This approach reduces the controllability problem to the study of a spectral problem, which is solved by using the Paley-Wiener method introduced by Rosier. Then, the issue is to establish when a certain quotient of entire functions still turns out to be an entire function. It can be viewed as a problem of factoring an entire function that, depending on the control configuration, leads to the study of a transcendental equation. Finally, by using the contraction mapping theorem, we derive the local controllability for the full system.
Paper Structure (36 sections, 26 theorems, 365 equations)

This paper contains 36 sections, 26 theorems, 365 equations.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Linear Systems
  4. The IBVP for the KdV equation
  5. The system of two couple KdV equations
  6. The adjoint system
  7. Nonlinear System
  8. Observability Properties
  9. Case A: Observability inequality with four control configurations
  10. Further Configurations of Four Controls
  11. Case B: Observability inequality with three controls configuration
  12. Further Configuration of Three Controls
  13. Exact Controllability
  14. Linear System
  15. Nonlinear System
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $T>0$, $L>0$ and $r\neq 0$. Then, there exists $\delta=\delta(t,L)>0$, such that, for any $(u_0,v_0)$, $(u_T,v_T) \in\mathcal{X}:=(L^2(0,L))^2$ verifying there exist controls $(h_0,h_1,h_2)$ and $(g_0,g_1,g_2)$ in $\mathcal{H}:=H^{\frac{1}{3}}(0,T)\times L^2(0,T) \times H^{-\frac{1}{3}}(0,T)$, with the following configurations for four controls: such that the system kdv-inputs under conditi

Theorems & Definitions (49)

  • Definition 1.1
  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.1
  • ...and 39 more