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Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

Kuntal Bhandari, Subrata Majumdar

TL;DR

The paper addresses null-controllability for a mixed KS-KdV-parabolic and elliptic system on $(0,1)$ with a single interior control. It develops global Carleman estimates for the adjoint, leading to observability inequalities and a control cost bounded by $Ce^{C/T}$ for the linearized models. The nonlinear systems are shown to be small-time locally null-controllable using the source-term method and a Banach fixed point argument, both when the control acts in KS-KdV or in the elliptic equation. The results extend KS-type controllability to a parabolic-elliptic coupling and establish a robust framework for single-control strategies in mixed PDE systems with explicit cost estimates.

Abstract

This paper deals with the null-controllability of a system of {\em mixed parabolic-elliptic pdes} at any given time $T>0$. More precisely, we consider the \textit{Kuramoto-Sivashinsky--Korteweg-de Vries equation} coupled with a second order elliptic equation posed in the interval $(0,1)$. We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the \textit{Carleman approach}, we provide the existence of a control with the explicit cost $Ce^{C/T}$ with some constant $C>0$ independent in $T$. Then, applying the source term method followed by the \textit{Banach fixed point theorem}, we conclude the small-time local null-controllability result of the nonlinear systems.

Local null-controllability of a system coupling Kuramoto-Sivashinsky-KdV and elliptic equations

TL;DR

The paper addresses null-controllability for a mixed KS-KdV-parabolic and elliptic system on with a single interior control. It develops global Carleman estimates for the adjoint, leading to observability inequalities and a control cost bounded by for the linearized models. The nonlinear systems are shown to be small-time locally null-controllable using the source-term method and a Banach fixed point argument, both when the control acts in KS-KdV or in the elliptic equation. The results extend KS-type controllability to a parabolic-elliptic coupling and establish a robust framework for single-control strategies in mixed PDE systems with explicit cost estimates.

Abstract

This paper deals with the null-controllability of a system of {\em mixed parabolic-elliptic pdes} at any given time . More precisely, we consider the \textit{Kuramoto-Sivashinsky--Korteweg-de Vries equation} coupled with a second order elliptic equation posed in the interval . We first show that the linearized system is globally null-controllable by means of a localized interior control acting on either the KS-KdV or the elliptic equation. Using the \textit{Carleman approach}, we provide the existence of a control with the explicit cost with some constant independent in . Then, applying the source term method followed by the \textit{Banach fixed point theorem}, we conclude the small-time local null-controllability result of the nonlinear systems.
Paper Structure (22 sections, 17 theorems, 156 equations)

This paper contains 22 sections, 17 theorems, 156 equations.

Table of Contents

  1. Introduction and main results
  2. Bibliographic comments and motivations
  3. The systems under study
  4. Controllability results of the nonlinear systems
  5. Strategy of proofs
  6. Controllability results of the associated linearized systems
  7. Notations
  8. Paper organization
  9. Well-posedness of the linearized systems
  10. Control acting in the KS-KdV equation
  11. A global Carleman estimate
  12. Some useful notations
  13. Observability inequality and null-controllability of the linearized model
  14. Null controllability of the nonlinear system: control in KS-KdV equation
  15. The source term method
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let be $(a, b)\in \mathbb R^2$ with $b\neq 0$. Then, the system nonlinear-KS--bd--in is small-time locally null-controllable around the equilibrium $(0,0)$, that is to say, for any given time $T>0$, there is a $R>0$ such that for chosen initial data $u_0\in H^{-1}(0,1)$ with $\|u_0\|_{H^{-1}(0,1)}\l

Theorems & Definitions (32)

  • Theorem 1.1: Control on KS-KdV equation
  • Theorem 1.2: Control on elliptic equation
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • ...and 22 more