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Insensitizing controls for a quasi-linear parabolic equation with diffusion depending on gradient of the state

Dany Nina Huaman, Miguel R. Nuñez-Chávez

TL;DR

The paper addresses insensitizing controls for a quasi-linear parabolic equation with diffusion depending on the gradient by recasting the problem as null controllability for a cascade system. It develops Carleman estimates for the adjoint system and proves local null controllability of a nonlinear optimal system via Liusternik’s theorem, yielding existence of insensitizing controls under small data and a nonempty intersection of control and observation regions. The main contribution extends Xu Liu’s open case to gradient-dependent diffusion and outlines 1D refinements, higher-dimensional obstacles for $N\ge 4$, and numerical strategies for practical computation. This advances the theoretical foundation of insensitizing controls in nonlinear diffusion settings and suggests avenues for applications and computation in controlled diffusion processes.

Abstract

In this paper, a quasi-linear parabolic equation with a diffusion term dependent on the gradient to the state with Dirichlet boundary conditions is considered. The goal of this paper is to prove the existence of control that insensitizes the system under study which is the case that Xu Liu left open in 2012. It is well known that the insensitizing control problem is equivalent to a null controllability result for a cascade system, which is obtained by duality arguments, Carleman estimates, and the Right Inverse mapping theorem. Also, some possible extensions and open problems concerning other quasi-linear systems are presented.

Insensitizing controls for a quasi-linear parabolic equation with diffusion depending on gradient of the state

TL;DR

The paper addresses insensitizing controls for a quasi-linear parabolic equation with diffusion depending on the gradient by recasting the problem as null controllability for a cascade system. It develops Carleman estimates for the adjoint system and proves local null controllability of a nonlinear optimal system via Liusternik’s theorem, yielding existence of insensitizing controls under small data and a nonempty intersection of control and observation regions. The main contribution extends Xu Liu’s open case to gradient-dependent diffusion and outlines 1D refinements, higher-dimensional obstacles for , and numerical strategies for practical computation. This advances the theoretical foundation of insensitizing controls in nonlinear diffusion settings and suggests avenues for applications and computation in controlled diffusion processes.

Abstract

In this paper, a quasi-linear parabolic equation with a diffusion term dependent on the gradient to the state with Dirichlet boundary conditions is considered. The goal of this paper is to prove the existence of control that insensitizes the system under study which is the case that Xu Liu left open in 2012. It is well known that the insensitizing control problem is equivalent to a null controllability result for a cascade system, which is obtained by duality arguments, Carleman estimates, and the Right Inverse mapping theorem. Also, some possible extensions and open problems concerning other quasi-linear systems are presented.
Paper Structure (17 sections, 14 theorems, 187 equations)

This paper contains 17 sections, 14 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Statements of the Main Results
  3. Preliminary results
  4. Reformulation of the insensitizing problem:
  5. Carleman estimates for \ref{['adjoint-1']}
  6. Null controllability for the linear system \ref{['linear-1']}
  7. Some additional estimates of the state
  8. Proof of the main result
  9. Locally null controllability of optimal system \ref{['eq-char']}
  10. Proof of Theorem \ref{['main-t']}
  11. Some Additional Comments and Questions
  12. Insensitizing controls for a quasi-linear parabolic equation with diffusion depending on gradient of the state in one dimension
  13. Insensitizing controls for the system \ref{['EC1']} when $N\geq 4$
  14. Numerical Results
  15. Well-possessedness for the system \ref{['EC1']}
  16. ...and 2 more sections

Key Result

Theorem 1

Assume that $\omega\cap \mathcal{O}\neq \emptyset$ and $y_{0}=0.$ Then, there exist two positive constants $\tilde{M}$ and $\delta$ depending only on $N,\, \Omega,\, T,\, M,\, a_{0}$ and $a_{1}$, such that for any $\xi\in X_{0}$ satisfying one can find a control function $u\in X_{0}$ with $\mathrm{supp}\ u\subset \omega\times[0,T]$, which insensitizes the functional $\Phi$ in the sense of Definit

Theorems & Definitions (31)

  • Definition 1
  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Remark 1
  • Remark 2
  • ...and 21 more