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Controls insensitizing the norm of solution of a Schrödinger type system with mixed dispersion

Roberto de A. Capistrano Filho, Thiago Yukio Tanaka

Abstract

The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schrödinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schrödinger cascade type system with zero-order coupling which is equivalent to the insensitizing control problem. Precisely, employing a new Carleman estimate, we first obtain a null controllability result for the linearized system around zero, and then the null controllability for the nonlinear case is extended using an inverse mapping theorem.

Controls insensitizing the norm of solution of a Schrödinger type system with mixed dispersion

Abstract

The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schrödinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schrödinger cascade type system with zero-order coupling which is equivalent to the insensitizing control problem. Precisely, employing a new Carleman estimate, we first obtain a null controllability result for the linearized system around zero, and then the null controllability for the nonlinear case is extended using an inverse mapping theorem.

Paper Structure

This paper contains 18 sections, 12 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Insensitizing control problems for PDEs
  4. Main results
  5. Heuristic and structure of the manuscript
  6. Carleman estimates
  7. Null controllability results
  8. Null controllability: Linear case
  9. Null controllability: Nonlinear case
  10. Further comments and open issues
  11. Null condition of the initial data
  12. About the nonlinear terms
  13. About the sentinel functional
  14. N--dimensional case
  15. Well-posedness
  16. ...and 3 more sections

Key Result

Theorem \oldthetheorem

Assume that $\omega \cap \mathcal{O} \neq \emptyset$ and $u_{0} \equiv 0$. There exists a constant $C>0$ and $\delta>0$ such that for any $f$ satisfying one can find a control $h(x,t)=:h\in L^2(q_T)$ which insensitizes the functional $J$ defined by def:jfunc, in the sense of Definition def:insen.

Theorems & Definitions (26)

  • Definition 1: Insensitizing controls
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • proof : Proof of Theorem \ref{['propcarleman']}
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 16 more