Controls insensitizing the norm of solution of a Schr\"odinger type system with mixed dispersion

TL;DR

This paper establishes the existence of insensitizing controls for a nonlinear fourth-order Schr"odinger system with mixed dispersion, using Carleman estimates and null controllability techniques.

Contribution

It introduces a novel Carleman estimate to prove null controllability for a coupled fourth-order Schr"odinger system, enabling insensitizing control results for the nonlinear equation.

Findings

Proved null controllability for the linearized system.

Extended controllability to the nonlinear case.

Established existence of insensitizing controls.

Abstract

The main goal of this manuscript is to prove the existence of insensitizing controls for the fourth-order dispersive nonlinear Schr\"odinger equation with cubic nonlinearity. To obtain the main result we prove a null controllability property for a coupled fourth-order Schr\"odinger 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.