Exact Controllability for a Schr\"odinger equation with dynamic boundary conditions

TL;DR

This paper establishes the exact controllability of a Schrödinger equation with mixed boundary conditions, using new Carleman estimates tailored to the domain's geometry, enabling boundary and distributed control.

Contribution

It introduces novel Carleman estimates for the Schrödinger equation with dynamic boundary conditions, proving exact controllability with boundary control on a subset of the boundary.

Findings

Proved boundary controllability with control on Dirichlet boundary part.

Established distributed controllability of the Schrödinger system.

Developed Carleman estimates adapted to the domain geometry.

Abstract

In this paper, we study the controllability of a Schr\"odinger equation with mixed boundary conditions on disjoint subsets of the boundary: dynamic boundary condition of Wentzell type, and Dirichlet boundary condition. The main result of this article is given by new Carleman estimates for the associated adjoint system, where the weight function is constructed specially adapted to the geometry of the domain. Using these estimates, we prove the exact controllability of the system with a boundary control acting only in the part of the boundary where the Dirichlet condition is imposed. Also, we obtain a distributed exact controllability result for the system.