Exact boundary controllability of the linear Biharmonic Schr\"odinger equation with variable coefficients

TL;DR

This paper proves that the linear fourth-order Schrödinger equation with variable coefficients can be exactly controlled through boundary manipulation, using spectral analysis and nonharmonic Fourier series, for any positive time.

Contribution

It establishes the exact boundary controllability of a variable coefficient fourth-order Schrödinger equation, a novel result in control theory for such complex PDEs.

Findings

Exact controllability at any positive time T

Control acts on the first spatial derivative at the boundary

Spectral analysis and nonharmonic Fourier series are effective tools

Abstract

In this paper, we study the exact boundary controllability of the linear fourth-order Schr\"odinger equation, with variable physical parameters and clamped boundary conditions on a bounded interval. The control acts on the first spatial derivative at the left endpoint. We prove that this control system is exactly controllable at any time $T>0$. The proofs are based on a detailed spectral analysis and on the use of nonharmonic Fourier series.