Controllability for Schr\"odinger type system with mixed dispersion on compact star graphs

TL;DR

This paper establishes exact controllability results for the biharmonic Schr"odinger equation with mixed dispersion on star graphs, demonstrating control in the energy space via Neumann boundary conditions for sufficiently large time.

Contribution

It proves the first exact controllability result for a Schr"odinger type system with mixed dispersion on star graphs, with controls applied only through Neumann boundary conditions.

Findings

Exact controllability holds for T > T_min.

Control is achieved in the energy space.

Results depend on graph structure and boundary conditions.

Abstract

In this work we are concerned with solutions to the linear Schr\"odinger type system with mixed dispersion, the so-called biharmonic Schr\"odinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure $\mathcal{G}$ for $T>T_{min}$, with $$T_{min}=\sqrt{ \frac{ \overline{L} (L^2+\pi^2)}{\pi^2\varepsilon(1- \overline{L} \varepsilon)}},$$ when the couplings and the controls appear only on the Neumann boundary conditions.