Controllability for Schrödinger type system with mixed dispersion on compact star graphs

Roberto de A. Capistrano–Filho; Márcio Cavalcante; Fernando A. Gallego

doi:10.3934/eect.2022019

Article Contents

2023, Volume 12, Issue 1: 1-19. Doi: 10.3934/eect.2022019

This issue Previous Article Next Article Stability properties for a problem of light scattering in a dispersive metallic domain

Controllability for Schrödinger type system with mixed dispersion on compact star graphs

1.
Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil
2.
Instituto de Matemática, Universidade Federal de Alagoas, Maceió, Alagoas, Brazil
3.
Departamento de Matematicas y Estadística, Universidad Nacional de Colombia - Sede Manizales, Manizales, Colombia

^* Corresponding author: Roberto de A. Capistrano–Filho
^* Corresponding author: Roberto de A. Capistrano–Filho

Received: November 30, 2021

Revised: February 28, 2022

Early access: April 2022

Published: February 2023

Capistrano–Filho was supported by CNPq grants 408181/2018-4 and 307808/2021-1, CAPES grants 88881.311964/2018-01 and 88881.520205/2020-01, MATHAMSUD 21- MATH-03 and Propesqi (UFPE). Cavalcante was supported by CNPq 310271/2021-5 and CAPES-MATHAMSUD 88887.368708/2019-00. Gallego was supported by MATHAMSUD 21-MATH-03 and the 100.000 Strong in the Americas Innovation Fund

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Abstract

In this work we are concerned with solutions to the linear Schrödinger type system with mixed dispersion, the so-called biharmonic Schrödinger 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.

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Mathematics Subject Classification: Primary: 35R02, 35Q55, 35G30, 93B05, 93B07.

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Figure 1. A compact graph with $ N+1 $ edges

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