Existence of unattainable states for Schrödinger type flows on the half-line

Date
2023-12-01
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Source Title
IMA Journal of Mathematical Control and Information
Print ISSN
0265-0754
Electronic ISSN
1471-6887
Publisher
Oxford University Press
Volume
40
Issue
4
Pages
789 - 803
Language
en_US
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Abstract

We prove that the solutions of the Schrödinger and biharmonic Schrödinger equations do not have the exact boundary controllability property on the half-line by showing that the associated adjoint models lack observability. We consider the framework of L2 boundary controls with data spaces H−1(R+) and H−2(R+) for the classical and biharmonic Schrödinger equations, respectively. The lack of controllability on the half-line contrasts with the corresponding dynamics on a finite interval for a similar regularity setting. Our proof is based on an argument that uses the sharp fractional time trace estimates for solutions of the adjoint models. We also make several remarks on the connection of controllability and temporal regularity of spatial traces.

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