Vol. 2, No. 1, 2020

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Characterization by observability inequalities of controllability and stabilization properties

Emmanuel Trélat, Gengsheng Wang and Yashan Xu

Vol. 2 (2020), No. 1, 93–122
DOI: 10.2140/paa.2020.2.93
Abstract

Given a linear control system in a Hilbert space with a bounded control operator, we establish a characterization of exponential stabilizability in terms of an observability inequality. Such dual characterizations are well known for exact (null) controllability. Our approach exploits classical Fenchel duality arguments and, in turn, leads to characterizations in terms of observability inequalities of approximate null controllability and of α-null controllability. We comment on the relationships among those various concepts, at the light of the observability inequalities that characterize them.

Keywords
observability inequality, stabilizability, controllability
Mathematical Subject Classification 2010
Primary: 93B05, 93B07, 93C20
Milestones
Received: 30 April 2019
Revised: 19 June 2019
Accepted: 22 July 2019
Published: 9 November 2019
Authors
Emmanuel Trélat
Sorbonne Université
CNRS
Université de Paris
Inria
Laboratoire Jacques-Louis Lions
Paris
France
Gengsheng Wang
Center for Applied Mathematics
Tianjin University
Tianjin
China
Yashan Xu
School of Mathematical Sciences
Fudan University
KLMNS
Shanghai
China