A SPECTRAL CRITERION FOR EXACT DETECTABILITY OF A CLASS OF CONTINUOUS-TIME AND DISCRETE-TIME SYSTEMS WITH PERIODIC COEFFICIENTS

Abstract: This paper investigates exact detectability of systems with periodic coefficients in finite-dimensional real ordered Hilbert spaces, extending the classical framework of positive systems, which have applications in many fields. A spectral PBH-type criterion for detectability is established for both discrete and continuous-time systems in a unified treatment. We further propose a Barbasin–Krasovskii-type criterion for exponential stability by showing that the existence of a solution to a dual system, combined with detectability, ensures stability. The obtained results provide a Lyapunov-like framework and open directions for studying stability in optimal control.

Keywords: detectability, exponential stability, Popov-Belevich-Hautus type criterion, Barbasin-Krasovski-type criterion.

MSC: 93A99, 93B28, 47A50, 47A60, 47A65.

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DOI       10.56082/annalsarscimath.2026.1.31

^†Vasile.Dragan@imar.ro, Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, P.O.Box 1-764, RO-014700, Bucharest, Romania & Academy of the Romanian Scientists, 3 Ilfov, 050044, Bucharest, Romania
^‡lucian.popa@uab.ro, Department of Computing, Mathematics and Electronics ”1 Decembrie 1918” University of Alba Iulia, 510009 – Alba Iulia, Romania & Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091, Brasov, Romania
^§samir.aberkane@univ-lorraine.fr, Université de Lorraine, CRAN, UMR 7039, Campus Sciences, BP 70239, Vandoeuvre-les-Nancy Cedex, 54506, France CNRS, CRAN, UMR 7039, France
^¶efcosta@icmc.usp.br, Instituto de Ciências Matemáticas e de Computação Universidade de São Paulo, São Carlos-SP, CEP 13566-590, Brazil

PUBLISHED in

Annals Academy of Romanian Scientists Series on Mathematics and Its Application,

Volume 18 no 1, 2026

ISSN ONLINE 2066 – 6594
ISSN PRINT 2066 – 5997