Regularity and Stability Sets for Families of Sequences of Matrices
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Regularity and Stability Sets for Families of Sequences of Matrices Luis Barreira1
· Claudia Valls1
Received: 29 May 2019 / Revised: 4 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider the notions of Lyapunov regularity and of Lyapunov stability and asymptotic stability for a dynamics defined by a continuous 1-parameter family of sequences of matrices. In particular, we identify all classes of sets that can be the regularity set, the stability set and the asymptotic stability set of any such family. Moreover, we construct explicitly families of sequences of matrices whose regularity set, stability set or asymptotic stability set is a given set in those classes. Keywords Lyapunov regularity · Sequences of matrices · Stability Mathematics Subject Classification Primary 37D99
1 Introduction We consider the notions of Lyapunov regularity and of Lyapunov stability for a continuous 1-parameter family of nonautonomous dynamics with discrete time defined by a family of sequences of matrices An (θ ), with n ∈ N. The two properties and the extensive development of their study owe much to Lyapunov himself who in his thesis introduced the first notion and contributed substantially to the stability theory (see [17] for a recent edition of the thesis). Our main aim is to discuss what sets can be the regularity sets and the stability sets of such 1-parameter families. The regularity set (respectively, stability set) of a family of sequences of matrices An (θ ) is the set of all numbers θ for which the dynamics defined by the sequence is regular (respectively, stable). Before proceeding we describe briefly the importance of the notions of (Lyapunov) regularity and (Lyapunov) stability, and how they are closely connected. The relevance of the
Partially supported by FCT/Portugal through the Project UID/MAT/04459/2013.
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Luis Barreira [email protected] Claudia Valls [email protected]
1
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal
123
Journal of Dynamics and Differential Equations
stability theory is well established, particularly in connection with applications, and its theory is extensively developed. In particular, Lyapunov functions and Lyapunov exponents are often considered in connection with the study of stability and have been much developed, with many variants of the notions and results. Among the first accounts of the theory of Lyapunov functions are the books [10,13,16]. In a related direction, the notion of hyperbolicity plays a central role in a large part of the stability theory, leading for example to the construction of topological conjugacies and invariant manifolds under sufficiently small perturbations of a linear dynamics (for related discussions see for example the books [8,11,21] and the references therein). We also consider the importance of the theory of Lyapunov regularity (we refer to [6,7] for detailed descriptions). First note that one can easily verify
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