Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-03T16:11:43.282Z Has data issue: false hasContentIssue false
  • English
  • Français

On exponential dichotomy in Banach spaces

Published online by Cambridge University Press:  17 April 2009

Mihail Megan  and
Petre Preda
Mihail Megan
Affiliation:
Department of Mathematics, University of Timisoara, 1900 – Timisoara, RS Romania.
Petre Preda
Affiliation:
Department of Mathematics, University of Timisoara, 1900 – Timisoara, RS Romania.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study the exponential dichotomy property for linear systems, the evolution of which can be described by a semigroup of class C0 on a Banach space. We define the class of (p, q) dichotomic semigroups and establish the connections between the dichotomy concepts and admissibility property of the pair (Lp, Lq) for linear control systems. The obtained results are generalizations of well-known results of W.A. Coppel, J.L. Massera and J.J. Schäffer, K.J. Palmer.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 2 , April 1981 , pp. 293 - 306
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Balakrishnan, A.V., Applied functional analysis (Applications of Mathematics, 3. Springer-Verlag, New York, Heidelberg, Berlin, 1976).Google Scholar
[2]Coppel, W.A., Dichotomies in stability theory (Lecture Notes in Mathematics, 629. Springer-Verlag, Berlin, Heidelberg, New York, 1978).CrossRefGoogle Scholar
[3]Hillie, Einar, Phillips, Ralph S., Functional analysis and semi-groups, revised edition (American Mathematical Society Colloquium Publications, 31. American Mathematical Society, Providence, Rhode Island, 1957).Google Scholar
[4]Massera, José Luis, Schäffer, Juan Jorge, Linear differential equations and function spaces (Pure and Applied Mathematics, 21. Academic Press, New York and London, 1966).Google Scholar
[5]Megan, M., “On exponential stability of linear control systems in Hilbert spaces”, An. Univ. Timisoaro Ser. Sti. Mat. 14 (1976), 125–130.Google Scholar
[6]Megan, Mihail, “On the input-output stability of linear controllable systems”, Canad. Math. Bull. 21 (1978), 187–195.CrossRefGoogle Scholar
[7]Palmer, K.J., “Two linear systems criteria for exponential dichotomy”, Ann. Mat. Pura Appl. (to appear).Google Scholar
[8]Perron, Oskar, “Die Stabilitätsfrage bei Differentialgleichungen”, Math. Z. 32 (1930), 703–728.CrossRefGoogle Scholar
[9]Silverman, L.M. and Anderson, B.D.O., “Controllability, observability and stability of linear systems”, SIAM J. Control Optim. 6 (1968), 121–130.CrossRefGoogle Scholar