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IMA Journal of Mathematical Control and Information 1988 5(4):269-283; doi:10.1093/imamci/5.4.269
© 1988 by Institute of Mathematics and its Applications
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Uniform Stability for Time-Varying Infinite-Dimensional Discrete Linear Systems

C. S. KUBRUSLY

Department of Research and Development, National Lab. for Scientific Comp.-LNCC R. Lauro Müller 455, Rio de Janeiro, 22290, Brazil
Department of Electrical Engineering, Catholic University-PUC/RJ R. Marques de S. Vicente 209, Rio de Janeiro, 22453, Brazil

Stability for time-varying discrete linear systems in a Banach space is investigated. On the one hand is established a fairly complete collection of necessary and sufficient conditions for uniform asymptotic equistability for input-free systems. This includes uniform and strong power equistability, and uniform and strong lp-equistability, among other technical conditions which also play an essential role in stability theory. On the other hand, it is shown that uniform asymptotic equistability for input-free systems is equivalent to each of the following concepts of uniform stability for forced systems: lp-input lp-state, eo-input eo-state, bounded-input bounded-state, lp-input bounded-state (with p>1), eo-input bounded-state, and convergent-input bounded-state; these are also equivalent to their nonuniform counterparts. For time-varying convergent systems, the above is also equivalent to convergent-input convergent-state stability. The proofs presented here are all ‘lementary’ in the sense that they are based essentially only on the Banach–Steinhaus theorem.

Keywords: Discrete-time systems; linear systems; infinite-dimensional systems; stability theory; time-varying systems.


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