Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

• Published in: Journal of difference equations and applications Vol. 19; no. 6; pp. 952 - 980
• Main Authors: Ungureanu, V.M., Dragan, V.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis Group 01.06.2013; Taylor & Francis Ltd
• Subjects: Banach space; Criteria; Differential equations; discrete-time systems; Evolution; evolution operators; Mathematical analysis; Operators; positive operators; Representations; Riccati equation; Stability; Stochastic systems; Theorems
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236198.2012.704369

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert-Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.