Stability of discrete-time positive evolution operators on ordered Banach spaces and applications |
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Authors: | VM Ungureanu V Dragan |
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Institution: | 1. Department of Mathematics , University Constantin Brancu?i , Blvd Republicii, No. 1, Targu Jiu, Gorj , Romania vio@utgjiu.ro;3. Institute of Mathematics of the Romanian Academy , PO Box 1-764, RO-014700 , Bucharest , Romania |
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Abstract: | 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. |
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Keywords: | positive operators evolution operators stability discrete-time systems Riccati equation |
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