Global solutions of a class of discrete-time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control

2012 ◽  
Vol 34 (2) ◽  
pp. 164-190 ◽  
Author(s):  
V. M. Ungureanu ◽  
V. Dragan ◽  
T. Morozan
Keyword(s):  
Banach Spaces ◽  
Stochastic Control ◽  
Nonlinear Equations ◽  
Discrete Time ◽  
Global Solutions ◽  
Riccati Equations ◽  
Ordered Banach Spaces

scholarly journals Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Viorica Mariela Ungureanu
Keyword(s):  
Banach Spaces ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Trace Class ◽  
Algebraic Riccati Equation ◽  
Bounded Operators ◽  
Linear Quadratic ◽  
Infinite Dimensional ◽  
Ordered Banach Spaces

We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.

scholarly journals Stability criteria for positive linear discrete-time systems

Positivity ◽  
2021 ◽  
Author(s):  
Jochen Glück ◽  
Andrii Mironchenko
Keyword(s):  
Banach Spaces ◽  
Discrete Time ◽  
Positive Cone ◽  
Stability Criteria ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Discrete Time Systems ◽  
Small Gain ◽  
Time Systems ◽  
Ordered Banach Spaces

AbstractWe prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain.

Stein iterations for the coupled discrete-time Riccati equations

2009 ◽  
Vol 71 (12) ◽  
pp. 6244-6253 ◽  
Author(s):  
Ivan Ganchev Ivanov
Keyword(s):  
Discrete Time ◽  
Riccati Equations
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