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Global solutions of a class of discrete-time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control. (English) Zbl 1273.93044

Summary: In this paper, we consider the problem of existence of certain global solutions for general discrete-time backward nonlinear equations, defined on infinite dimensional ordered Banach spaces. This class of nonlinear equations includes as special cases many of the discrete-time Riccati equations arising both in deterministic and stochastic optimal control problems. On the basis of a linear matrix inequalities approach, we give necessary and sufficient conditions for the existence of maximal, stabilizing, and minimal solutions of the considered discrete-time backward nonlinear equations. As an application, we discuss the existence of stabilizing solutions for discrete-time Riccati equations of stochastic control and filtering on Hilbert spaces. The tools provided by this paper show that a wide class of nonlinear equations can be treated in a uniform manner.

MSC:

93B25 Algebraic methods
93E03 Stochastic systems in control theory (general)
93C55 Discrete-time control/observation systems
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References:

[1] Abou-Kaudil, Matrix Riccati Equations in Control and Systems Theory (2003) · Zbl 1027.93001
[2] Damm, Rational Matrix Equations in Stochastic Control 297 (2004)
[3] Dragan, A class of discrete time generalized Riccati equations, Journal of Difference Equations and Applications 16 (4) pp 291– (2010) · Zbl 1185.93074
[4] Dragan, Mathematical Methods in Robust Control of Discrete Time Linear Stochastic Systems (2010) · Zbl 1183.93001
[5] Morozan, Discrete time Riccati equations connected with quadratic control for linear systems with independent random perturbations, Revue Roumaine de Mathématiques Pures Appliquées 37 (3) pp 233– (1992) · Zbl 0763.93053
[6] Morozan, Stability radii of some discrete-time systems with independent random perturbations, Stochastic Analysis and Applications 15 (3) pp 375– (1997) · Zbl 0898.93035
[7] Ungureanu, Quadratic control problem for linear discrete-time varying systems with multiplicative noise in Hilbert spaces, Mathematical Reports 1 7 (57) pp 73– (2005) · Zbl 1097.93039
[8] Ungureanu, Optimal control for linear discrete time systems with Markov perturbations in Hilbert spaces, IMA Journal of Mathematical Control and Information 26 (1) pp 105– (2009) · Zbl 1159.93036
[9] El Bouhtouri, H -type control for discrete-time stochastic systems, International Journal of Robust and Nonlinear Control 13 (9) pp 923– (1999) · Zbl 0934.93022
[10] Cao, Stochastic stabilizability and H control for discrete-time jump linear systems with time delay, Journal of the Franklin Institute 336 pp 1263– (1999) · Zbl 0967.93095
[11] Costa, Full information H -control for discrete-time infinite Markov jump parameter, Journal of Mathematical Analysis and Applications 202 (2) pp 578– (1996) · Zbl 0862.93025
[12] Costa, State feedback H -control for discrete-time infinite-dimensional stochastic bilinear systems, Journal of Mathematical Systems, Estimation and Control 6 (2) pp 1– (1996) · Zbl 0844.93036
[13] Bensoussan, Representation and Control of Infinite Dimensional Systems II (1992)
[14] Daduna, Queueing Networks with Discrete Time Scale 2046 (2001) · Zbl 0988.60094
[15] Costa, Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems, Automatica 43 pp 587– (2007) · Zbl 1115.49021
[16] Do Val, Receding horizon control of jump linear systems and a macroeconomic policy problem, Journal of Economic Dynamics and Control 23 pp 1099– (1999) · Zbl 0962.91058
[17] Fragoso, Optimal control for continuous time LQ-problems with infinite Markov jump parameters, SIAM Journal on Control and Optimization 40 pp 270– (2001) · Zbl 1058.93058
[18] Dragan, Linear quadratic optimization problems for some discrete-time stochastic linear systems, Mathematical Reports 11 4 (61) pp 307– (2009) · Zbl 1212.93324
[19] Dragan V Stoica A Riccati type equations for stochastic systems with jumps Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems 2008 753 765
[20] Hu, Stochastic stability and robust control for sampled-data systems with Markovian jump parameters, Journal of Mathematical Analysis and Applications 313 pp 504– (2006) · Zbl 1211.93131
[21] Ichikawa, Linear Time Varying Systems and Sampled Data Systems 265 (2001) · Zbl 0971.93003
[22] Costa, A generalized multi-period mean-variance portfolio optimization with Markov switching parameters, Automatica 44 pp 2487– (2008) · Zbl 1157.91356
[23] Dragan, Discrete time linear equations defined by positive operators on ordered Hilbert spaces, Revue Roumaine de Mathématiques Pures Appliquées 53 (2-3) pp 131– (2008)
[24] de Souza C Fragoso M H filtering for discrete-time systems with Markovian jumping parameters Proceedings of the 36th Conference on Decision and Control 1997 2181 2186
[25] Dragan, Exponential stability for discrete time linear equations defined by positive operators, Integral Equations and Operator Theory 54 (4) pp 465– (2006) · Zbl 1094.39008
[26] Costa, On the detectability and observability of discrete-time Markov jump linear systems, Systems & Control Letters 44 pp 135– (2001) · Zbl 0986.93008
[27] Li, Detectability and observability of discrete-time stochastic systems and their applications, Automatica 45 pp 1340– (2009) · Zbl 1162.93321
[28] Zabczyk, On optimal stochastic control of discrete-time parameter systems in Hilbert spaces, SIAM Journal on Control and Optimization 13 pp 1217– (1975) · Zbl 0313.93067
[29] Ungureanu, Mean stability of a stochastic difference equation, Annales Polonici Mathematici 1 (93) pp 33– (2008) · Zbl 1141.37037
[30] Ungureanu, Mean square error synchronization in networks with ring structure, Taiwanese Journal of Mathematics 14 (6) pp 2405– (2010) · Zbl 1252.37044
[31] Corach, Generalized Schur complements and oblique projections, Linear Algebra and its Applications 341 pp 259– (2002) · Zbl 1015.47014
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