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

Abstract

SUMMARYIn 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.

Output‐based H2 optimal controllers for a class of discrete‐time stochastic linear systems with periodic coefficients

“….t; i/B T v .t; i/ (B.10).t; i/ 2 Z S. Proceeding as in the proof of Lemma 24 in[17] one obtains:Lemma B.2 Assume: (a) The assumption H 1 is fulfilled. (b) For each t 2 Z C , P t is a nondegenerate stochastic matrix.Under these conditions the following hold:(i) The system (B.1) is stochastic detectable if the sequence of linear and positive operators ¹ Q ….t /º t 2Z is stabilizable in the sense of Definition B.1.…”

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

“…In this paper, we investigate the solution properties for a class of MAREs of control associated with autonomous infinite-dimensional DTLSs with MN and infinite MJs. These MAREs are time-invariant versions of the infinitedimensional GDTREs studied in [8]. For a detailed treatment of finite-dimensional GDTREs, the reader can consult [6,7] and the references therein.…”

“…The problem of existence of SSs for infinite-dimensional GDTREs associated with stabilizable DTLSs with infinite MJs was investigated in [3,5,9,10] under either stochastic detectability or observability hypotheses. In [8], a set of necessary and sufficient conditions for the existence of SSs is expressed in terms of feasibility of some linear matrix inequalities (LMIs) system.…”

Discrete-time jump linear systems with Markov chain in a general state space.

“…For instance, Costa and Fragoso (1995, Remark 6) showed that for the case in which the state space of the Markov chain is countably infinite, mean square and stochastic stability are no longer equivalent. Due to that, the case in which the Markov chain is countably infinite has received some attention (see Costa et al, 2005b;Ungureanu, 2009;Ungureanu et al, 2013).…”

“…These authors have also derived conditions for the existence and uniqueness of a positive semi-definite solution to the infinite coupled algebraic Riccati equations (ICARE) via the concepts of (stochastic) stabilizability and detectability. Ungureanu et al (2013), Ungureanu (2009) and Costa et al (2005b) have also dealt with the LQ optimal control of discrete-time MJLS with Markov chain in an infinite countable state space. Kordonis and Papavassilopoulos (2014) have characterized the finite and infinitehorizon LQ optimal control problems for MJLS with Markov chain in a general state space in terms of appropriate Riccati integral equations.…”