Discrete-time linear equations defined by positive operators

Abstract

In this chapter we study a class of discrete-time deterministic linear equations, namely discrete-time equations defined by sequences of positive linear operators acting on ordered Hilbert spaces. As we show in Chapter 3 such equations play a crucial role in the derivation of some useful criteria for exponential stability in the mean square of the stochastic systems considered in this book.

The results proved in this chapter also provide some powerful devices that help us to prove the existence of some global solutions, maximal solutions, minimal solutions, and stabilizing solutions of a large class of nonlinear equations including Riccati-type equations.

We want to mention that the results of this chapter may be successfully used to derive the solution of some control problems for deterministic positive systems with applications in economy, finances, biology, and so on. Such applications exceed the purpose of this monograph.