Stability criteria for positive linear discrete-time systems

Abstract

We 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.

ISS small-gain criteria for infinite networks with linear gain functions

“…Our small-gain theorem relies on Proposition 9 below. In this proposition, we show that the small-gain condition is equivalent to uniform global exponential stability of the monotone discrete-time system induced by the gain operator, which is of great significance on its own, and extends several known criteria for linear discrete-time systems summarized in [14]. Moreover, this proposition provides a linear path of strict decay through which we construct an ISS Lyapunov function for the overall network.…”

A Lyapunov-based ISS small-gain theorem for infinite networks of nonlinear systems

“…Assume to the contrary that there is s ∈ + ∞ \{0} such that Γs ≥ (1−ε)s for a fixed but arbitrary ε ∈ (0, 1). Then also Rs ≥ Γs ≥ (1 − ε)s, where R is the right-shift operator from [25,Ex. 3.15].…”

“…3.15]. However, in view of [25,Ex. 3.15], R satisfies the strong small-gain condition with any ε ∈ (0, 1), and we come to a contradiction.…”

“…In particular, the method for the construction of the paths of strict decay for the gain operator, which we adopted from [22] and successfully employed here, relies strongly on the max formulation of the ISS property for subsystems. In case that the internal gains are all linear, and the ISS property for subsystems is given in a sum formulation, the gain operator is linear, and characterizations of the condition r(Γ) < 1 have been developed in [25]. In particular, they result in a construction of a path of strict decay for the linear gain operator.…”

“…Another important open problem concerns the relationships between the existence of a path of strict decay and other properties of the gain operator, such as robust strong and uniform small-gain conditions, the monotone limit property, uniform global asymptotic stability of the induced discrete-time system, etc. This would not only provide a better understanding of the applicability of the nonlinear ISS small-gain theorems, but also would provide powerful small-gain type criteria for uniform global asymptotic stability of nonlinear discrete-time systems induced by positive operators (for the finite-dimensional case, see [27], and for linear infinite-dimensional discrete-time systems, see [25]).…”