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.
Highlights
Positive systems occur frequently in the modeling, analysis and control of dynamical systems, for instance, in chemical engineering, compartmental systems and ecological systems [17]
In [22] characterizations for the negativity of the spectral bound of resolvent positive operators have been studied, which are continuous-time counterparts of the results studied in this paper
After a brief description of ordered Banach spaces and positivity, we focus on the stability analysis of positive linear discrete-time evolution equations
Summary
Positive systems occur frequently in the modeling, analysis and control of dynamical systems, for instance, in chemical engineering, compartmental systems and ecological systems [17]. 3.1, we introduce several novel stability properties, most notably the uniform small-gain condition, and characterize the exponential stability of positive linear discrete-time systems in terms of such properties (Theorem 3.1). Though not all, of the equivalences in Theorem 3.1 have been shown in [39] to hold even in the non-linear case; these results are related to the input-to-state stability of control systems with inputs and to so-called small-gain theorems. 4. There we discuss several results on uniform stability of (non-positive) linear systems which are scattered throughout the literature; we include various references which point the reader to further related results. For bounded linear operators on real Banach spaces, spectral properties are defined by means of complexification; for details, we refer to the subsection on complexifications at the end of Sect. For bounded linear operators on real Banach spaces, spectral properties are defined by means of complexification; for details, we refer to the subsection on complexifications at the end of Sect. 2
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