Stability of positive linear switched systems on ordered Banach spaces

Abstract

We provide sufficient criteria for the stability of positive linear switched systems on ordered Banach spaces. The switched systems can be generated by finitely many bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we assume an appropriate interior point of the cone, whereas in the continuous-time case an appropriate interior point of the dual cone is sufficient for stability. This is an extension of the concept of linear Lyapunov functions for positive systems to the setting of infinite-dimensional partially ordered spaces. We illustrate our results with examples.

Introduction

Stability problems of switched systems have been studied in various works for the past years. Most authors focus on switched systems in ${\mathbb{R}}^d$, see e.g.  [1], [2] and references therein. However, stability problems of switched systems in infinite-dimensional spaces have been addressed less frequently, see e.g.  [3]. Usually, there are two approaches when dealing with switched systems: Either analysing stability behaviour of the switched system under arbitrary switching or finding a switching signal which generates a stable solution, referred to as stabilizability of the switched system.

In this paper, we want to investigate the stability behaviour of positive linear switched systems. These systems have recently attracted a large interest due to the extensive possibilities of their application, e.g. in biology, economy or engineering (see e.g.  [4], [5] and the references therein). In the literature, stability of switched linear systems is proved by using different methods, e.g. the joint spectral radius  [6], [7], [8] or quadratic Lyapunov functions  [1]. Note that continuous-time switched systems can be equivalently formulated as differential inclusions, see e.g.  [9], [10], [11], [12], [13].

Considering positivity in ${\mathbb{R}}^d$ with respect to the standard order induced by the cone ${\mathbb{R}}_{+}^d≔\{{(x_1,\dots ,x_d)}^{\top }\in {\mathbb{R}}^d;x_i\ge 0\text{ for }i=1,\dots ,d\}$, the existence of a linear Lyapunov function (LLF) is sufficient for the stability of a positive switched system, since the trajectories are restricted to the cone. Such an LLF can be constructed by means of an interior point of ${\mathbb{R}}_{+}^d$ which is mapped appropriately (see e.g.  [14]). In  [15], [16], the existence of an LLF is related to the product property of matrices. An algorithm to construct an LLF is given in  [17] using Collatz–Wielandt sets. Concerning positive linear switched systems with respect to polyhedral cones, we refer to  [18] for stability criteria using an LLF and to  [19] for a copositivity approach. For positive systems with respect to more general cones in ${\mathbb{R}}^d$, stability is investigated in  [20], [21], where extremal norms are used.

In the present work we establish sufficient criteria for the stability of positive linear switched systems generated by bounded operators in infinite-dimensional spaces with a general class of order-inducing cones. In the discrete-time case, we exploit the idea of the existence of an appropriately mapped interior point of the cone to ensure stability of the switched system. A dual concept using a uniformly positive linear functional, i.e. an interior point of the dual cone, is established to prove stability in the continuous-time case. This is an extension of the concept of LLF cited above to infinite-dimensional partially ordered spaces. Even in the setting of finite-dimensional spaces ordered by arbitrary cones to our knowledge similar results have not been known so far.

The paper is structured as follows: In Section  2, we collect basic notions and concepts concerning ordered vector spaces and positive operators. Section  3 deals with positive linear discrete-time switched systems. We provide a criterion for the stability of such systems in order unit spaces and relate it to the joint spectral radius. In Section  4 we consider positive linear continuous-time switched systems on ordered Banach spaces. The stability of the system is ensured by the existence of a suitable uniformly positive linear functional. The main results are illustrated by examples.