Observations on the stability properties of cooperative systems

Abstract

We extend two fundamental properties of positive linear time-invariant (LTI) systems to homogeneous cooperative systems. Specifically, we demonstrate that such systems are -stable, meaning that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed homogeneous cooperative system is globally asymptotically stable (GAS) for any non-negative delay if and only if the system is GAS for zero delay.

Stability and positivity of equilibria for subhomogeneous cooperative systems

“…This property is commonly referred to as D-stability. It was shown in [15] [2] that a nonlinear analogue of D-stability property also holds for a significant class of nonlinear positive systems; namely cooperative systems that are homogeneous [2]. We shall show here that these results can be further extended to subhomogeneous cooperative systems (we define these formally in the following two sections).…”

“…A restricted form of this concept was considered in [15] for a specific class of homogeneous systems. A more general definition was then introduced in [2] wherein D-stability results for arbitrary homogeneous cooperative systems were established.…”

“…We shall show here that these results can be further extended to subhomogeneous cooperative systems (we define these formally in the following two sections). It should be pointed out that the methods used in [15] [2] relied heavily on an extension of the PerronFrobenius theorem given in [1]. In contrast, our approach here makes use of the Knaster-Kuratowski-Mazurkiewicz (KKM) theorem [9] and is inspired by recent applications of this result to small-gain conditions and input-to-state stability in [3] [19].…”

$\mu $ -Stability of Nonlinear Positive Systems With Unbounded Time-Varying Delays

“…To overcome this disadvantage, there have been some excellent works on nonlinear positive systems in the literature. For example, [23] is a pioneering work on generalizing the corresponding analysis from positive linear systems to homogeneous cooperative and irreducible systems; [24] shows that a constant-delayed homogeneous cooperative system is globally asymptotically stable (GAS) for all nonnegative delays if and only if the system is GAS for zero delay; [25] shows that GAS and cooperative systems, homogeneous of any order with respect to arbitrary dilation maps are Dstable with constant time delay; [26] investigates the exponential stability of homogeneous positive systems of degree one with bounded time-varying delays; and [27] generalizes the homogeneous positive systems to any degree, and the bounded time-varying delay to be unbounded, the asymptotic and µ-stability are discussed for both continuous-time and discretetime systems.…”

Robust stability of positive linear time delay systems under time-varying perturbations

“…Recently, problems of stability and robust stability of positive systems have attracted a lot of attention from researchers, see e.g. [2,11,12,[23][24][25][26][27][28][29][30][31][32][33] and references therein.…”