Sub-homogeneous positive monotone systems are insensitive to heterogeneous time-varying delays

We show that a sub-homogeneous positive monotone system with bounded heterogeneous time-varying delays is globally asymptotically stable if and only if the corresponding delay-free system is globally asymptotically stable. The proof is based on an extension of a delay-independent stability result for monotone systems under constant delays by Smith to systems with bounded heterogeneous time-varying delays. Under the additional assumption of positivity and sub-homogeneous vector fields, we establish the aforementioned delay insensitivity property and derive a novel test for global asymptotic stability. If the system has a unique equilibrium point in the positive orthant, we prove that our stability test is necessary and sufficient. Specialized to positive linear systems, our results extend and sharpen existing results from the literature.Comment: Submitted to the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS), 201

Asymptotic Stability and Decay Rates of Homogeneous Positive Systems With Bounded and Unbounded Delays

There are several results on the stability of nonlinear positive systems in the presence of time delays. However, most of them assume that the delays are constant. This paper considers time-varying, possibly unbounded, delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, we present a necessary and sufficient condition for delay-independent stability of continuous-time positive systems whose vector fields are cooperative and homogeneous. We show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays. For various classes of time delays, we are able to derive explicit expressions that quantify the decay rates of positive systems. We also provide the corresponding counterparts for discrete-time positive systems whose vector fields are non-decreasing and homogeneous.Comment: SIAM Journal on Control and Optimizatio

Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions

We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time-delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state-of-the-art