Abstract
For monotone systems evolving on the positive orthant of $\mathbb{R}^n_+$ two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function.
Highlights
IntroductionIn this paper we consider dynamical systems defined on Rn+ := [0, ∞)n via the differential equation x = f (x) (1)
In this paper we consider dynamical systems defined on Rn+ := [0, ∞)n via the differential equation x = f (x) (1)with f : Rn+ → Rn locally Lipschitz continuous and f (0) = 0
Monotone systems appear in a variety of real-world scenarios, such as chemical reaction networks [9], gene expression [28] and general systems biology [41], as well as traffic networks [7]
Summary
In this paper we consider dynamical systems defined on Rn+ := [0, ∞)n via the differential equation x = f (x) (1). Monotonicity and the equilibrium at the origin imply that system (1) leaves Rn+ invariant in forward time. (For a monotone system with equilibrium in x0, the set {y ∈ Rn : x0 ≤ y} must be positively invariant, so we might as well shift coordinates to get x0 = 0.). If V is continuously differentiable for points x = 0 a sufficient. This publications subsumes material presented at and published in the proceedings of the IEEE Conferences on Decision and Control in 2013 [31] and 2014 [24].
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