Separability of Lyapunov functions for contractive monotone systems

Abstract

We consider constructing Lyapunov functions for systems that are both monotone and contractive with respect to a weighted one norm or infinity norm. This class of systems admits separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a single argument. In either case, two classes of separable Lyapunov functions exist: the first class is separable along the system's state, and the second class is separable along components of the system's vector field. The latter case is advantageous for many practically motivated systems for which it is difficult to measure the system's state but easier to measure the system's velocity or rate of change. We provide several examples to demonstrate our results.

A contractive approach to separable Lyapunov functions for monotone systems

“…Note that (28) and (30) are state separable Lyapunov functions while (29) and (31) are flow separable Lyapunov functions. Corollaries 1 and 2 were previously reported in [7].…”

On resilient control of dynamical flow networks

“…Exponential stability of (5) can instead be certified by weighted l 1 -distances of the form i ω i |x i − x * i |, for positive values of the ω i 's that depend on 6 the specific values of the entries of L. In fact, existence of such sum-as well as max-separable Lyapunov functions is equivalent to stability for positive (not only compartmental) systems [12,Proposition 1], and this is at the heart of the scalability properties of these systems. (See [13] for extensions of this result to nonlinear monotone systems.) In contrast, the unweighted l 1 -distance can be used to prove global asymptotic stability of compartmental systems (via LaSalle's theorem) using structural properties only, i.e., properties that are independent of the specific values of the entries of the compartmental matrix −L, but depend just on their sign pattern.…”

Approximating periodic trajectories of contractive systems

“…Often it is useful to work with scaled vector norms (see, e.g. [18,19]). Let | · | * : R n → R + be some vector norm, and let µ * : R n×n → R denote its induced matrix measure.…”

“…This means that the Jacobian A of (19) satisfies µ 2,P (A) ≤ −η, where µ 2,P is the matrix measure induced by the scaled Euclidean norm |z| 2,P := |P z| 2 (see (4)). Thus, (19) is contractive with respect to this scaled norm with contraction rate η, and every solution of (19) converges to the unique T -periodic solution γ(t) of (19). Letū := 1…”