Title: A Contractive Approach to Separable Lyapunov Functions for Monotone Systems,
Abstract: Monotone systems preserve a partial ordering of states along system
trajectories and are often amenable to separable Lyapunov functions that are
either the sum or the maximum of a collection of functions of a scalar
argument. In this paper, we consider constructing separable Lyapunov functions
for monotone systems that are also contractive, that is, the distance between
any pair of trajectories exponentially decreases. The distance is defined in
terms of a possibly state-dependent norm. When this norm is a weighted
one-norm, we obtain conditions which lead to sum-separable Lyapunov functions,
and when this norm is a weighted infinity-norm, symmetric conditions lead to
max-separable Lyapunov functions. In addition, we consider two classes of
Lyapunov functions: 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. In addition, we present an algorithm based
on sum-of-squares programming to compute such separable Lyapunov functions. We
provide several examples to demonstrate our results.