Abstract:
We consider monotone systems defined by ODEs on the positive orthant
in $\mathbb{R}^n$. These systems appear in various areas of
application, and we will discuss in some level of detail one of these
applications related to large-scale systems stability analysis.
Lyapunov functions are frequently used in stability analysis of
dynamical systems. For monotone systems so called sum- and
max-separable Lyapunov functions have proven very successful. One can
be written as a sum, the other as a maximum of functions of scalar
arguments.
We will discuss several constructive existence results for both
types of Lyapunov function. To some degree, these functions can be
associated with left- and right eigenvectors of an appropriate
mapping. However, and perhaps surprisingly, examples will demonstrate
that stable systems may admit only one or even neither type of
separable Lyapunov function.