Linear monotone systems admit Lyapunov functions that are separated into a sum or a maximum over functions depending on one state only, which is a useful feature for large-scale system analysis. Under certain conditions these functions can be constructed using the leading eigenvectors (i.e., the eigenvectors corresponding to the eigenvalue with the largest real part) of the drift matrix. In this paper, our goal is to extend some of these results to the nonlinear setting. In order to do so, we employ the Koopman operator, which allows for a linear infinite-dimensional description of a nonlinear system. Since the Koopman operator is linear, we can compute its eigenfunctions, which can be seen as infinite-dimensional eigenvectors. We show that the leading eigenfunction of a Koopman operator associated with a monotone system is a Lyapunov function. However, this Lyapunov function is not necessarily sum-separable in contrast to the linear case. We also show that a recently proposed max-separable Lyapunov function can be written in terms of the leading eigenfunction under certain conditions. This allows to characterize a subset of monotone systems for which these functions exist, thus further developing the existing results. We illustrate our theoretical findings on examples, which in particular show that global monotonicity is not necessary for existence of max-separable Lyapunov functions, and discuss future research directions.
['Aivar Sootla']
Construction of max-separable Lyapunov functions for monotone systems using the Koopman operator