Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

Abstract

Abstract-In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory. We exploit the framework of the Koopman operator, which provides a linear infinite-dimensional description of nonlinear dynamical systems and spectral operator-theoretic notions such as eigenvalues and eigenfunctions. The sublevel sets of the dominant eigenfunction form a family of nested forwardinvariant sets and the basin of attraction is the largest of these sets. The boundaries of these sets, called isostables, allow studying temporal properties of the system. Our first observation is that the dominant eigenfunction is increasing in every variable in the case of monotone systems. This is a strong geometric property which simplifies the computation of isostables. We also show how variations in basins of attraction can be bounded under parametric uncertainty in the vector field of monotone systems. Finally, we study the properties of the parameter set for which a monotone system is multistable. Our results are illustrated on several systems of two to four dimensions.

Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty

“…since it takes more time to reach C − β with the same optimal control signal. Now using (13) we get V (y, ν, β, p 1 ) ≥ V (z, µ, β, p 2 ) ≥ V (z, µ, α, p 2 ) and complete the proof.…”

Solving Optimal Control Problems for Monotone Systems Using the Koopman Operator

“…One of the fundamental results for this chapter is the description of spectral properties of monotone systems using the Koopman operator [31].…”

“…Closed-loop solutions, such as solutions arising from approximate dynamic programming, are preferable due to their inherent ability to deal with measurement noise, modeling errors and disturbances. While we also constructed a closed-loop control law (8) and there are efficient methods for estimating basins of attraction for monotone systems [31], this solution does not offer much insight into the "geometry" of the problem. Therefore one of the main outcomes of these studies was not a solution, but a series of questions: considering all the benefits of open-loop controls 7, is it possible to formulate an optimal control problem, which gives (7) (or (8)) as optimal solutions?…”

“…If the premise of the point (ii) in Lemma 1 holds, then the level sets of r are the graphs of strictly decreasing functions, a result which simplifies their computations as discussed in [33]. The algorithms developed in [31], [30] and [13] can be used to estimate these level sets. Proof.…”

Data-Driven Inference of High-Accuracy Isostable-Based Dynamical Models in Response to External Inputs

“…An alternative strategy is to identify Koopman eigenfunctions of the autonomous, unforced system and work directly in the resulting eigenfunction basis [19]. This is the approach taken by the isostable reduction framework [20], [21], [22], [23], [24] that uses a basis of the most slowly decaying eigenfunctions associated with the Koopman operator. In many applications an accurate reduced order model can be obtained using a small number of isostable coordinates [22], [24], [25] making control and analysis possible in a substantially reduced order setting.…”