Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator

TL;DR

This paper develops a control method using temporal pulses and Koopman operator eigenfunctions to switch bistable monotone systems between steady states, with applications in biomedical fields.

Contribution

It introduces the switching function concept related to Koopman eigenfunctions, extending control capabilities for bistable systems without feedback.

Findings

Computed pulse sets for targeted state switching.

Estimated convergence times to steady states.

Extended previous monotone system control results.

Abstract

In this paper, we further develop a recently proposed control method to switch a bistable system between its steady states using temporal pulses. The motivation for using pulses comes from biomedical and biological applications (e.g. synthetic biology), where it is generally difficult to build feedback control systems due to technical limitations in sensing and actuation. The original framework was derived for monotone systems and all the extensions relied on monotone systems theory. In contrast, we introduce the concept of switching function which is related to eigenfunctions of the so-called Koopman operator subject to a fixed control pulse. Using the level sets of the switching function we can (i) compute the set of all pulses that drive the system toward the steady state in a synchronous way and (ii) estimate the time needed by the flow to reach an epsilon neighborhood of the target steady state. Additionally, we show that for monotone systems the switching function is also monotone in some sense, a property that can yield efficient algorithms to compute it. This observation recovers and further extends the results of the original framework, which we illustrate on numerical examples inspired by biological applications.