Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator

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 di cult 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 xed 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
ow 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 e cient 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.