Title: An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator,
Abstract: In many applications, and in systems/synthetic biology, in particular, it is
desirable to compute control policies that force the trajectory of a bistable
system from one equilibrium (the initial point) to another equilibrium (the
target point), or in other words to solve the switching problem. It was
recently shown that, for monotone bistable systems, this problem admits
easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step
functions of fixed length and fixed magnitude). In this paper, we develop this
idea further and formulate a problem of convergence to an equilibrium from an
arbitrary initial point. We show that this problem can be solved using a static
optimization problem in the case of monotone systems. Changing the initial
point to an arbitrary state allows to build closed-loop, event-based or
open-loop policies for the switching/convergence problems. In our derivations
we exploit the Koopman operator, which offers a linear infinite-dimensional
representation of an autonomous nonlinear system. One of the main advantages of
using the Koopman operator is the powerful computational tools developed for
this framework. Besides the presence of numerical solutions, the
switching/convergence problem can also serve as a building block for solving
more complicated control problems and can potentially be applied to
non-monotone systems. We illustrate this argument on the problem of
synchronizing cardiac cells by defibrillation. Potentially, our approach can be
extended to problems with different parametrizations of control signals since
the only fundamental limitation is the finite time application of the control
signal.