Solving Optimal Control Problems for Monotone Systems Using the Koopman Operator

Koopman operator theory offers numerous techniques for analysis and control of complex systems. In particular, in this chapter we will argue that for the problem of convergence to an equilibrium, the Koopman operator can be used to take advantage of the geometric properties of controlled systems, thus making the optimal solutions more transparent, and easier to analyze and implement. The motivation for the study of the convergence problem comes from biological applications, where easy-to-implement and easy-to-analyze solutions are of particular value. At the moment, theoretical results have been developed for a class of nonlinear systems called monotone systems. However, the core ideas presented here can be applied heuristically to non-monotone systems. Furthermore, the convergence problem can serve as a building block for solving other control problems such as switching between stable equilibria or inducing oscillations. These applications are illustrated in biologically inspired numerical examples.