Solving Optimal Control Problems for Monotone Systems Using the Koopman Operator

peer reviewedaudience: researcher, professional, studentKoopman 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 analyse and implement. The motivation for the study of the convergence problem comes from biological applications, where easy-to-implement and easy-to-analyse 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 on biologically inspired numerical examples

Symmetry-based model reduction for approximate stochastic analysis

International audienceFor models of cell-to-cell communication, with many reactions and species per cell, the computational cost of stochastic simulation soon becomes intractable. Deterministic methods, while computationally more efficient, may fail to contribute reliable approximations for those models. In this paper, we suggest a reduction for models of cell-to-cell communication, based on symmetries of the underlying reaction network. To carry out a stochastic analysis that otherwise comes at an excessive computational cost, we apply a moment closure (MC) approach. We illustrate with a community effect, that allows synchronization of a group of cells in animal development. Comparing the results of stochastic simulation with deterministic and MC approximation, we show the assets of our approach. The reduction presented here is potentially applicable to a broad range of highly regular systems