A Fokker-Planck Strategy to Control Stochastic Processes

Alfio Borzi'
University of Würzburg
Tuesday 9th September 2014
Aula Saleri 6° Piano - Dipartimento di Matematica del Politecnico di Milano
An efficient framework for the optimal control of probability density functions (PDF) of multidimensional stochastic processes and piecewise deterministic processes is presented. This framework is based on Fokker-Planck-type equations that govern the time evolution of the PDF of stochastic processes. In this approach, the control objectives may require to follow a given PDF trajectory or to minimize an expectation functional. Theoretical results concerning the forward and the optimal control problems are provided. In the case of stochastic (Ito) processes, the Fokker-Planck equation is of parabolic type and it is shown that under appropriate assumptions the open-loop bilinear control function is unique. In the case of piecewise deterministic processes (PDP), the Fokker-Planck equation consists of a first-order hyperbolic system. Discretization schemes are discussed that guarantee positivity and conservativeness of the forward solution. The proposed control framework is validated with multidimensional biological, quantum mechanical, and financial models.