|Abstract:|| 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.|