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# EMS Surveys in Mathematical Sciences

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**Volume 5, Issue 1/2, 2018, pp. 65–98**

**DOI: 10.4171/EMSS/27**

Published online: 2018-06-18

A Fokker–Planck control framework for stochastic systems

Mario Annunziato^{[1]}and Alfio Borzì

^{[2]}(1) Università degli Studi di Salerno, Fisciano, Italy

(2) Universität Würzburg, Germany

A new framework for the optimal control of probability density functions (PDF) of stochastic processes is reviewed. This framework is based on Fokker–Planck (FP) partial differential equations that govern the time evolution of the PDF of stochastic systems and on control objectives that may require to follow a given PDF trajectory or to minimize an expectation functional.

Corresponding to different stochastic processes, different FP equations are obtained. In particular, FP equations of parabolic, fractional parabolic, integro parabolic, and hyperbolic type are discussed. The corresponding optimization problems are deterministic and can be formulated in an open-loop framework and within a closed-loop model predictive control strategy. The connection between the Dynamic Programming scheme given by the Hamilton–Jacobi–Bellman equation and the FP control framework is discussed. Under appropriate assumptions, it is shown that the two strategies are equivalent. Some applications of the FP control framework to different models are discussed and its extension in a mean-field framework is elucidated.

*Keywords: *Fokker–Planck equations, Hamilton–Jacobi–Bellman equation, stochastic process, hybrid systems, Markov renewal processes, probability density function, mean-field approach, optimal control theory, model predictive control, numerical analysis, scientific computing

Annunziato Mario, Borzì Alfio: A Fokker–Planck control framework for stochastic systems. *EMS Surv. Math. Sci.* 5 (2018), 65-98. doi: 10.4171/EMSS/27