A Fokker–Planck control framework for multidimensional stochastic processes

An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of termina...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2013-01, Vol.237 (1), p.487-507
Main Authors: Annunziato, M., Borzì, A.
Format: Article
Language:English
Subjects:
Computation
Fokker–Planck equation
Mathematical models
Model predictive control
Multidimensional stochastic process
Optimal control
Optimal control theory
Optimization
Probability density function
Probability density functions
Stochastic processes
Strategy
Tracking
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Summary:An efficient framework for the optimal control of probability density functions (PDFs) of multidimensional stochastic processes is presented. This framework is based on the Fokker–Planck equation that governs the time evolution of the PDF of stochastic processes and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open-loop optimality systems in a receding-horizon control strategy. Many theoretical results concerning the forward and the optimal control problem are provided. In particular, it is shown that under appropriate assumptions the open-loop bilinear control function is unique. The resulting optimality system is discretized by the Chang–Cooper scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with a stochastic Lotka–Volterra model and a noised limit cycle model.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2012.06.019