Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls
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This paper is devoted to the analysis of a bilinear optimal control problem subject to the Fokker–Planck equation. The control function depends on time and space and acts as a coefficient of the advection term. For this reason, suitable integrability properties of the control function are required to ensure well posedness of the state equation. Under these low regularity assumptions and for a general class of objective functionals, we prove the existence of optimal controls. Moreover, for common quadratic cost functionals of tracking and terminal type, we derive the system of first-order necessary optimality conditions.
KeywordsBilinear control Fokker–Planck equation Optimal control theory Optimization in Banach spaces Probability density function Stochastic optimal control
Mathematics Subject Classification35Q84 35Q93 49J20 49K20
The authors wish to express their gratitude to Lars Grüne for suggesting them this very interesting subject and for many helpful comments. They would also like to thank Alfio Borzì for very helpful discussions and the referees for their valuable comments that helped to improve the manuscript. This work was partially supported by the EU under the 7th Framework Program, Marie Curie Initial Training Network FP7-PEOPLE-2010-ITN SADCO, GA 264735-SADCO, by the DFG Project Model Predictive Control for the Fokker–Planck equation, GR 1569/15-1, and by the INdAM through the GNAMPA Research Project 2015 “Analisi e controllo di equazioni a derivate parziali nonlineari.” Most of the results proved in this paper have been announced in a less general setting in a proceedings of the 2nd IFAC Workshop on Control of Systems Governed by Partial Differential Equations.
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