Optimal control of the Fokker-Planck equation with space-dependent controls. (English) Zbl 1373.35311
Summary: 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.
MSC:
| 35Q84 | Fokker-Planck equations |
| 35Q93 | PDEs in connection with control and optimization |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49K20 | Optimality conditions for problems involving partial differential equations |
Keywords:
bilinear control; Fokker-Planck equation; optimal control theory; optimization in Banach spaces; probability density function; stochastic optimal controlReferences:
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