Optimal Control of the Fokker-Planck Equation with State-Dependent Controls

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Fleig, Arthur ; Guglielmi, Roberto:
Optimal Control of the Fokker-Planck Equation with State-Dependent Controls.
Bayreuth ; Linz , 2016 . - 21 S.

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Abstract

For a large class of stochastic processes, the evolution of the underlying probability density function is prescribed by a forward Kolmogorov equation, also called Fokker-Planck equation, a second-order parabolic partial differential equation. In this manner, an optimal control problem subject to an Itô stochastic differential equation can be rendered deterministic by recasting it as an optimal control problem for the Fokker-Planck equation, which we study in this work. In this setting, the control acts as a coefficient of the state variable in the advection term, i.e., it is of bilinear type. This optimal control problem has been firstly studied by Annunziato and Borzì (2010, 2013) for constant or time dependent controls. We extend the analysis to the case of time and space dependent controls, which allows to consider a wider variety of possible objectives. In order to deduce existence of nonnegative solutions for the state equation we require suitable integrability assumptions on the coefficients of the Fokker-Planck equation and thus on the control function. Therefore, the optimization takes place in a Banach space. We develop a systematic analysis of the existence of optimal controls and derive the system of first order necessary optimality conditions.