Bilinear Optimal Control of the Fokker-Planck Equation*

The optimal tracking problem of the probability density function of a stochastic process can be expressed in terms of an optimal bilinear control problem for the Fokker-Planck equation, with the control in the coefficient of the divergence term. We analyze the case of time and space dependent controls. 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. Furthermore, we establish the existence of optimal controls and we derive the associated first order necessary optimality conditions.