First and second order optimality conditions for the control of Fokker-Planck equations

In this article we study an optimal control problem subject to the Fokker-Planck equation ∂ t ρ − ν ∆ div( ρB [ u ]) = 0 The variable is time-dependent and possibly multidimensional, function B depends on space control. cost functional of tracking type includes a quadratic regularization term For problem, prove existence controls first order necessary conditions. Main emphasis placed second sufficient