First and Second Order Optimality Conditions for the Control of Fokker-Planck Equations

Abstract

In this article we study an optimal control problem subject to the Fokker-Planck equationThe control variable u is time-dependent and possibly multidimensional, and the function B depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic (Tikhonov regularizing) term on the control. For this problem we prove existence of optimal control, first order necessary conditions, and second order necessary and sufficient conditions.