First and second order optimality conditions for the control of Fokker-Planck equations
M. Soledad Aronna, Fredi Tröltzsch
Abstract
In this article we study an optimal control problem subject to the Fokker-Planck equation ∂ t ρ − ν ∆ ρ − div( ρB [ u ]) = 0 The 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 regularization term on the control. For this problem, we prove existence of optimal controls and first order necessary conditions. Main emphasis is placed on second order necessary and sufficient conditions.
Topics & Concepts
Fokker–Planck equationOptimal controlControl functionRegularization (linguistics)Control variableVariable (mathematics)MathematicsApplied mathematicsQuadratic equationOrder (exchange)Term (time)Control theory (sociology)Control (management)Mathematical optimizationComputer scienceMathematical analysisPhysicsPartial differential equationEconomicsArtificial intelligenceGeometryStatisticsFinanceQuantum mechanicsStability and Controllability of Differential EquationsStochastic processes and financial applicationsNumerical methods in inverse problems