Control strategies for the Fokker−Planck equation

Abstract

Using a projection-based decoupling of the Fokker-Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup.Mathematics Subject Classification. 35Q35, 49J20, 93D05, 93D15.

Model reduction of controlled Fokker–Planck and Liouville–von Neumann equations

“…The intuitive idea now is splitting the state into the direct sum of the asymptotically stable subspace and the eigenspace associated with the eigenvalue 0. Since a straightforward implementation in general will destroy the sparsity pattern of the matrices, we suggest to use a particular decomposition that has been introduced in a similar setup in [17]. Define the matrix…”

Taylor expansions of the value function associated with a bilinear optimal control problem

“…In this section, we show that assumptions (A1)-(A4) are satisfied for a concrete infinite-dimensional bilinear optimal control problem. Following the setup discussed in [5], we focus on the controlled Fokker-Planck equation ∂ρ ∂t =ν∆ρ + ∇ · (ρ∇G) + u∇ · (ρ∇α) in Ω × (0, ∞),…”

“…In order to consider (101) as a stabilization problem of the form (P ), we introduce a state variable y := ρ − ρ ∞ as the deviation to the stationary distribution. As discussed in [5], this yields a system of the formẏ…”

L2-Tracking of Gaussian Distributions via Model Predictive Control for the Fokker–Planck Equation

“…A complete characterization of possible boundary conditions for d = 1 can be found in the work of Feller [9]. In the multidimensional case, one possible choice is the zero-flux boundary condition n · j(x, t) = 0 on ∂Ω × (0, T ), where j denotes the probability flux and n is the unit normal vector to the surface ∂Ω, see [3,4]. With this, the conservation of mass property in (4) holds.…”

“…These piecewise constant control sequences fit well with the notation of Σ(k) introduced in the beginning of Section 5. 4 All simulations were carried out with such controls. Otherwise one should specify how to evaluate the stage cost (26) in every MPC step.…”