A fractional Fokker-Planck control framework for subdiffusion processes

Abstract

Summary An efficient framework for the optimal control of the probability density function of a subdiffusion process is presented. This framework is based on a fractional Fokker–Planck equation that governs the time evolution of the PDF of the subdiffusion process and on tracking objectives of terminal configuration of the desired PDF. The corresponding optimization problems are formulated as a sequence of open‐loop optimality systems in a model predictive control strategy. The resulting optimality system with fractional evolution operators is discretized by a suitable scheme that guarantees positivity of the forward solution. The effectiveness of the proposed computational framework is validated with numerical experiments. Copyright © 2015 John Wiley & Sons, Ltd.

On an optimal control problem of time-fractional advection-diffusion equation

“…[30,2,37,38,27]. Optimal control problems of time-fractional diffusion type equations and numerical schemes have been considered by Antil et al in [4] and by Annunziato et al in [3]. Subdiffusion processes deviate from classical Gaussian process in that the motion of particles are interrupted by long sojourns, possibly due to trapping effects.…”

Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

“…In addition we denote by convention (I 1−α (0,t] m)(T ) = I 1−α (0,T ] m. System (1) is derived in [28] (see also [5]), where the Fokker-Planck equation (4) is considered in its weak formulation via a general fractional Sobolev space framework. The well-posedness of the coupled system is obtained using compactness and Schauder fixed-point argument.…”

“…Time fractional advection-diffusion equations have been extensively used to study evolution of the probability of particles governed by subdiffusion behaviour. This refers to the trajectory of a single particle modeled by the non-Markovian stochastic process where the mean square displacement is no longer linear [5,6,24,23,31]. The nonlocal character of fractional time derivative makes it very useful to describe memory effects and derive the power law decay structure of energy in a diffusion-transport PDE system [13,17,18,29].…”

A Fokker–Planck control framework for stochastic systems

“…However, while the FP equation has been considered for long time to model the time evolution of stochastic processes, it is only recently that a control framework for these processes based on the FP equation has been proposed; see [5] for an earlier publication. Following this publication, the Authors of this review have considerably developed this topic [5,6,7,8,9,10,12,29,65,66,95,96,108,109,116] and witnessed a surge of research work in this field focusing on FP models and related control problems; see, e.g., [25,28,56,57,58,74,77,78,125,127].…”

“…Concerning Itō stochastic processes, we refer to [6] for application of the FP control framework to a stochastic Lotka-Volterra model, to [7] for the FP control of a stochastic quantum spin model, to [108,109] for the control of crowd motion and to [77,78] for that of the statistics of the spike emission of a neural membrane. For stochastic Itō systems that include random jumps, e.g., for finance modelling, and sparsity of the control we quote [66] and [9] for sub-diffusion models. Concerning other stochastic systems, the FP control approach has been applied successfully also to PDP models such as [8], to the optimization of antibiotic subtilin production [116], and to discrete random walks [29].…”