时间分数阶Fokker-Planck方程最优控制问题的逼近
Approximation of an optimal control problem for the time-fractional Fokker-Planck equation
摘要：
本文研究了具有分数阶时间导数的偏微分方程组的数值逼近。利用凸对偶方法，研究了具有不定常漂移的时间分数阶Fokker-Planck方程的最优控制问题.该系统由一个时间分数向后的Hamilton-Jacobi-Bellman方程和一个向前的Fokker-Planck方程组成，可以用来描述被囚禁在反常扩散区的粒子几率密度的演化.我们近似Caputo衍生物的系统通过L1格式和哈密顿量的有限差分。构造了求解Fokker-Planck方程的格式，使得PDE系统的对偶结构在离散水平上得以保持。我们证明了该方案的适定性和收敛到连续问题的解。
英文摘要：
In this paper, we study the numerical approximation of a system of PDEs with fractional time derivatives. This system is derived from an optimal control problem for a time-fractional Fokker-Planck equation with time dependent drift by convex duality argument. The system is composed by a time-fractional backward Hamilton-Jacobi-Bellman and a forward Fokker-Planck equation and can be used to describe the evolution of probability density of particles trapped in anomalous diffusion regimes. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed such that the duality structure of the PDE system is preserved on the discrete level. We prove the well posedness of the scheme and the convergence to the solution of the continuous problem.
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