论文标题
时间分数Fokker-Planck方程的最佳控制问题的近似
Approximation of an optimal control problem for the time-fractional Fokker-Planck equation
论文作者
论文摘要
在本文中，我们研究了具有分数时间衍生物的PDE系统的数值近似。该系统源自凸二次参数的时间分数fokker-planck方程的最佳控制问题。该系统由时间折叠的后向汉密尔顿 - 雅各比 - 贝尔曼（Hamilton-Jacobi-Bellman）和正向fokker-planck方程组成，可用于描述被困在异常扩散方案中的粒子的概率密度的演变。我们通过L1方案和Hamiltonian通过有限的差异来近似系统中的Caputo衍生物。构建了Fokker-Planck方程的方案，以便在离散级别保留PDE系统的二元性结构。我们证明了该方案的良好姿势以及对连续问题的解决方案的收敛性。
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.
