分布阶最优控制问题的Petrov-Galerkin谱方法数值模拟

SPECTRALPETROV-GALERKIN APPROXIMATION OF OPTIMAL CONTROL PROBLEM GOVERNED BY DISTRIBUTED ORDER FRACTIONAL DIFFERENTIAL EQUATION

研究了一类分布阶微分方程最优控制问题的Petrov-Galerkin谱方法数值模拟.首先利用拉格朗日泛函推导出一阶最优性条件,然后利用第一类和第二类广义雅可比多项式作为基函数逼近状态和伴随状态变量,基于先最优后离散的策略构造了Petrov-Galerkin谱方法离散格式.最后讨论了离散格式的数值实现,并给出了数值算例.结果表明收敛速度与解的正则性有关,说明了该方法具有指数收敛速度,验证了Petrov-Galerkin离散格式的稳定性和有效性.

In this paper we investigate a spectral Petrov-Galerkin discretization of optimal control problem governed by distributed order fractional differential equation.First order optimality condition is derived by using Lagrangian functional.A spectral Petrov-Galerkin scheme is constructed based on first optimize then discretize approach,where the Jacobi polyfractonomials of first kind and second kind are used as the basis functions to approximate the state and adjoint state variable,respectively.Numerical implementations are discussed and numerical example is given to illustrate the effectiveness of the discrete scheme.