抛物型最优控制问题的三次B样条有限元方法

对一类四阶非线性抛物方程最优控制问题提出一种三次B样条有限元方法。状态变量和对偶状态变量用具有更好光滑性的分片三次B样条连续函数进行逼近,控制变量由分片常数函数进行逼近。这样得到的状态变量和对偶状态变量的数值解二阶连续可微。建立最优性系统的全离散格式,并用迭代法进行求解。最后建立数值算例,验证方法的有效性。

Cubic B-spline finite element method for parabolic optimal control problems

A cubic B-spline finite element method is proposed for optimal control problems governed by a class of fourth-order nonlinear parabolic equations. The state and co-state variables are discretized by piecewise cubic B-spline continuous functions which have better smoothness and the control variable is approximated by piecewise constant functions. The numerical solutions of the state and co-state variables thus obtained are second-order continuously differentiable. A fully discrete scheme of the optimality system is established and solved by an iterative method. Finally, some numerical examples are presented to verify the effectivity of the proposed method.