一阶双曲最优控制问题的数值分析

Numerical Analysis for First-Order Hyperbolic Optimal Control Problems

考虑了一阶双曲方程约束的最优分布控制问题,利用拉格朗日乘子方法给出一阶双曲最优控制问题的最优性条件,即状态方程、伴随方程和变分不等式。双曲方程中对流占主导,标准的有限体积元方法会产生物理震荡,该文采用高阶迎风有限体积元方法和变分离散相结合的方法对最优性条件进行数值离散。分别对最优性解的控制、状态和伴随给出了误差估计的结果。数值实验验证了方法的有效性和误差分析的结果。

In this paper,the optimal distributed control problem with constraints for first-order hyperbolic equations is considered.By using the Lagrange multiplier method,the optimality conditions of the first-order hyperbolic optimal control problem are given,which consist of the state equation,the adjoint equation and the variational inequality.Convection is dominant in hyperbolic equation,and the standard finite volume element method will produce physical oscillation.In this paper,the high-order upwind finite volume element method and the variational discretization method are used to discretize the optimality conditions.The error estimates for the control,state and adjoint of the optimal solutions are given.Numerical experiments verify the effectiveness of the method and the results of error analysis.