In this paper, finite point method (FPM) and meshless weighted least squares (MWLS) method are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The FPM scheme uses shape function constructed by moving least square (MLS) approximation to discretize the equations, while the MWLS scheme employs both MLS approximation and penalty terms to solve the same problem. Error estimates for the FPM scheme are presented and numerical results are provided to examine the impact of parameters and validate the efficiency of the proposed schemes. The extended model (Navier–Stokes equations) shows the ability of our algorithm to handle complex problems. Our explorative work shows the flexibility and great potential of the meshless methods in optimal control problems.

提出了用有限点法（FPM）和无网格加权最小二乘（MWLS）法求解椭圆方程控制的狄利克雷边界最优控制问题。FPM方案使用通过移动最小二乘（MLS）近似构造的形状函数来离散化方程，而MWLS方案同时使用MLS近似和惩罚项来解决相同的问题。提出了FPM方案的误差估计，并提供了数值结果以检查参数的影响并验证所提出方案的效率。扩展模型（Navier–Stokes方程）显示了我们算法处理复杂问题的能力。我们的探索性工作表明了无网格方法在最优控制问题中的灵活性和巨大潜力。