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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. [ABSTRACT FROM AUTHOR]