We suggest a method of $C^2$ surface extension with the aid of well-controlled functions. The extended surface is $C^2$ continuous along the old boundary. The function of the extension surface is obtained by replacing the monomials in the quadratic Taylor polynomial of the given surface-representing function by other functions subject to some boundary conditions. We present several sets of control functions. In order to illustrate our suggestion, it is shown that surfaces with a circular boundary and a square boundary can be extended using several base functions.