We consider the problem of controlling the heat flow at a boundary in the framework of the linear diffusion equation in one spatial dimension. By the use of equations based on the theory of differential forms, it has been possible to derive linear relations between the moments of the boundary functions and those of the control function; these relations play the role of state equations in the optimal control problem considered, which is that of approaching in an optimal manner a given heat flow pattern at a boundary. We then introduce an approximation scheme that can be used for the numerical estimation of the optimal control. The approach can be extended to the control of non-linear diffusion equations with polynomial non-linearities.