compact support with respect to the time variable or the effect of adding numerical dissipative singular terms. But the main contribution of this paper is to develop the numerical version of the so-called transmutation method that allows writing the control of a heat process in terms of the corresponding control of the associated wave process, by means of a "time convolution" with a one-dimensional controlled fundamental heat solution. This method, although it can be proved to converge, is also subtle in its computational implementation. Indeed, in one hand, it requires using convergent numerical schemes for the control of the wave equation, a problem that, as mentioned above, is delicate in itself. But it also needs computing an accurate approximation of a controlled fundamental heat solution, an issue that requires its own analysis and significant numerical and computational new developments. These methods are thoroughly illustrated and discussed along the paper, accompanied by some numerical experiments in one space dimension that show the subtlety of the issue. These experiments allow comparing the efficiency of the various methods. This is done in the case where the control is distributed in some subdomain of the domain where the heat process evolves but similar results and numerical experiments could be derived for other cases such as the one in which the control acts on the boundary. The techniques we employ here can also be adapted to the multi-dimensional case.

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