Controlling heat or diffusion processes is a problem that appears in a systematic 
manner in many engineering applications. Information propagates at an infinite speed thus making the control mechanisms to be more efficient than in wave processes where the finite speed of propagation requires a minimal control time and suitable geometric configurations for the location of the controllers. But, simultaneously, in the case of the heat equation, because of the very strong time irreversibility, the space of controllable data is hard to identify. For the same reason, controls are hard to compute numerically. In this lecture we address these two issues using the method of transmutation or the socalled Kannai transform. A good understanding of the corresponding problems in the context  of waves allows getting satisfactory results for the heat equation too.
But he subject is full of interesting open problems.