Backstepping-method for parabolic systems with in-domain actuation

State feedback design for linear parabolic systems with in-domain actuation is considered. This is achieved by generalizing the backstepping approach originally developed for boundary controlled systems by means of an additional state transform that associates the system under consideration with an equivalent boundary controlled system. This state transform is determined by exploiting the flatness property of the system under consideration, i.e., the parametrization of its solutions by the trajectory of a so-called flat output.