Backstepping design for parabolic systems with in-domain actuation and Robin boundary conditions

State feedback design for linear parabolic systems with in-domain actuation and general Robin boundary conditions is considered. To this end the system is shown to be state equivalent to a boundary controlled system. By means of the well established backstepping transformation this latter system is feedback equivalent to a stable parabolic equation. Within the contribution previous results concerning systems with Neumann boundary conditions are generalized by means of functional analytic methods. Existence of the involved transformations is discussed by means of the Fredholm theory while a late lumping approach is proposed for the numerical implementation.