Backstepping control for parabolic PDEs with in-domain actuation

The backstepping method is a systematic design tool for boundary control of various types of partial differential equations (PDEs). There has been no attempt to apply it to PDEs whose input is not at the boundary. In this paper, we consider a problem of feedback stabilization of 1-dimensional parabolic (unstable) PDEs with internal actuation based on the backstepping method. Since such a PDE can not be converted to a stable PDE by state feedback and the state transformation used in backstepping, an additional transformation is introduced. Under a certain condition, the newly proposed transformation moves the input from the interior of the domain to the boundary. This enables us to cancel the residual term that causes the open-loop instability by using the input. Furthermore, this transformation is continuously invertible. Therefore, a stabilizing state feedback for the original PDE is derived through the inverse transformation. The results are demonstrated by a numerical simulation.