Stabilization of a coupled transport-diffusion system with boundary input

We consider the problem of stabilizing a coupled transport-diffusion system with boundary input. The system is described by two linear transport-diffusion equations and is not asymptotically stable. In order to stabilize the system with boundary input, sensor influence functions are assumed to be located at interior of the domain. First, we formulate the system as an evolution equation with unbounded output operators in a Hilbert space, using variable transformation. Next, we derive a reduced-order model with a finite-dimensional state variable for the infinite-dimensional system. Then, a stabilizing controller is constructed for the reduced-order model under an additional assumption. It is shown that the finite-dimensional controller together with a residual mode filter plays a role of a finite-dimensional stabilizing controller for the original infinite-dimensional system, if the order of the residual mode filter is chosen sufficiently large. Finally, the validity of the design method is demonstrated through a numerical simulation.