Controller design for distributed parameter systems with time delays in the boundary feedbacks via the backstepping method

For a distributed parameter system with an input delay in the boundary, a feedback control law is presented by means of the backstepping method. The square integrability of input signal is verified based on the target system. Then, the boundedness and invertibility of the corresponding backstepping transformation are proved under the regularity of system and the admissibility of feedback operator. Thus, the resulting closed-loop system is shown to be exponentially stable. Finally, as an application, a numerical simulation of a one-dimensional Schrödinger equation with a delay input is carried out, and the simulation results demonstrate the effectiveness of the suggested control law.