Input Delay Compensation in Prescribed-Time of Boundary-Actuated Reaction-Diffusion PDEs

We consider the problem of prescribed-time stabilization of a reaction-diffusion PDE having constant input delay at the boundary. We rely on recent advances in time-varying backstepping which, for the delay-free case, allows one to shift the parabolic operator's eigenvalues increasingly to the left as time advances. For systems with input delay, this time-varying damping approach generates a hyperbolic delay equation cascaded into a time-varying parabolic one, where a careful equalization between time and delay time is required to resolve the differences in PDE classes. The resulting time-varying control gains simultaneously exhibit diffusive (parabolic) behavior with respect to delay time and advective (hyperbolic) behavior with respect to time. We present a bounded, delay-compensated time-varying control law which stabilizes the plant within a time which can be prescribed independently of the system's initial conditions.