  This paper studies the robustness of a PDE backstepping delay-compensated
boundary controller for a reaction-diffusion partial differential equation
(PDE) with respect to a nominal delay subject to stochastic error disturbance.
The stabilization problem under consideration involves random perturbations
modeled by a finite-state Markov process that further obstruct the actuation
path at the controlled boundary of the infinite-dimension plant. This scenario
is useful to describe several actuation failure modes in process control.
Employing the recently introduced infinite-dimensional representation of the
state of an actuator subject to stochastic input delay for ODEs (Ordinary
Differential Equations), we convert the stochastic input delay into $r+1$
unidirectional advection PDEs, where $r$ corresponds to the number of jump
states. Our stability analysis assumes full-state measurement of the spatially
distributed plant's state and relies on a hyperbolic-parabolic PDE cascade
representation of the plant plus actuator dynamics. Integrating the plant and
the nominal stabilizing boundary control action, all while considering
probabilistic delay disturbances, we establish the proof of mean-square
exponential stability as well as the well-posedness of the closed-loop system
when random phenomena weaken the nominal actuator compensating effect. Our
proof is based on the Lyapunov method, the theory of infinitesimal operator for
stability, and $C_0$-semigroup theory for well-posedness. Our stability result
refers to the $L^2$-norm of the plant state and the $H^2$-norm of the actuator
state...
