  This paper addresses the problem of input-to-state stabilization for a class
of parabolic equations with time-varying coefficients, as well as Dirichlet and
Robin boundary disturbances. By using time-invariant kernel functions, which
can reduce the complexity in control design and implementation, an
observer-based output feedback controller is designed via backstepping. By
using the generalized Lyapunov method, which can be used to handle Dirichlet
boundary terms, the input-to-state stability of the closed-loop system under
output feedback control, as well as the state estimation error system, is
established in the spatial $L^\infty$-norm. Numerical simulations are conducted
to confirm the theoretical results and to illustrate the effectiveness of the
proposed control scheme.
