  This paper considers the backstepping design of state feedback controllers
for coupled linear parabolic partial integro-differential equations (PIDEs) of
Volterra-type with distinct diffusion coefficients, spatially-varying
parameters and mixed boundary conditions. The corresponding target system is a
cascade of parabolic PDEs with local couplings allowing a direct specification
of the closed-loop stability margin. The determination of the state feedback
controller leads to kernel equations, which are a system of coupled linear
second-order hyperbolic PIDEs with spatially-varying coefficients and rather
unusual boundary conditions. By extending the method of successive
approximations for the scalar case to the considered system class, the
well-posedness of these kernel equations is verified by providing a
constructive solution procedure. This results in a systematic method for the
backstepping control of coupled parabolic PIDEs as well as PDEs. The
applicability of the new backstepping design method is confirmed by the
stabilization of two coupled parabolic PIDEs with Dirichlet/Robin unactuated
boundaries and a coupled Neumann/Dirichlet actuation.
