  This paper considers the backstepping state feedback control of coupled
linear parabolic PDEs with spatially varying coefficients and bilateral
actuation. By making use of the folding technique, a system representation with
unilateral actuation is obtained, allowing to apply the standard backstepping
transformation. To ensure the regularity of the solution, the folded system is
subject to unusual folding boundary conditions, which lead to additional
boundary couplings between the PDEs. Therefore, the solution of the
corresponding kernel equations determining the transformations is a very
challenging problem. A systematic approach to derive the corresponding integral
equations is proposed, allowing to solve them with the method of successive
approximations. By making use of a Volterra and a Volterra-Fredholm
transformation, the closed-loop system is mapped into a cascade of stable
parabolic systems. This allows a simple proof of exponential stability in the
$L_2$-norm with the decay rate as design parameter. The bilateral state
feedback stabilization of an unstable system of two coupled parabolic PDEs and
the comparison to the application of an unilateral controller demonstrates the
results of the paper.
