Bilateral backstepping control of coupled linear parabolic PDEs with spatially varying coefficients

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.