Backstepping PDE Design: A Convex Optimization Approach

Abstract

Backstepping design for boundary linear partial differential equation (PDE) is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to nonstrict feedback structures. Based on the compactness of the Volterra- and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using sum-of-squares decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the $\mathcal{L}^2$ -norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs.