A Convex Optimization Approach for Backstepping PDE Design: Volterra and Fredholm Operators

TL;DR

This paper introduces a convex optimization method for designing backstepping controllers for boundary linear PDEs, utilizing polynomial approximations of Volterra and Fredholm operators and SOS decomposition to ensure system stability.

Contribution

It presents a novel convex optimization framework for PDE backstepping design that leverages polynomial kernel approximations and semidefinite programming.

Findings

Successfully stabilizes certain parabolic and hyperbolic PDEs.

Demonstrates the effectiveness of SOS-based kernel optimization.

Highlights limitations in approximation accuracy for complex PDEs.

Abstract

Backstepping design for boundary linear 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 non-strict 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(SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L2-norm. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs.