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

2017/10/10 by Pedro Ascencio, Alessandro Astolfi, Ascencio, Pedro +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Electrical engineering #Stability and Controllability of Differential Equations #Systems and Control (eess.SY) #electronic engineering #information engineering

paper · pdf · doi:10.48550/arxiv.1710.03723

openalex publication_date 2017/10/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Semidefinite programming relaxations for semialgebraic problems
• Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes

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