A Convex Optimization Approach for Backstepping PDE Design: Volterra and Fredholm Operators
For control theorists working on PDE backstepping, this provides a convex optimization framework that handles non-strict feedback structures, though it is an incremental extension of existing SOS methods.
This paper formulates backstepping design for boundary linear PDEs as a convex optimization problem, using polynomial approximation of Volterra and Fredholm operators and Sum-of-Squares decomposition to solve Kernel-PDEs with sufficient precision for L2-norm stability.
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.