Backstepping Control of Coupled Linear Parabolic PIDEs with Spatially-Varying Coefficients

TL;DR

This paper develops a systematic backstepping control method for coupled linear parabolic PIDEs with spatially-varying coefficients, enabling stabilization of complex systems with mixed boundary conditions.

Contribution

It extends the backstepping approach to coupled parabolic PIDEs with spatially-varying parameters, providing a constructive solution for kernel equations and stability analysis.

Findings

Successfully stabilizes coupled parabolic PIDEs with mixed boundary conditions.

Provides a constructive method for solving kernel equations in backstepping design.

Demonstrates applicability through stabilization examples.

Abstract

This paper considers the backstepping design of state feedback controllers for coupled linear parabolic partial integro-differential equations (PIDEs) of Volterra-type with distinct diffusion coefficients, spatially-varying parameters and mixed boundary conditions. The corresponding target system is a cascade of parabolic PDEs with local couplings allowing a direct specification of the closed-loop stability margin. The determination of the state feedback controller leads to kernel equations, which are a system of coupled linear second-order hyperbolic PIDEs with spatially-varying coefficients and rather unusual boundary conditions. By extending the method of successive approximations for the scalar case to the considered system class, the well-posedness of these kernel equations is verified by providing a constructive solution procedure. This results in a systematic method for the backstepping control of coupled parabolic PIDEs as well as PDEs. The applicability of the new backstepping design method is confirmed by the stabilization of two coupled parabolic PIDEs with Dirichlet/Robin unactuated boundaries and a coupled Neumann/Dirichlet actuation.