Stabilization of a Class of Large-Scale Systems of Linear Hyperbolic PDEs via Continuum Approximation of Exact Backstepping Kernels

TL;DR

This paper demonstrates that stabilization of large-scale linear hyperbolic PDE systems can be effectively achieved by approximating the exact backstepping kernels with those derived from a continuum model, simplifying computation.

Contribution

It introduces a novel approach to approximate backstepping kernels for large-scale PDEs using a continuum limit, preserving stability and reducing computational complexity.

Findings

Backstepping kernels for large systems can be approximated by continuum kernels.

The approximation preserves exponential stability of the system.

Computational complexity does not scale with the number of PDE components.

Abstract

We establish that stabilization of a class of linear, hyperbolic partial differential equations (PDEs) with a large (nevertheless finite) number of components, can be achieved via employment of a backstepping-based control law, which is constructed for stabilization of a continuum version (i.e., as the number of components tends to infinity) of the PDE system. This is achieved by proving that the exact backstepping kernels, constructed for stabilization of the large-scale system, can be approximated (in certain sense such that exponential stability is preserved) by the backstepping kernels constructed for stabilization of a continuum version (essentially an infinite ensemble) of the original PDE system. The proof relies on construction of a convergent sequence of backstepping kernels that is defined such that each kernel matches the exact backstepping kernels (derived based on the original, large-scale system), in a piecewise constant manner with respect to an ensemble variable; while showing that they satisfy the continuum backstepping kernel equations. We present a numerical example that reveals that complexity of computation of stabilizing backstepping kernels may not scale with the number of components of the PDE state, when the kernels are constructed on the basis of the continuum version, in contrast to the case in which they are constructed on the basis of the original, large-scale system. In addition, we formally establish the connection between the solutions to the large-scale system and its continuum counterpart. Thus, this approach can be useful for design of computationally tractable, stabilizing backstepping-based control laws for large-scale PDE systems.