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

Core Concepts

Stabilization of a class of large-scale systems of linear hyperbolic PDEs can be achieved by employing a backstepping-based control law constructed for stabilization of a continuum version of the PDE system, rather than the original large-scale system.

Abstract

The key idea of the presented approach is to approximate the exact backstepping kernels, derived for stabilization of a large-scale system of n+1 linear hyperbolic PDEs, using the backstepping kernels constructed for stabilization of a continuum version of the original system.
The proof consists of three main steps:

Constructing a sequence of backstepping kernels that match the exact kernels in a piecewise constant manner with respect to an ensemble variable, while satisfying the continuum kernel equations.
Showing that this sequence of approximate kernels converges to the continuum backstepping kernel as n goes to infinity.
Establishing stability of the closed-loop system employing the approximate control law, by treating the difference between the exact and approximate control gains as a perturbation that becomes arbitrarily small as n increases.

The authors also formally establish the connection between the solutions to the large-scale system and its continuum counterpart. This approach can be useful for designing computationally tractable, stabilizing backstepping-based control laws for large-scale PDE systems.

Stats

The authors present a numerical example that reveals the computational complexity of the stabilizing backstepping kernels may not scale with the number of components of the PDE state, when the kernels are constructed based on the continuum version, in contrast to the case where they are constructed based on the original, large-scale system.

Quotes

"Stabilization of a class of large-scale systems of linear hyperbolic PDEs 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."
"The key idea of our approach is to construct approximate backstepping kernels for stabilization of the large-scale (nevertheless, with a finite number of components) system relying on the continuum backstepping kernels developed in [27] for a continuum version of the original, large-scale system."

Key Insights Distilled From

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

by Jukka-Pekka ... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19455.pdf

Deeper Inquiries

How can the presented approach be extended to other classes of large-scale PDE systems beyond the considered class of linear hyperbolic PDEs

The presented approach can be extended to other classes of large-scale PDE systems by adapting the continuum approximation technique to the specific characteristics of the new system. For different types of PDEs, such as parabolic or elliptic PDEs, the continuum approximation may involve different transformations and considerations based on the underlying dynamics and boundary conditions of the system. By identifying the key parameters and properties of the new PDE system, a similar methodology can be applied to construct approximate backstepping kernels for stabilization. This extension would require a thorough analysis of the system's behavior, stability properties, and the continuum limit to ensure the effectiveness of the approach.

What are the potential limitations or drawbacks of the continuum approximation approach compared to other methods for computing stabilizing backstepping-based control laws for large-scale PDE systems

One potential limitation of the continuum approximation approach compared to other methods for computing stabilizing backstepping-based control laws for large-scale PDE systems is the accuracy of the approximation. While the continuum approximation provides a computationally tractable way to stabilize large-scale systems, there may be inherent errors introduced by the simplification of the system into a continuum form. These errors could impact the stability and performance of the control laws, especially in systems where precise control is crucial. Additionally, the complexity of deriving the continuum approximation and ensuring its validity for a wide range of PDE systems could be a drawback compared to more direct methods tailored to specific system dynamics.

Can the connection between the solutions of the large-scale system and its continuum counterpart be further exploited to develop efficient numerical schemes for computing the stabilizing control laws

The connection between the solutions of the large-scale system and its continuum counterpart can be further exploited to develop efficient numerical schemes for computing the stabilizing control laws by leveraging the concept of convergence. By understanding how the solutions of the large-scale system converge to the solutions of the continuum system as the number of components increases, numerical algorithms can be designed to approximate the stabilizing control laws with increasing accuracy. This convergence property can guide the development of adaptive numerical methods that adjust the level of approximation based on the system size, leading to more efficient and accurate computations of the control laws. Additionally, the continuum connection can inform the design of optimization algorithms that exploit the relationship between the large-scale and continuum systems to streamline the computation of stabilizing control laws.