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

Jukka-Pekka Humaloja, Nikolaos Bekiaris-Liberis

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.

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

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.
Paper Structure (20 sections, 8 theorems, 71 equations, 7 figures)

This paper contains 20 sections, 8 theorems, 71 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature
  4. Contributions
  5. Organization
  6. Notation
  7. Stabilization of Large-Scale Systems of Linear Hyperbolic PDEs via Exact Backstepping Kernels
  8. Stabilization of a Continuum of Linear Hyperbolic PDEs via Continuum Backstepping
  9. Stabilization of the Finite Large-Scale System via Continuum Approximation of Exact Kernels
  10. Statement of the Main Result
  11. Proof of Theorem \ref{['thm:stab']}
  12. Proof of Theorem \ref{['thm:stab']} Using a Lyapunov Functional
  13. Numerical Example
  14. Stabilization via Approximate Kernels
  15. Comparison with Exact Control Kernels
  16. ...and 5 more sections

Key Result

Theorem IV.1

Consider an $n+1$ system eq:n+1, eq:nbcuy with parameters $\lambda_i,\mu,W_i,\theta_i,\sigma_{i,j},q$ for $i,j=1,2,\ldots,n$ satisfying Assumption ass:n+1. Let the parameters $\lambda,\mu, W, \theta, \sigma, q$ satisfy Assumption ass:inf and relations eq:afn. Then, if $n$ is sufficiently large, the where $\left(\widetilde{k}^i\right)_{i=1}^{n+1}$ are given in eq:akn, with $\left(k,\bar{k}\right)$

Figures (7)

  • Figure 1: Schematic view of the continuum PDE system \ref{['eq:inf']}, \ref{['eq:cbcuy']}. Boundary terms are denoted in magenta and in-domain terms are denoted in blue.
  • Figure 2: The controls $U(t)$ based on the approximate control law \ref{['eq:aUn']} when $n=2,\ldots,6$.
  • Figure 3: The solution component $u^n(t,x)$ when $n=2,\ldots,5$.
  • Figure 4: The controls $U(t)$ based on the approximate control law \ref{['eq:aUn']} when $n=6,10,15,20$.
  • Figure 5: The solution component $u^n(t,x)$ when $n=6,10,15,20$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Remark II.2
  • Theorem IV.1
  • Lemma IV.2
  • proof
  • Lemma IV.3
  • proof
  • Remark IV.4
  • Lemma IV.5
  • proof
  • Remark IV.6
  • ...and 9 more