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On Computation of Approximate Solutions to Large-Scale Backstepping Kernel Equations via Continuum Approximation

Jukka-Pekka Humaloja, Nikolaos Bekiaris-Liberis

TL;DR

This work develops computational tools to construct backstepping-based stabilizing kernels for continua of linear hyperbolic PDEs and, by extension, large-scale PDE systems. It introduces a triple power-series method for the continuum kernel PDEs with rigorous convergence guarantees under analytic parameters, plus an order-reduction option in the ensemble variable to reduce complexity. It also identifies a special class of parameter configurations that admit closed-form continuum solutions, enabling exact kernel gains in certain cases. The paper validates the approaches with numerical experiments, showing that continuum kernels yield stabilizing feedback for large-scale systems with computational costs that do not grow with the number of state components, and demonstrates substantial accuracy and efficiency advantages over solving the full large-scale kernel equations.

Abstract

We provide two methods for computation of continuum backstepping kernels that arise in control of continua (ensembles) of linear hyperbolic PDEs and which can approximate backstepping kernels arising in control of a large-scale, PDE system counterpart (with computational complexity that does not grow with the number of state components of the large-scale system). In the first method, we provide explicit formulae for the solution to the continuum kernels PDEs, employing a (triple) power series representation of the continuum kernel and establishing its convergence properties. In this case, we also provide means for reducing computational complexity by properly truncating the power series (in the powers of the ensemble variable). In the second method, we identify a class of systems for which the solution to the continuum (and hence, also an approximate solution to the respective large-scale) kernel equations can be constructed in closed form. We also present numerical examples to illustrate computational efficiency/accuracy of the approaches, as well as to validate the stabilization properties of the approximate control kernels, constructed based on the continuum.

On Computation of Approximate Solutions to Large-Scale Backstepping Kernel Equations via Continuum Approximation

TL;DR

This work develops computational tools to construct backstepping-based stabilizing kernels for continua of linear hyperbolic PDEs and, by extension, large-scale PDE systems. It introduces a triple power-series method for the continuum kernel PDEs with rigorous convergence guarantees under analytic parameters, plus an order-reduction option in the ensemble variable to reduce complexity. It also identifies a special class of parameter configurations that admit closed-form continuum solutions, enabling exact kernel gains in certain cases. The paper validates the approaches with numerical experiments, showing that continuum kernels yield stabilizing feedback for large-scale systems with computational costs that do not grow with the number of state components, and demonstrates substantial accuracy and efficiency advantages over solving the full large-scale kernel equations.

Abstract

We provide two methods for computation of continuum backstepping kernels that arise in control of continua (ensembles) of linear hyperbolic PDEs and which can approximate backstepping kernels arising in control of a large-scale, PDE system counterpart (with computational complexity that does not grow with the number of state components of the large-scale system). In the first method, we provide explicit formulae for the solution to the continuum kernels PDEs, employing a (triple) power series representation of the continuum kernel and establishing its convergence properties. In this case, we also provide means for reducing computational complexity by properly truncating the power series (in the powers of the ensemble variable). In the second method, we identify a class of systems for which the solution to the continuum (and hence, also an approximate solution to the respective large-scale) kernel equations can be constructed in closed form. We also present numerical examples to illustrate computational efficiency/accuracy of the approaches, as well as to validate the stabilization properties of the approximate control kernels, constructed based on the continuum.
Paper Structure (16 sections, 3 theorems, 50 equations, 5 figures, 7 tables, 1 algorithm)

This paper contains 16 sections, 3 theorems, 50 equations, 5 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Large-Scale Kernel Equations and Their Continuum Approximation
  3. Power Series Approach to Solving Continuum Kernel Equations
  4. Convergence of Power Series Representation
  5. Computation of Approximate Solution with Truncated Power Series
  6. Computational Complexity of the Power Series-Based Kernels Approximation
  7. On Order Reduction with Respect to $y$
  8. On Explicit Solution to Continuum Kernel Equations
  9. Numerical Experiments: Power Series-Based Computation for Example 1
  10. Application to Stabilization of a Large-Scale System
  11. Parameters and Their Continuum Approximation
  12. Computation of Continuum Kernels with Power Series
  13. Stabilization Using Power Series-Based Approximate Continuum Kernels
  14. Conclusions and Future Work
  15. Proof of Proposition \ref{['prop:esol']}
  16. ...and 1 more sections

Key Result

Theorem 1

If the parameters of eq:kc, eq:kcbc are analytic on polydisks with radii larger than one, so that they can be represented as the series in eq:ppc, and $|\lambda(x,y)| > 0$ for all $x,y \in [0,1]$, $|\mu(x)| > 0$ for all $x \in [0,1]$, then the series defined in eq:kps, eq:kbps converge. That is, th

Figures (5)

  • Figure 1: Polynomial fits of order $M=2,\ldots,6$ to the $q_i$ data.
  • Figure 2: The control gain $k(1,\xi,y)$ approximated by full-order (in $y$) power series \ref{['eq:kpsa']} for $N = 6,10,15,20,25,30$.
  • Figure 3: The control gain $k(1,\xi,y)$ approximated by reduced-order (in $y$) power series \ref{['eq:kpsay']} for $N = 6,10,15,20,25,30$ and $N_y=2$.
  • Figure 4: The controls obtained with approximate continuum kernels computed with full-order (in $y$) power series \ref{['eq:solpsa']} of order $N = 6,10,15,20,25$ and by solving \ref{['eq:kn']}, \ref{['eq:knbc']} for the $n+1$ kernels.
  • Figure 5: The controls obtained with approximate continuum kernels computed with reduced-order (in $y$) power series \ref{['eq:kpsay']} of order $N = 6,10,15,20,25$ and $N_y=2$ and by solving \ref{['eq:kn']}, \ref{['eq:knbc']} for the $n+1$ kernels.

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4