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A Mixed-FEM approximation with uniform conservation of the exponential stability for a class of anisotropic port-Hamiltonian system and its application to LQ control

Luis A. Mora, Kirsten Morris

TL;DR

The paper develops a mixed finite-element discretization for a class of anisotropic boundary-damped port-Hamiltonian systems that uniformly preserves exponential stability, yielding a mesh-size independent decay rate bound via a multiplier-based Lyapunov framework. By formulating the discretization in co-energy variables and constructing structure-preserving global matrices, the authors demonstrate a discrete port-Hamiltonian model with provable energy decay that remains robust as the mesh is refined. The piezoelectric beam serves as a concrete example, and the MFEM approach is shown to support reliable infinite-horizon LQ control, with the discrete controller converging to the continuous controller as the discretization is refined, outperforming standard FE methods. Key theoretical contributions include explicit discrete operator identities ensuring port-Hamiltonian structure, concrete sufficiency conditions (D1/D2) for uniform stability, and detailed lemmas on spectral properties and inner-product computations that facilitate energy-based analyses. Practically, this yields stable, scalable numerical schemes for controlled distributed-parameter systems with anisotropy, enabling consistent controller synthesis across discretization orders.

Abstract

In this manuscript, we present a mixed finite element discretization for a class of boundary-damped anisotropic port-Hamiltonian systems. Using a multiplier method, we demonstrate that the resulting approximation model uniformly preserves the exponential stability of the uncontrolled system, establishing a lower bound for the exponential decay rate that is independent of the mesh size. This property is illustrated through the spatial discretization of a piezoelectric beam. Furthermore, we show how the uniform preservation of exponential stability by the proposed model aids in the convergence of controllers derived from an infinite-time linear quadratic control design, in comparison to models obtained from the standard finite-element method.

A Mixed-FEM approximation with uniform conservation of the exponential stability for a class of anisotropic port-Hamiltonian system and its application to LQ control

TL;DR

The paper develops a mixed finite-element discretization for a class of anisotropic boundary-damped port-Hamiltonian systems that uniformly preserves exponential stability, yielding a mesh-size independent decay rate bound via a multiplier-based Lyapunov framework. By formulating the discretization in co-energy variables and constructing structure-preserving global matrices, the authors demonstrate a discrete port-Hamiltonian model with provable energy decay that remains robust as the mesh is refined. The piezoelectric beam serves as a concrete example, and the MFEM approach is shown to support reliable infinite-horizon LQ control, with the discrete controller converging to the continuous controller as the discretization is refined, outperforming standard FE methods. Key theoretical contributions include explicit discrete operator identities ensuring port-Hamiltonian structure, concrete sufficiency conditions (D1/D2) for uniform stability, and detailed lemmas on spectral properties and inner-product computations that facilitate energy-based analyses. Practically, this yields stable, scalable numerical schemes for controlled distributed-parameter systems with anisotropy, enabling consistent controller synthesis across discretization orders.

Abstract

In this manuscript, we present a mixed finite element discretization for a class of boundary-damped anisotropic port-Hamiltonian systems. Using a multiplier method, we demonstrate that the resulting approximation model uniformly preserves the exponential stability of the uncontrolled system, establishing a lower bound for the exponential decay rate that is independent of the mesh size. This property is illustrated through the spatial discretization of a piezoelectric beam. Furthermore, we show how the uniform preservation of exponential stability by the proposed model aids in the convergence of controllers derived from an infinite-time linear quadratic control design, in comparison to models obtained from the standard finite-element method.

Paper Structure

This paper contains 9 sections, 9 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Distributed-parameters systems anisotropic port-Hamiltonian systems.
  3. Mixed-FEM numerical model.
  4. Uniform exponential stability
  5. Physical parameters: required properties.
  6. Example: Piezoelectric beam
  7. Application to linear quadratic control
  8. Proof of Lemma \ref{['lem:4']}.
  9. Proof of Lemma \ref{['lem:inner']}.

Key Result

Theorem 1

Mora2023 Consider $\boldsymbol{u}(t)=0$. If Assumption assump:1 holds, then, defining and with $\mu_{\Psi}= \max eig(\Psi)$, $\Psi=^\top \Theta^{-1}(x_r) $, $\eta_K=\min eig(K)$, $\eta_\Theta=\min_{x\in[x_l,x_r]} eig(\Theta(x))$, and $\mu_{P_1}=\sqrt{\max eig(P_1^{-2})},$ system eq:PHS_2--eq:BC_a is exponentially stable with a decay rate bounded by

Figures (4)

  • Figure 1: Mixed finite elements basis and test functions
  • Figure 2: Piezoelectric beam
  • Figure 3: Maximum real part of the eigenvalues, $\sigma_{max}$, for the approximated model in open- and closed-loop, with $\rho_0=\tau_0=1$ and $\kappa_1=0.05$. (a): using the standard finite-element methods. (b): using the MFEM scheme proposed.
  • Figure 4: Behavior of $K_d=h[\mathbf{k}_p \quad \mathbf{k}_q]$ for different values of $N$, considering the FE and MFEM model.

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • ...and 11 more