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Extension theory via boundary triplets for infinite-dimensional implicit port-Hamiltonian systems

Hannes Gernandt, Friedrich Philipp, Till Preuster, Manuel Schaller

TL;DR

This work develops a boundary-triplet framework for range representations of linear relations, enabling direct characterization of self-adjoint, skew-adjoint, and maximally dissipative intermediate extensions via boundary data defined on $\operatorname{dom}[\mathcal{P}\mathcal{S}]$. It establishes a coercivity-based main result that ties the extendibility of $\operatorname{ran}[\mathcal{P}\mathcal{S}]$ to boundary relations $\Theta$ in the boundary space $\mathcal{G}\times\mathcal{G}$, and links these extensions with Lagrangian subspaces central to port-Hamiltonian geometry. The theory is then specialized to implicit port-Hamiltonian systems on 1D domains with matrix differential operators, yielding explicit boundary-triplet constructions from coefficient matrices and trace data, including the Dzektser equation, the biharmonic wave equation, and an elastic rod with non-local elasticity. These results provide a rigorous, operator-theoretic route to well-posedness, boundary control/observation, and energy-based modeling for generalized pH PDEs in infinite dimensions.

Abstract

The solution of constrained linear partial-differential equations can be described via parametric representations of linear relations. To study these representations, we provide a novel definition of boundary triplets for linear relations in range representations where the associated boundary map is defined on the domain of the parameterizing operators rather than the relation itself. This allows us to characterize all boundary conditions such that the underlying dynamics is represented by a self-adjoint, skew-adjoint or maximally dissipative relation. The theoretical results are applied to a class of implicit port-Hamiltonian systems on one-dimensional spatial domains. More precisely, we explicitly construct a boundary triplet which solely depends on the coefficient matrices of the involved matrix differential operators and we derive the associated Lagrangian subspace. We exemplify our approach by means of the Dzektser equation, the biharmonic wave equation, and an elastic rod with non-local elasticity condition.

Extension theory via boundary triplets for infinite-dimensional implicit port-Hamiltonian systems

TL;DR

This work develops a boundary-triplet framework for range representations of linear relations, enabling direct characterization of self-adjoint, skew-adjoint, and maximally dissipative intermediate extensions via boundary data defined on . It establishes a coercivity-based main result that ties the extendibility of to boundary relations in the boundary space , and links these extensions with Lagrangian subspaces central to port-Hamiltonian geometry. The theory is then specialized to implicit port-Hamiltonian systems on 1D domains with matrix differential operators, yielding explicit boundary-triplet constructions from coefficient matrices and trace data, including the Dzektser equation, the biharmonic wave equation, and an elastic rod with non-local elasticity. These results provide a rigorous, operator-theoretic route to well-posedness, boundary control/observation, and energy-based modeling for generalized pH PDEs in infinite dimensions.

Abstract

The solution of constrained linear partial-differential equations can be described via parametric representations of linear relations. To study these representations, we provide a novel definition of boundary triplets for linear relations in range representations where the associated boundary map is defined on the domain of the parameterizing operators rather than the relation itself. This allows us to characterize all boundary conditions such that the underlying dynamics is represented by a self-adjoint, skew-adjoint or maximally dissipative relation. The theoretical results are applied to a class of implicit port-Hamiltonian systems on one-dimensional spatial domains. More precisely, we explicitly construct a boundary triplet which solely depends on the coefficient matrices of the involved matrix differential operators and we derive the associated Lagrangian subspace. We exemplify our approach by means of the Dzektser equation, the biharmonic wave equation, and an elastic rod with non-local elasticity condition.

Paper Structure

This paper contains 16 sections, 28 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Extension theory for parametric representations of linear relations
  3. Main result: Self-adjointness and dissipativity via boundary triplets
  4. Proof of Theorem \ref{['thm:self-adj_extension']}
  5. An Example: Dzektser equation
  6. Correspondence of boundary triplets and Lagrangian subspaces
  7. Boundary triplets for skew-symmetric operators
  8. Application to implicit port-Hamiltonian systems on one-dimensional spatial domains
  9. Basic properties of matrix differential operators
  10. Construction of boundary triplets for matrix differential operators
  11. Characterization of skew-adjoint intermediate extensions of matrix differential operators in one spatial dimension
  12. Application to generalized implicit port-Hamiltonian systems
  13. Dzektser equation revisited
  14. Biharmonic wave equation as a port-Hamiltonian system
  15. Elastic rod with non-local elasticity condition
  16. ...and 1 more sections

Key Result

Theorem 2.3

Let Assumption ass:ass1 hold, let $(\mathcal{G}, \Gamma_0, \Gamma_1)$ be a boundary triplet for $\operatorname{ran} \left[\right]$, let $\Theta \subset \mathcal{G} \times \mathcal{G}$, and define an operator $A_\Theta: \mathcal{X} \supset \operatorname{dom} A_\Theta \to \mathcal{X} \times \mathcal{X Then, $H_\Theta = \operatorname{ran} A_\Theta \subset \mathcal{X} \times \mathcal{X}$ is self-adjoi

Theorems & Definitions (61)

  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • proof
  • Proposition 2.8
  • ...and 51 more