The interconnection of port-Hamiltonian systems is usually understood as structure-preserving due to the geometric fact that the composition of the underlying Dirac structures produces again a Dirac structure. Pointwise, this is doubtless true, but when comparing dynamical systems, their trajectories are usually compared, and trajectories usually obey some regularity assumptions. It is shown that both classical and weak solutions do not necessarily exhibit a one-to-one correspondence between solutions of the interconnected port-Hamiltonian system, and the dynamical system realising the interconnection. In the mostly studied cases of constant Dirac structures or port-Hamiltonian ODE systems, both systems are behaviourally equivalent; for non-constant Dirac structures, a sufficient condition is given.

The energy-based modelling framework proposed by Altmann and Schulze is demonstrated to have two distinct representations as port-Hamiltonian systems. The ambiguity is expressed in the role of the algebraic variable, and is distinct from the usual ambiguity in the geometric representation of coordinate representations of classical port-Hamiltonian systems. A straightforward extension to systems with feedthrough is given.

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While the cost functional of minimal energy supply is intrinsic to the port-Hamiltonian structure, the necessary conditions of optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason is the linear dependence on the control, which makes the problem of determining the optimal control as a function of the state and the adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity of the (differential-algebraic) optimality system by using the interplay of the cost functional and the dynamics. In case of the optimality DAE being characterized by a regular matrix pencil, we fully determine the control on the singular arc. In case of singular matrix pencils of the optimality system, we propose an approach to compute rank-minimal quadratic perturbations of the objective such that the optimal control problem becomes regular. We illustrate the applicability of our results by a general second-order mechanical system and a discretized boundary-controlled heat equation.

We study the geometric structure of the drift dynamics of Irreversible port-Hamiltonian systems. This drift dynamics is defined with respect to a product of Poisson brackets, reflecting the interconnection structure and the constitutive relations of the irreversible phenomena occuring in the system. We characterise this product of Poisson brackets using a covariant 4-tensor and an associated function. We derive various conditions for which this 4-tensor and the associated function may be reduced to a product of almost Poisson brackets.

The definition of conservative-irreversible functions is extended to smooth manifolds. Local representation of these functions is studied and reveals that they can not necessarily be given as the weighted product of almost Poisson brackets, but as the sum of such. The biquadratic functions induced by conservative-irreversible functions are studied and demonstrate a possibility for an algebraic framework on arbitrary and in particular complex algebras.

This note presents a notion of generalised metriplectic systems which includes not necessarily holonomic constraints using methods from port-Hamiltonian systems theory. Metriplectic systems can be written as particular dissipative Hamiltonian systems with the exergy as Hamiltonian. Within the generalised dissipative Hamiltonian systems, a generalisation of metriplectic systems is identified.

We study port-control for metriplectic systems. Using the well-known representation of metriplectic systems as dissipative Hamiltonian systems with the exergy as Hamiltonian function, the corresponding port-controlled Hamiltonian systems are considered as exergy-controlled metriplectic systems. The applicability of the machinery of port-Hamiltonian systems theory, in particular interconnection of exergy controlled metriplectic systems, is studied. Copyright (c) 2025 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

We study the geometric structure of port-Hamiltonian systems. Starting with the intuitive understanding that port-Hamiltonian systems are “in between” certain closed Hamiltonian systems, the geometric structure of port-Hamiltonian systems must be “in between” the geometric structures of the latter systems. These are Courant algebroids; and hence the geometric structures should be related by Courant algebroid morphisms. Using this idea, we propose a definition of an intrinsic geometric structure and show that it is unique, if it exists.

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x ), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x ) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156 ) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x ) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156 ) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

We give insight in the structure of port-Hamiltonian systems as control systems in between two closed Hamiltonian systems. Using the language of category theory, we identify systems with their behavioural representation and view a port-control structure with desired structural properties on a given closed system as an extension of this system which itself may be embedded in a “larger” closed system. The latter system describes the nature of the ports (e.g. Hamiltonian, metriplectic etc.). This point of view allows us to describe meaningful port-control structures for a large family of systems, which is illustrated with Hamiltonian and metriplectic systems.

The present work is a successor of [Ilchmann, Kirchhoff 2022] on generic controllability and of [Ilchmann, Kirchhoff 2023] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive new results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

Irreversible Port Hamiltonian Systems are a deviation from Port Hamiltonian Systems which embeds the definition of the irreversible phenomena taking place in the system. They are defined with respect to a quasi-Poisson bracket which ensures the positiveness of the entropy generation and is expressed in terms of the total entropy of the system. The port maps, however, associated with the conjugated port variables, were poorly justified and lacked any physical characterization. In this paper, we suggest a novel definition of the port maps which allows to recover not only the energy balance equation (when the Hamiltonian equals the total energy of the system) but also a entropy balance equation including the irreversible entropy creation term at the interface (the port) of the system in addition to the entropy creation term due to internal irreversible phenomena.

The present note is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and stabilizability of linear differential-algebraic equations. We resolve the drawback that genericity is considered in the unrestricted set of system matrices (E,A,B)∈ℝ^ℓ××ℝ^ℓ× n×ℝ^ℓ× m , while for relative genericity we allow the restricted set Σ _ℓ ,n,m^≤ r := {(E,A,B)∈ℝ^ℓ× n×ℝ^ℓ× n×ℝ^ℓ× m | rk _ℝ E ≤ r} , where r∈ℕ . Our main results are characterizations of generic controllability and generic stabilizability in Σ _ℓ ,n,m^≤ r in terms of the numbers ℓ , n, m, r .

The new concept of relative generic subsets is introduced. It is shown that the set of controllable linear finite-dimensional port-Hamiltonian systems is a relative generic subset of the set of all linear finite-dimensional port-Hamiltonian systems. This implies that a random, continuously distributed port-Hamiltonian system is almost surely controllable.

We investigate genericity of various controllability and stabilizability concepts of linear, time-invariant differential-algebraic systems. Based on well-known algebraic characterizations of these concepts (see the survey article by Berger and Reis (in: Ilchmann A, Reis T (eds) Surveys in differential-algebraic equations I, Differential-Algebraic Equations Forum, Springer, Berlin, pp 1–61. https://doi.org/10.1007/978-3-642-34928-7_1 )), we use tools from algebraic geometry to characterize genericity of controllability and stabilizability in terms of matrix formats.