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Bernhard Maschke

Publications and source records attributed to Bernhard Maschke.

At least 19 recordsLinked to original sources
arXiv2025

Infinite-dimensional port-Hamiltonian systems with a stationary interface

We consider two systems of two conservation laws that are defined on complementary, one-dimensional spatial intervals and coupled by an interface as a single port-Hamiltonian system. In case of a fixed interface position, we characterize the boundary and interface conditions for which the associated port-Hamiltonian operator generates a contraction semigroup. Furthermore, we present sufficient conditions for the exponential stability of the generated $C_0$-semigroup. The results are illustrated by the example of two acoustic waveguides coupled by a membrane interface.

math.AP↗
arXiv2024

Existence of solutions to port-Hamiltonian systems: initial value problems and optimal control

We investigate the existence of solutions of reversible and irreversible port-Hamiltonian systems. To this end, we utilize the associated exergy, a function that is composed of the system's Hamiltonian and entropy, to prove global existence in time for bounded control functions. The results are then leveraged to prove existence of solutions of energy- and entropy-optimal control problems. Last, we explore model predictive control tailored to irreversible port-Hamiltonian systems by means of a numerical case study with a heat exchanger network.

math.OC↗
arXiv2024

Optimal control of port-Hamiltonian systems: energy, entropy, and exergy

We consider irreversible and coupled reversible-irreversible nonlinear port-Hamiltonian systems and the respective sets of thermodynamic equilibria. In particular, we are concerned with optimal state transitions and output stabilization on finite-time horizons. We analyze a class of optimal control problems, where the performance functional can be interpreted as a linear combination of energy supply, entropy generation, or exergy supply. Our results establish the integral turnpike property towards the set of thermodynamic equilibria providing a rigorous connection of optimal system trajectories to optimal steady states. Throughout the paper, we illustrate our findings by means of two examples: a network of heat exchangers and a gas-piston system.

math.OC↗
arXiv2024

Remarks on the geometric structure of port-Hamiltonian systems

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.

math-ph↗
arXiv2024

Port-Hamiltonian systems with energy and power ports

We extend the port-Hamiltonian framework defined with respect to a Lagrangian submanifold and a Dirac structure by augmenting the Lagrangian submanifold with the space of external variables. The new pair of conjugated variables is called energy port. We show that in the most general case, the extension describes constrained Hamiltonian systems whose Hamiltonian function depends on inputs.

math.OC↗
arXiv2024

Generating functions for irreversible Hamiltonian systems

The definition of conservative-irreversible functions is extended to smooth manifolds. The local representation of these functions is studied and reveals that not each conservative-irreversible function is given by the weighted product of almost Poisson brackets. The biquadratic functions given by conservative-irreversible functions are studied and reveal a possibility for an algebraic framework on arbitrary and in particular complex algebras.

math-ph↗
arXiv2024

Jet space extensions of infinite-dimensional Hamiltonian systems

We analyze infinite-dimensional Hamiltonian systems corresponding to partial differential equations on one-dimensional spatial domains formulated with formally skew-adjoint Hamiltonian operators and nonlinear Hamiltonian density. In various applications, the Hamiltonian density can depend on spatial derivatives of the state such that these systems can not straightforwardly be formulated as boundary port-Hamiltonian system using a Stokes-Dirac structure. In this work, we show that any Hamiltonian system of the above class can be reformulated as a Hamiltonian system on the jet space, in which the Hamiltonian density only depends on the extended state variable itself and not on its derivatives. Consequently, well-known geometric formulations with Stokes- Dirac structures are applicable. Additionally, we provide a similar result for dissipative systems. We illustrate the developed theory by means of the the Boussinesq equation, the dynamics of an elastic rod and the Allen-Cahn equation.

math.AP↗
arXiv2023

Linear Boundary Port-Hamiltonian Systems with Implicitly Defined Energy

In this paper we extend the previously introduced class of boundary port-Hamiltonian systems to boundary control systems where the variational derivative of the Hamiltonian functional is replaced by a pair of reciprocal differential operators. In physical systems modelling, these differential operators naturally represent the constitutive relations associated with the implicitly defined energy of the system and obey Maxwell's reciprocity conditions. On top of the boundary variables associated with the Stokes-Dirac structure, this leads to additional boundary port variables and to the new notion of a Stokes-Lagrange subspace. This extended class of boundary port-Hamiltonian systems is illustrated by a number of examples in the modelling of elastic rods with local and non-local elasticity relations. Finally it shown how a Hamiltonian functional on an extended state space can be associated with the Stokes-Lagrange subspace, and how this leads to an energy balance equation involving the boundary variables of the Stokes-Dirac structure as well as of the Stokes-Lagrange subspace.

math.OC↗
arXiv2023

A case study of port-Hamiltonian systems with a moving interface

We model two systems of two conservation laws defined on complementary spatial intervals and coupled by a moving interface as a single non-autonomous port-Hamiltonian system, and provide sufficient conditions for its Kato-stability. An example shows that these conditions are quite restrictive. The more general question under which conditions an evolution family is generated remains open.

math.AP↗
arXiv2023

On the Generating Functions of Irreversible port-Hamiltonian Systems

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 quasi-Poisson brackets, reflecting the interconnection structure and the constitutive relations of the irreversible phenomena occuring in the system. We characterize this product of quasi-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 quasi-Poisson brackets.

math.DS↗
arXiv2023

Port maps of Irreversible Port Hamiltonian Systems

Irreversible Port Hamiltonian Systems are departure of Port Hamiltonian Systems as they are generated not only by a Hamiltonian function but also by an entropy function and defined with respect to a quasi-Poisson bracket which embeds the definition of the irreversible phenomena taking place in the system. However the port map, consisting in the input map and the output map were poorly justified and lacked any physical consistency. 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 at the interface (the port) of the system in addition to the entropy creation term due to internal irreversible phenomena.

math.DS↗
arXiv2022

Optimal control of thermodynamic port-Hamiltonian Systems

We consider the problem of minimizing the entropy, energy, or exergy production for state transitions of irreversible port-Hamiltonian systems subject to control constraints. Via a dissipativity-based analysis we show that optimal solutions exhibit the manifold turnpike phenomenon with respect to the manifold of thermodynamic equilibria. We illustrate our analytical findings via numerical results for a heat exchanger.

math.OC↗
arXiv2022

Optimal control of port-Hamiltonian descriptor systems with minimal energy supply

We consider the singular optimal control problem of minimizing the energy supply of linear dissipative port-Hamiltonian descriptor systems subject to control and terminal state constraints. To this end, after reducing the problem to an ODE with feed-through term, we derive an input-state turnpike towards a subspace for optimal control of generalized port-Hamiltonian ordinary differential equations. We study the reachability properties of the system and prove that optimal states exhibit a turnpike behavior with respect to the conservative subspace. By means of the port-Hamiltonian structure, we show that, despite control constraints, this turnpike property is global in the initial state. Further, we characterize the class of dissipative Hamiltonian matrices and pencils.

math.OC↗
arXiv2021

Differential operator Dirac structures

As shown in earlier work, skew-adjoint linear differential operators, mapping efforts into flows, give rise to Dirac structures on a bounded spatial domain by a proper definition of boundary variables. In the present paper this is extended to pairs of linear differential operators defining a formally skew-adjoint relation between flows and efforts. Furthermore it is shown how the underlying repeated integration by parts operation can be streamlined by the use of two-variable polynomial calculus. Dirac structures defined by formally skew adjoint operators and differential operator effort constraints are treated within the same framework. Finally it is sketched how the approach can be also used for Lagrangian subspaces on bounded domains.

math.OC↗
arXiv2021

Boundary controlled irreversible port-Hamiltonian systems

Boundary controlled irreversible port-Hamiltonian systems (BC-IPHS) on 1-dimensional spatial domains are defined by extending the formulation of reversible BC-PHS to irreversible thermodynamic systems controlled at the boundaries of their spatial domains. The structure of BC-IPHS has clear physical interpretation, characterizing the coupling between energy storing and energy dissipating elements. By extending the definition of boundary port variables of BC-PHS to deal with the dissipative terms, a set of boundary port variables are defined such that BC-IPHS are passive with respect to a given set of conjugated inputs and outputs. As for finite dimensional IPHS, the first and second principle are satisfied as a structural property. Several examples are given to illustrate the proposed approach.

eess.SY↗
arXiv2021

Control of port-Hamiltonian systems with minimal energy supply

We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.

math.OC↗