08.05.2024 - 4 pm (CET) - Paul Kotyczka (TU Munich)

Title: 

Geometric integration and discrete-time port-Hamiltonian systems

Abstract: 

The interest of this talk is to show possibilities to preserve “structure” when continuous-time port-Hamiltonian (PH) models are translated via numerical integration to the discrete-time domain. On the example of a simple
(mechanical) Hamiltonian system with one degree of freedom, we first illustrate symplecticity, i.e., area preservation in the phase plane, of the flow as an underlying structural property, from which energy conservation
is derived. Consequently, we give examples for numerical integration schemes that are symplectic or energy-conserving. 

Both families of integrators can be used for the definition of discrete-time PH systems, where the definitions of discrete-time port variables play a fundamental role to describe energy transfer over the system boundary. We highlight similarities and differences using the two paths, in particular based on the discrete-time energy balance equations.

Finally, we give two examples from our recent research, where discrete-time models of geometrically nonlinear systems – elastic continua and beams – are obtained with structure-preserving methods.

31.05.2024 - 11 am (CET)  - Jacquelien Scherpen (U Groningen)

03.07.2024 - 4 pm (CET) - Silke Glas (U Twente)

04.09.2024 - 4 pm (CET) - Yann Le Gorrec (FEMTO-ST, Besançon)

02.10.2024 - 4 pm (CET) - TBA 

06.11.2024 - 4 pm (CET) - Dorothée Normand-Cyrot (Laboratoire des Signaux et Systèmes, Paris)

04.12.2024 - 4 pm (CET) - TBA 

08.01.2025 - 4 pm (CET) - Claudia Totzeck (BU Wuppertal)

31.01.2024 - Volker Mehrmann (TU Berlin)    Slides

Title: 

Port-Hamiltonian systems: algebraic, geometric and operator theoretic representations

Abstract: 

Different representations of dissipative Hamiltonian and port-Hamiltonian differential-algebraic equations (DAE) systems are presented and compared. Using global geometric and algebraic points of view, translations between the different representations are presented. The results also apply in the Hilbert space setting of linear operator equations. Characterizations are also derived when a general DAE system can be transformed into one of these structured representations. Approaches for computing the structural information and the described transformations are derived that can be directly implemented as numerical methods. The results are demonstrated with a large number of examples.

Joint work partly with Arjan van der Schaft and partly with Hans Zwart

06.03.2024 - Denis Matignon (ISAE-SUPAERO, Toulouse)    Slides

Title: 

The  partitioned finite element method for port-Hamiltonian systems: a structure-preserving discretization for boundary controlled wave and heat PDEs.

Abstract: 

Boundary controlled and observed wave and heat PDEs can be recast as port-Hamiltonian systems on an n-D domain, starting from physical principles and allowing for a power balance which proves most useful when interconnecting such subsystems. 

A mixed finite element method ensures the preservation of these properties at the discrete level: this will be introduced with a primer on the finite element method (FEM); then, some optimal convergence results will be provided and illustrated on the 2D inhomogeneous and anisotropic wave equation.

Finally, the effectiveness of PFEM will finally be illustrated when capturing refined asymptotic behaviours of the coupled heat-wave PDE system in different geometric configurations.

Joint work partly with Ghislain Haine.

03.04.2024 - Bernhard Maschke (U Lyon)  Slides

Title: 

The geometry of the state space of physical systems and the consequences on the definition of Port-Hamiltonian systems.

Abstract: 

In the first part, we recall the geometric structure of the state space of physical systems. Indeed, for Thermodynamical systems, it is well accepted that the system is first defined by its so-called equilibrium properties. These properties are defined by a set of relations among the extensive and intensive variables, the Thermodynamic Phase variables, which should satisfy the Gibbs' equations. Actually Gibbs' equations define a Legendre submanifold of the Thermodynamic Phase Space which is generated by a family of functions, called thermodynamic functions. This Legendre submanifolds actually defines the state space of the system.

A similar construction holds for Hamiltonian systems arising for mechanical systems or electro-mechanical systems' models, when instead of defining a Hamiltonian function, one considers the reciprocal constitutive relations relating the energy and the co-energy variables. These reciprocal relations define a Lagrangian submanifold of the cotangent space of the energy variables (the space of energy and the co-energy variables).

In the second part of the talk, we shall draw the consequence of the definition of the state space Lagrange or Legendre submanifolds for Hamiltonian and port Hamiltonian systems. Indeed, defining the state space as a submanifold of some phase space,  corresponds to an implicit definition of the Hamiltonian dynamics. For irreversible Thermodynamic systems, one defines a contact Hamiltonian system on the Thermodynamic Phase Space, leaving invariant some Legendre submanifold. For Hamiltonian systems defined on Lagrange submanifolds, one defines a implicit Hamiltonian system restricted to some Lagrange submanifold. 

We shall finally present some ongoing work, how this geometric perspective of the state space of physical systems, leads to define a novel class of Port Hamiltonian systems equipped with a new type of port variables, derived from the definition of Lagrange or Legendre submanifolds. We shall illustrate the work with various simple examples taken from physical and engineering systems.