Dirac structures and their composition on Hilbert spaces

Dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures. We study these structures and their properties from an operator-theoretic point of view. In particular, we find necessary and sufficient conditions for the composition of two Dirac structures to be a Dirac structure and we show that they can be seen as Lagrangian (hyper-maximal neutral) subspaces of Krein spaces. Moreover, special emphasis is laid on Dirac structures associated with operator colligations. It turns out that this class of Dirac structures is linked to boundary triplets and that this class is closed under composition.

[1]  A. Schaft,et al.  On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems , 1999 .

[2]  Arjan van der Schaft,et al.  Interconnection of port-Hamiltonian systems and composition of Dirac structures , 2007, Autom..

[3]  Arjan van der Schaft.  Interconnection and geometry , 1999 .

[4]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[5]  Hans Zwart,et al.  Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators , 2005, SIAM J. Control. Optim..

[6]  Boundary relations and generalized resolvents of symmetric operators , 2006, math/0610299.

[7]  V. Derkach,et al.  The extension theory of Hermitian operators and the moment problem , 1995 .

[8]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[9]  J. Ball,et al.  Conservative State-Space Realizations of Dissipative System Behaviors , 2006 .

[10]  Mikael Kurula,et al.  Towards input/output-free modelling of linear infinite-dimensional systems in continuous time , 2010 .

[11]  O. Staffans,et al.  Well-Posed State/Signal Systems in Continuous Time , 2010 .

[12]  J. Bognár,et al.  Indefinite Inner Product Spaces , 1974 .

[13]  R. Redheffer.  On a Certain Linear Fractional Transformation , 1960 .

[14]  S. Hassi,et al.  Boundary relations and their Weyl families , 2006 .

[15]  J. Malinen,et al.  Impedance Passive and Conservative Boundary Control Systems , 2007 .

[16]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[17]  V. Derkach,et al.  Generalized resolvents and the boundary value problems for Hermitian operators with gaps , 1991 .

[18]  Mikael Kurula,et al.  On Passive and Conservative State/Signal Systems , 2010 .

[19]  Yu. L. Shmul'yan.  The theory of linear relations and spaces with an indefinite metric , 1976 .

[20]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[21]  T. Azizov,et al.  Linear Operators in Spaces with an Indefinite Metric , 1989 .

[22]  M. Langer,et al.  Boundary value problems for elliptic partial differential operators on bounded domains , 2007 .

[23]  V. Gorbachuk,et al.  Boundary Value Problems for Operator Differential Equations , 1990 .

[24]  J. A. Villegas,et al.  A Port-Hamiltonian Approach to Distributed Parameter Systems , 2007 .

[25]  Goran Golo,et al.  Interconnection structures in port-based modelling: tools for analysis and simulation , 2002 .

[26]  Irene Dorfman,et al.  Dirac Structures and Integrability of Nonlinear Evolution Equations , 1993 .

[27]  A. Schaft,et al.  Hamiltonian formulation of distributed-parameter systems with boundary energy flow , 2002 .

[28]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.