The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is derived. It is shown that this structure is invariant under equivalence transformations, and that it is adequate also for the modeling of high-index descriptor systems. The regularization procedure for descriptor systems to make them suitable for simulation and control is modified to deal with the port-Hamiltonian structure. The relevance of the new structure is demonstrated with several examples.

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order √ ε, where ε is the machine precision, the new method computes the eigenvalues to full possible accuracy. Mathematics Subject Classification (1991): 65F15

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, H∞ optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/or skew-Hamiltonian structure. We extend the recently developed methods for Hamiltonian matrices to the general case of skew-Hamiltonian/Hamiltonian pencils. The algorithms circumvent problems with skew-Hamiltonian/Hamiltonian matrix pencils that lack structured Schur forms by embedding them into matrix pencils that always admit a structured Schur form. The rounding error analysis of the resulting algorithms is favorable. For the embedded matrix pencils, the algorithms use structure preserving unitary matrix computations and are strongly backwards stable, i.e., they compute the exact structured Schur form of a nearby matrix pencil with the same structure.

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