Motivated from linear-quadratic optimal control problems for dierential-algebraic equations (DAEs), we study the functional analytic properties of the operator associated with the necessary optimality boundary value problem and show that it is associated with a self-conjugate operator and a self-adjoint pair of matrix functions. We then study general self-adjoint pairs of matrix valued functions and derive condensed forms under orthogonal congruence transformations that preserve the self-adjointness. We analyze the relationship between self-adjoint DAEs and Hamiltonian systems with symplectic ows. We also show how to extract self-adjoint and Hamiltonian reduced systems from derivative arrays.

In the automated modeling of multi-physics dynamical systems, frequently different subsystems are coupled together via interface or coupling conditions. This approach often results in large-scale high-index differential-algebraic equations (DAEs). Since the direct numerical simulation of these kinds of systems leads to instabilities and possibly non-convergence of numerical methods, a regularization or remodeling of such systems is required. In many simulation environments, a structural method that analyzes the system based on its sparsity pattern is used to determine the index and an index-reduced system model. However, this approach is not reliable for certain problem classes, and in particular not suited for coupled systems of DAEs. We present a new approach for the regularization of coupled dynamical systems that combines the Signature method (Σ-method) for the structural analysis with algebraic regularization techniques. This allows to handle structurally singular systems and also enables a proper treatment of redundancies or inconsistencies in the system.

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Automated modeling of multi-physics dynamical systems often results in large-scale high-index differential-algebraic equations (DAEs). Since direct numerical simulation of such systems leads to instabilities and possibly non-convergence of numerical methods, a regularization or remodeling is required. In many simulation environments, a structural analysis based on the sparsity pattern of the system is used to determine the index and an index-reduced system model. Here, usually the Pantelides algorithm in combination with the Dummy Derivative Method is used. We present a new approach for the regularization of DAEs that is based on the Signature method (Σ-method).

Motivated by the structure which arises in the port-Hamiltonian formulation of constraint dynamical systems, structure preserving condensed forms for skew-adjoint differential-algebraic equations (DAEs) are derived. Moreover, structure preserving condensed forms under constant rank assumptions for linear port-Hamiltonian differential-algebraic equations are developed. These condensed forms allow for the further analysis of the properties of port-Hamiltonian DAEs and to study, e.g., existence and uniqueness of solutions or to determine the index. It can be shown that under certain conditions for regular port-Hamiltonian DAEs the strangeness index is bounded by $\mu\leq1$.