Volker Mehrmann

Differential-algebraic equations (DAEs) present a ubiquitous model class in the automated modeling, simulation, control and optimization of complex finite or infinite dimensional dynamical systems, in particular in the construction of digital twins. When combined with adequate space and time discretization methods they present the work-horse of developments in many engineering domains. However, this approach comes with certain challenges for the numerical methods and in particular in the coupling of systems from different physical domains with different scales or regularity requirements. We survey the modeling with DAEs and address how to overcome some of the challenges via the framework of port-Hamiltonian differential-algebraic systems (pHDAEs) and their simulation and optimization with space-time-model adaptive methods.

For a quadratic matrix polynomial dependent on parameters and a given tolerance [Formula: see text], the minimization of the [Formula: see text]-pseudospectral abscissa over the set of permissible parameter values is discussed, with applications in damping optimization and brake squeal reductions in mind. An approach is introduced that is based on nonsmooth and global optimization (or smooth optimization techniques such as BFGS if there are many parameters) equipped with a globally convergent criss-cross algorithm to compute the [Formula: see text]-pseudospectral abscissa objective when the matrix polynomial is small. For the setting when the matrix polynomial is large, a subspace framework is introduced, and it is argued formally that it solves the minimization problem globally. The subspace framework restricts the parameter-dependent matrix polynomial to small subspaces and thus solves the minimization problem for such restricted small matrix polynomials. It then expands the subspaces using the minimizers for the restricted polynomials. The proposed approach makes the global minimization of the [Formula: see text]-pseudospectral abscissa possible for a quadratic matrix polynomial dependent on a few parameters and for sizes up to at least a few hundreds. This is illustrated in several examples originating from damping optimization.

Numerical methods for developing port-Hamiltonian representations of general linear time-invariant systems are studied. The approach extends previous port-Hamiltonian characterizations to include the general non-minimal case and the case where the feedthrough term fails to have an invertible symmetric part. The resulting construction is able to identify infeasibility when the system fails to be port-Hamiltonian, and allows for the incorporation of perturbations in order to arrive at a nearby port-Hamiltonian system. Results are illustrated via numerical examples.

The concepts of hypocoercivity and hypocontractivity and their relationship are studied for semi-dissipative continuous-time and discrete-time evolution equations in a Hilbert space setting. New proofs for the characterization of the short-time decay of the solution from the initial value are presented, that in particular characterize the constants in the leading terms of the solution when expanded in time. Maximally coercive/contractive representations of hypocoercive and hypocontractive semi-dissipative systems are presented, as well as the effect of different representations on the error estimates for the numerical solution.

We study linear time-invariant Dissipative Hamiltonian (DH) systems arising in energy-based modeling of dynamical systems. An advantage of DH systems is that they are always stable due to the structure of their coefficient matrices, and, under further weak conditions, even asymptotically stable. In this paper, we discuss the computation of the stability radii for a given asymptotically stable DH system; i.e., the smallest structured perturbation that puts a DH system on the boundary of the region of asymptotic stability, so that it has purely imaginary eigenvalues. We obtain explicit computable formulas for various structured stability radii. For this, the problem of computing stability radii is reformulated in terms of minimizing the Rayleigh quotient of a Hermitian matrix or the sum of two generalized Rayleigh quotients of Hermitian semidefinite matrices. This reformulation results in the problem of minimizing the largest eigenvalue of an eigenvector-dependent Hermitian matrix or minimizing the smallest eigenvalue of a Hermitian matrix which depends on the eigenvector. It is also demonstrated (via numerical experiments) that, under structure-preserving perturbations, the asymptotic stability of a DH system is much more robust than under general perturbations, since the distance to instability is typically much larger when structure-preserving perturbations are considered. Finally, similar results are obtained for optimally robust representations of stable systems.

In the recently published paper “Eigenstructure Perturbations for a Class of Hamiltonian Matrices and Solutions of Related Riccati Inequalities” by V. Mehrmann and H. Xu an error has occured in the construction of a block diagonal form. A new construction is provided.