One of the most important contributions of Heinz Langer in the area of operator theory in Krein spaces is the introduction of the notion of definitizable operators and the construction of the corresponding spectral function. In this note we obtain a new characterization for the subclass of non-negative…
We consider irreversible and coupled reversible–irreversible nonlinear port-Hamiltonian systems and the respective sets of thermodynamic equilibria. In particular, we are concerned with optimal state transitions and output stabilization on finite-time horizons. We analyze a class of optimal control problems,…
It is well known that linear and non-linear dissipative port-Hamiltonian systems in finite dimensions admit an energy balance, relating the energy increase in the system with the supplied energy and the dissipated energy. The integrand in the dissipation term is then a function of the state variable.…
We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of…
San Diego, Calif. [u.a.]: Academic Pr., Elsevier Science, 2024-04-04
While Koopman-based techniques like extended Dynamic Mode Decomposition are nowadays ubiquitous in the data-driven approximation of dynamical systems, quantitative error estimates were only recently established. To this end, both sources of error resulting from a finite dictionary and only finitely-many…
We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to…
Cham (ZG): Springer International Publishing AG, 2022-12-21
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for…
We analyze strict dissipativity of generalized linear quadratic optimal control problems on Hilbert spaces. Here, the term “generalized” refers to cost functions containing both quadratic and linear terms. We characterize strict pre-dissipativity with a quadratic storage function via coercivity of a…
Frankfurt ; München [u.a.]: Elsevier BV, 2022-11-23
We consider the problem of minimizing the entropy, energy, or exergy production for state transitions of irreversible port-Hamiltonian systems subject to control constraints. Via a dissipativity-based analysis we show that optimal solutions exhibit the manifold turnpike phenomenon with respect to the…
Frankfurt ; München [u.a.]: Elsevier BV, 2022-11-23
The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic…
We consider the problem of minimizing the supplied energy of infinite-dimensional linear port-Hamiltonian systems and prove that optimal trajectories exhibit the turnpike phenomenon towards certain subspaces induced by the dissipation of the dynamics. The theoretical foundations are illustrated by means…
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n…
Cham (ZG): Springer International Publishing AG, 2021-02-22
We prove new spectral enclosures for the non-real spectrum of a class of 2x2 block operator matrices with self-adjoint operators A and D on the diagonal and operators B and -B* as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to J-frame…
The essential spectrum of operator pencils with bounded coefficients in a Hilbert space is studied. Sufficient conditions in terms of the operator coefficients of two pencils are derived which guarantee the same essential spectrum. This is done by exploiting a strong relation between an operator pencil…
We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare the number of Jordan chains of length at least n corresponding to some eigenvalue to each other. In the operator case, it was recently proved that the difference…
We introduce the notion of spectral points of type π+ and type π− of
closed operators A in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that A is symmetric with respect…
We define and characterize the Krein space numerical range W(A) and the Krein space co-numerical range Wco(A) of a non-negative operator A in a Krein space. It is shown that the non-zero spectrum of A is contained in the closure of W(A) Wco(A).