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Birgit Jacob

Publications and source records attributed to Birgit Jacob.

At least 55 records · Page 3Linked to original sources
arXiv2020

Pseudospectrum enclosures by discretization

A new method to enclose the pseudospectrum via the numerical range of the inverse of a matrix or linear operator is presented. The method is applied to finite-dimensional discretizations of an operator on an infinite-dimensional Hilbert space, and convergence results for different approximation schemes are obtained, including finite element methods. We show that the pseudospectrum of the full operator is contained in an intersection of sets which are expressed in terms of the numerical ranges of shifted inverses of the approximating matrices. The results are illustrated by means of two examples: the advection-diffusion operator and the Hain-Lüst operator.

math.SP↗
arXiv2020

Riesz bases of port-Hamiltonian systems

The location of the spectrum and the Riesz basis property of well-posed homogeneous infinite-dimensional linear port-Hamiltonian systems on a 1D spatial domain are studied. It is shown that the Riesz basis property is equivalent to the fact that system operator generates a strongly continuous group. Moreover, in this situation the spectrum consists of eigenvalues only, located in a strip parallel to the imaginary axis and they can decomposed into finitely many sets having each a uniform gap.

math.FA↗
arXiv2020

Remarks on input-to-state stability of collocated systems with saturated feedback

We investigate input-to-state stability (ISS) of infinite-dimensional collocated control systems subject to saturated feedback. Here, the unsaturated closed loop is dissipative and uniformly globally asymptotically stable. Under an additional assumption on the linear system, we show ISS for the saturated one. We discuss the sharpness of the conditions in light of existing results in the literature.

math.OC↗
arXiv2020

Dynamic iteration schemes and port-Hamiltonian formulation in coupled DAE circuit simulation

Electric circuits are usually described by charge- and flux-oriented modified nodal analysis. In this paper, we derive models as port-Hamiltonian systems on several levels: overall systems, multiply coupled systems and systems within dynamic iteration procedures. To this end, we introduce new classes of port-Hamiltonian differential-algebraic equations. Thereby, we additionally allow for nonlinear dissipation on a subspace of the state space. Both, each subsystem and the overall system, possess a port-Hamiltonian structure. A structural analysis is performed for the new setups. Dynamic iteration schemes are investigated and we show that the Jacobi approach as well as an adapted Gauss-Seidel approach lead to port-Hamiltonian differential-algebraic equations.

math.NA↗
arXiv2019

Krylov type methods exploiting the quadratic numerical range

The quadratic numerical range $W^2(A)$ is a subset of the standard numerical range of a linear operator which still contains its spectrum. It arises naturally in operators which have a $2 \times 2$ block structure, and it consists of at most two connected components, none of which necessarily convex. The quadratic numerical range can thus reveal spectral gaps, and it can in particular indicate that the spectrum of an operator is bounded away from $0$. We exploit this property in the finite-dimensional setting to derive Krylov subspace type methods to solve the system $Ax = b$, in which the iterates arise as solutions of low-dimensional models of the operator whose quadratic numerical ranges is contained in $W^2(A)$. This implies that the iterates are always well-defined and that, as opposed to standard FOM, large variations in the approximation quality of consecutive iterates are avoided, although $0$ lies within the convex hull of the spectrum. We also consider GMRES variants which are obtained in a similar spirit. We derive theoretical results on basic properties of these methods, review methods on how to compute the required bases in a stable manner and present results of several numerical experiments illustrating improvements over standard FOM and GMRES.

math.NA↗
arXiv2019

Stability radius for infinite-dimensional interconnected systems

The stability radius for finitely many interconnected linear exponentially stable well-posed systems with respect to static perturbations is studied. If the output space of each system is finite-dimensional, then a lower bound for the stability radius in terms of the norm of the corresponding transfer functions is given. Moreover, for regular linear systems with zero feedthrough operator and finite-dimensional output spaces a formula for the stability radius is developed.

math.FA↗
arXiv2019

On exact controllability of infinite-dimensional linear port-Hamiltonian systems

Infinite-dimensional linear port-Hamiltonian systems on a one-dimensional spatial domain with full boundary control and without internal damping are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. The main result shows that well-posed port-Hamiltonian systems, with state space $L^2((0,1);\mathbb C^n)$ and input space $\mathbb C^n$, are exactly controllable.

math.OC↗
arXiv2019

Well-posedness of a class of hyperbolic partial differential equations on the semi-axis

In this article we study a class of hyperbolic partial differential equations of order one on the semi-axis. The so-called port-Hamiltonian systems cover for instance the wave equation and the transport equation, but also networks of the aforementioned equations fit into this framework. Our main results firstly characterize the boundary conditions which turn the corresponding linear operator into the generator of a strongly continuous semigroup. Secondly, we equip the equation with inputs (control) and outputs (observation) at the boundary and prove that this leads to a well-posed boundary control system. We illustrate our results via an example of coupled transport equations on a network, that allows to model transport from and to infinity. Moreover, we study a vibrating string of infinite length with one endpoint. Here, we show that our results allow to treat cases where the physical constants of the string tend to zero at infinity.

math.FA↗
arXiv2019

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

We study a class of non-autonomous boundary control and observation linear systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by different fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

math.FA↗
arXiv2018

On continuity of solutions for parabolic control systems and input-to-state stability

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

math.FA↗
arXiv2018

Well-posedness of networks for 1-D hyperbolic partial differential equations

We consider the well-posedness of a class of hyperbolic partial differential equations on a one dimensional spatial domain. This class includes in particular infinite-dimensional networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. In the first part we assume a finite interval and in the second part, we consider partial differential equations on the semi-axis.

math.FA↗
arXiv2017

A Review on Realization Theory for Infinite-Dimensional Systems

We give an introduction to the realisation theory for infinite-dimensional systems. That is, we show that for any function $G$, analytic and bounded in the right half of the complex plane, there exists operators $A,B,C$ such that $G(s_1)-G(s_2) = (s_2-s_1) C(s_1 I-A)^{-1}(s_2 I-A)^{-1}B$. Here $A$ is the infinitesimal generator of a strongly continuous semigroup on a Hilbert space, and $B$ and $C$ are admissible input and output operators, respectively. Our results summarise and clarify the results as found in the literature, starting more than 40 years ago.

math.FA↗
arXiv2017

Zero Dynamics for Port-Hamiltonian Systems

The zero dynamics of infinite-dimensional systems can be difficult to characterize. The zero dynamics of boundary control systems are particularly problematic. In this paper the zero dynamics of port-Hamiltonian systems are studied. A complete characterization of the zero dynamics for a port-Hamiltonian systems with invertible feedthrough as another port-Hamiltonian system on the same state space is given. It is shown that the zero dynamics for any port-Hamiltonian system with commensurate wave speeds are well-defined, and are also a port-Hamiltonian system. Examples include wave equations with uniform wave speed on a network. A constructive procedure for calculation of the zero dynamics, that can be used for very large system order, is provided.

math.AP↗
arXiv2017

Infinite-dimensional input-to-state stability and Orlicz spaces

In this work, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can be characterized in terms of input-to-state stability with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions with respect to $L^\infty$ are equivalent.

math.OC↗
arXiv2017

Numerical Range and Quadratic Numerical Range for Damped Systems

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as $A_0$. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients $A_0$ and $D$ which improve earlier results for sectorial and selfadjoint $D$; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.

math.SP↗
arXiv2016

Variational principles for self-adjoint operator functions arising from second-order systems

Variational principles are proved for self-adjoint operator functions arising from variational evolution equations of the form \[ \langle\ddot{z}(t),y \rangle + \mathfrak{d}[\dot{z} (t), y] + \mathfrak{a}_0 [z(t),y] = 0. \] Here $\mathfrak{a}_0$ and $\mathfrak{d}$ are densely defined, symmetric and positive sesquilinear forms on a Hilbert space $H$. We associate with the variational evolution equation an equivalent Cauchy problem corresponding to a block operator matrix $\mathcal{A}$, the forms \[ \mathfrak{t}(λ)[x,y] := λ^2\langle x,y\rangle + λ\mathfrak{d}[x,y] + \mathfrak{a}_0[x,y], \] where $λ\in \mathbb C$ and $x,y$ are in the domain of the form $\mathfrak{a}_0$, and a corresponding operator family $T(λ)$. Using form methods we define a generalized Rayleigh functional and characterize the eigenvalues above the essential spectrum of $\mathcal{A}$ by a min-max and a max-min variational principle. The obtained results are illustrated with a damped beam equation.

math.FA↗
arXiv2015

$β$-admissibility of observation and control operators for hypercontractive semigroups

We prove a Weiss conjecture on $β$-admissibility of control and observation operators for discrete and continuous $γ$-hypercontractive semigroups of operators, by representing them in terms of shifts on weighted Bergman spaces and using a reproducing kernel thesis for Hankel operators. Particular attention is paid to the case $γ=2$, which corresponds to the unweighted Bergman shift.

math.AP↗