arXiv subjects

Birgit Jacob

Publications and source records attributed to Birgit Jacob.

68 records · Page 4Linked to original sources

arXiv2015

Optimal Actuator and Observation Location for Time-Varying Systems on a Finite-Time Horizon

The choice of the location of controllers and observations is of great importance for designing control systems and improving the estimations in various practical problems. For time-varying systems in Hilbert spaces, the existence and convergence of the optimal location based on linear-quadratic control on a finite-time horizon is studied. The optimal location of observations for improving the estimation of the state at the final time, based on Kalman filter, is considered as the dual problem to the LQoptimal problem of the control locations. Further, the existence and convergence of optimal locations of observations for improving the estimation at the initial time, based on Kalman smoother is discussed. The obtained results are applied to a linear advection-diffusion model.

math.OC↗

arXiv2015

Efficiency and Sensitivity Analysis of Observation Networks for Atmospheric Inverse Modelling with Emissions

The controllability of advection-diffusion systems, subject to uncertain initial values and emission rates, is estimated, given sparse and error affected observations of prognostic state variables. In predictive geophysical model systems, like atmospheric chemistry simulations, different parameter families influence the temporal evolution of the system.This renders initial-value-only optimisation by traditional data assimilation methods as insufficient. In this paper, a quantitative assessment method on validation of measurement configurations to optimize initial values and emission rates, and how to balance them, is introduced. In this theoretical approach, Kalman filter and smoother and their ensemble based versions are combined with a singular value decomposition, to evaluate the potential improvement associated with specific observational network configurations. Further, with the same singular vector analysis for the efficiency of observations, their sensitivity to model control can be identified by determining the direction and strength of maximum perturbation in a finite-time interval.

physics.ao-ph↗

arXiv2014

Asymptotics of Evolution Equations beyond Banach spaces

We study the asymptotics of strongly continuous operator semigroups defined on locally convex spaces in order to develop a stability theory for solutions of evolution equations beyond Banach spaces. In the classical case, there is only little choice for a semigroup's speed in approaching zero uniformly. Indeed, if a strongly continuous semigroup on a Banach space converges to zero uniformly at any speed then it converges already uniformly at exponential speed. Semigroups with this property are said to be exponentially stable. Leaving the Banach space setting, the situation changes entirely; for instance convergence to zero at a speed faster than any polynomial but not exponentially fast is possible. In this article we establish concepts of stability which refine the classical notions and allow to grasp the different kinds of asymptotic behavior. We give characterizations of the new properties, study their relations and consider generic examples like multiplication semigroups and shifts. In addition we apply our results to the transport and the heat equation on classical Fréchet function spaces.

math.AP↗

arXiv2014

Root locii for systems defined on Hilbert spaces

The root locus is an important tool for analysing the stability and time constants of linear finite-dimensional systems as a parameter, often the gain, is varied. However, many systems are modelled by partial differential equations or delay equations. These systems evolve on an infinite-dimensional space and their transfer functions are not rational. In this paper a rigorous definition of the root locus for infinite-dimensional systems is given and it is shown that the root locus is well-defined for a large class of infinite-dimensional systems. As for finite-dimensional systems, any limit point of a branch of the root locus is a zero. However, the asymptotic behaviour can be quite different from that for finite-dimensional systems. This point is illustrated with a number of examples. It is shown that the familiar pole-zero interlacing property for collocated systems with a Hermitian state matrix extends to infinite-dimensional systems with self-adjoint generator. This interlacing property is also shown to hold for collocated systems with a skew-adjoint generator.

math.AP↗

arXiv2014

$C_0$-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain

Hyperbolic partial differential equations on a one-dimensional spatial domain are studied. This class of systems includes models of beams and waves as well as the transport equation and networks of non-homogeneous transmission lines. The main result of this paper is a simple test for $C_0$-semigroup generation in terms of the boundary conditions. The result is illustrated with several examples.

math.AP↗

arXiv2014

On the right multiplicative perturbation of non-autonomous $L^p$-maximal regularity

This paper is devoted to the study of $L^p$-maximal regularity for non-autonomous linear evolution equations of the form \begin{equation*}\label{Multi-pert1-diss-non} \dot u(t)+A(t)B(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation*} where $\{A(t),\ t\in [0,T]\}$ is a family of linear unbounded operators whereas the operators $\{B(t),\ t\in [0,T]\}$ are bounded and invertible. In the Hilbert space situation we consider operators $A(t), \ t\in[0,T],$ which arise from sesquilinear forms. The obtained results are applied to parabolic linear differential equations in one spatial dimension.

math.FA↗

arXiv2013

Stability and Stabilization of Infinite-dimensional Linear Port-Hamiltonian Systems

Stability and stabilization of linear port-Hamiltonian systems on infinite-dimensional spaces are investigated. This class is general enough to include models of beams and waves as well as transport and Schrödinger equations with boundary control and observation. The analysis is based on the frequency domain method which gives new results for second order port-Hamiltonian systems and hybrid systems. Stabilizing controllers with colocated input and output are designed. The obtained results are applied to the Euler-Bernoulli beam.

math.AP↗

arXiv2012

On Laplace-Carleson embedding theorems

This paper gives embedding theorems for a very general class of weighted Bergman spaces: the results include a number of classical Carleson embedding theorems as special cases. We also consider little Hankel operators on these Bergman spaces. Next, a study is made of Carleson embeddings in the right half-plane induced by taking the Laplace transform of functions defined on the positive half-line (these embeddings have applications in control theory): particular attention is given to the case of a sectorial measure or a measure supported on a strip, and complete necessary and sufficient conditions for a bounded embedding are given in many cases.

math.FA↗

arXiv2012

Weighted multiple interpolation and the control of perturbed semigroup systems

In this paper the controllabillity and admissibility of perturbed semigroup systems are studied, using tools from the theory of interpolation and Carleson measures. In addition, there are new results on the perturbation of Carleson measures and on the weighted interpolation of functions and their derivatives in Hardy spaces, which are of interest in their own right.

math.FA↗

arXiv2012

Applications of Laplace-Carleson embeddings to admissibility and controllability

It is shown how results on Carleson embeddings induced by the Laplace transform can be use to derive new and more general results concerning the weighted admissibility of control and observation operators for linear semigroup systems with q-Riesz bases of eigenvectors. Next, a new Carleson embedding result is proved, which gives further results on weighted admissibility for analytic semigroups. Finally, controllability by smoother inputs is characterised by means of a new result about weighted interpolation.

math.OC↗

arXiv2009

Control of Volterra systems with scalar kernels

Volterra observations systems with scalar kernels are studied. New sufficient conditions for admissibility of observation operators are developed. The obtained results are applied to time-fractional diffusion equations of distributed order.

math.AP↗

arXiv2009

Admissibility and Controllability of diagonal Volterra equations with scalar inputs

This article studies Volterra evolution equations from the point of view of control theory, in the case that the generator of the underlying semigroup has a Riesz basis of eigenvectors. Conditions for admissibility of the system's control operator are given in terms of the Carleson embedding properties of certain discrete measures. Moreover, exact and null controllability are expressed in terms of a new interpolation question for analytic functions, providing a generalization of results known to hold for the standard Cauchy problem. The results are illustrated by examples involving heat conduction with memory.

math.OC↗

arXiv2008

Tangential interpolation in weighted vector-valued H^p spaces, with applications

In this paper, norm estimates are obtained for the problem of minimal-norm tangential interpolation by vector-valued analytic functions in weighted H^p spaces, expressed in terms of the Carleson constants of related scalar measures. Applications are given to the notion of p-controllability properties of linear semigroup systems and controllability by functions in certain Sobolev spaces.

math.FA↗

arXiv2007

Analyticity and Riesz basis property of semigroups associated to damped vibrations

Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix $A$ associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of $A$ in the phase space.

math.SP↗