Indefinite Sturm-Liouville operators in polar form. - In: Integral equations and operator theory, ISSN 1420-8989, Bd. 96 (2024), 2, S. 1-58
https://doi.org/10.1007/s00020-023-02746-3
Feedback rectifiable pairs and stabilization of switched linear systems. - In: Systems & control letters, ISSN 1872-7956, Bd. 186 (2024), 105755, S. 1-10
We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop subsystems share the same eigenstructure. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. In particular the proposed algorithm provides sets of eigenvalues and corresponding eigenvectors for the closed-loop subsystems that guarantee stability for arbitrary switching. Several examples illustrate the characteristics of the problem considered and the application of the proposed design procedure.
https://doi.org/10.1016/j.sysconle.2024.105755
The rank-one perturbation problem for linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (29 Seiten). - (Preprint ; M23,12)
We use the recently introduced Weyr characteristic of linear relations in Cn and its relation with the Kronecker canonical form of matrix pencils to describe their dimension. Then, this is applied to study one-dimensional perturbations of linear relations.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200305
On the essential spectrum of operator pencils. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (9 Seiten). - (Preprint ; M23,11)
For a closed densely defined linear operator A and a bounded linear operator B on a Banach space X whose essential spectrums are contained in disjoint sectors, we show that the essential spectrum of the associated operator pencil λA + B is contained in a sector of the complex plane whose boundaries are determined purely by the angles that define the two sectors, which contain the essential spectrums of A and B.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200291
Lower bounds for self-adjoint Sturm-Liouville operators. - In: Proceedings of the American Mathematical Society, ISSN 1088-6826, Bd. 151 (2023), 12, S. 5313-5323
https://doi.org/10.1090/proc/16523
Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (15 Seiten). - (Preprint ; M23,10)
The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when the coefficients are complex. Here the main result is a collection of various criteria which allow us to decide to which class of Sims' scheme a given Sturm-Liouville problem with complex coefficients belongs. This is subsequently applied to a second order differential equation defined on a ray in C which is motivated by the recent intensive research connected with PT-symmetric Hamiltonians.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200258
Functional models of symmetric and selfadjoint operators. - In: From complex analysis to operator theory: a panorama, (2023), S. 75-122
https://doi.org/10.1007/978-3-031-31139-0_7
Quantum particle under dynamical confinement: from quantum Fermi acceleration to high harmonic generation. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (15 Seiten). - (Preprint ; M23,09)
Quantum dynamics of a particle confined in a box with time-dependent wall is revisited by considering some unexplored aspects of the problem. In particular, the case of dynamical confinement in a time-dependent box in the presence of purely time-varying external potential is treated by obtaining exact solution. Also, some external potentials approving separation of space and time variables in the Schrödinger equation with time-dependent boundary conditions are classified. Time-dependence of the average kinetic energy and average quantum force are analyzed. A model for optical high harmonic generation in the presence of dynamical confinement and external linearly polarized monochromatic field is proposed.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200218
Perturbation and spectral theory for singular indefinite Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (26 Seiten). - (Preprint ; M23,08)
We study singular Sturm-Liouville operators of the form 1/r_j (-d/dx p_j d/dx +q_j), j=0,1, in L_2((a; b); rj ), where, in contrast to the usual assumptions, the weight functions r_j have different signs near the singular endpoints a and b. In this situation the associated maximal operators become self-adjoint with respect to indefnite inner products and their spectral properties differ essentially from the Hilbert space situation. We investigate the essential spectra and accumulation properties of nonreal and real discrete eigenvalues; we emphasize that here also perturbations of the indefinite weights r_j are allowed. Special attention is paid to Kneser type results in the indefinite setting and to L_1 perturbations of periodic operators.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200208
Feedback rectifiable pairs and stabilization of switched linear systems. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (12 Seiten). - (Preprint ; M23,07)
We address the feedback design problem for switched linear systems. In particular we aim to design a switched state-feedback such that the resulting closed-loop switched system is in upper triangular form. To this effect we formulate and analyse the feedback rectification problem for pairs of matrices. We present necessary and sufficient conditions for the feedback rectifiability of pairs for two subsystems and give a constructive procedure to design stabilizing state-feedback for a class of switched systems. Several examples illustrate the characteristics of the problem considered and the application of the proposed constructive procedure.
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2023200194