Spectrum analysis of LTI continuous-time systems with constant delays: A literature overview of some recent results

In recent decades, increasingly intensive research attention has been given to dynamical systems containing delays and those affected by the after-effect phenomenon. Such research covers a wide range of human activities and the solutions of related engineering problems often require interdisciplinary cooperation. The knowledge of the spectrum of these so-called time-delay systems (TDSs) is very crucial for the analysis of their dynamical properties, especially stability, periodicity, and dumping effect. A great volume of mathematical methods and techniques to analyze the spectrum of the TDSs have been developed and further applied in the most recent times. Although a broad family of nonlinear, stochastic, sampled-data, time-variant or time-varying-delay systems has been considered, the study of the most fundamental continuous linear time-invariant (LTI) TDSs with fixed delays is still the dominant research direction with ever-increasing new results and novel applications. This paper is primarily aimed at a (systematic) literature overview of recent (mostly published between 2013 to 2017) advances regarding the spectrum analysis of the LTI-TDSs. Specifically, a total of 137 collected articles-which are most closely related to the research area-are eventually reviewed. There are two main objectives of this review paper: First, to provide the reader with a detailed literature survey on the selected recent results on the topic and Second, to suggest possible future research directions to be tackled by scientists and engineers in the field. © 2013 IEEE.MSMT-7778/2014, FEDER, European Regional Development Fund; LO1303, FEDER, European Regional Development Fund; CZ.1.05/2.1.00/19.0376, FEDER, European Regional Development FundEuropean Regional Development Fund through the Project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; National Sustainability Program Project [LO1303 (MSMT-7778/2014)

Degenerate perturbation theory in thermoacoustics: High-order sensitivities and exceptional points

In this study, we connect concepts that have been recently developed in thermoacoustics, specifically, (i) high-order spectral perturbation theory, (ii) symmetry induced degenerate thermoacoustic modes, (iii) intrinsic thermoacoustic modes, and (iv) exceptional points. Their connection helps gain physical insight into the behaviour of the thermoacoustic spectrum when parameters of the system are varied. First, we extend high-order adjoint-based perturbation theory of thermoacoustic modes to the degenerate case. We provide explicit formulae for the calculation of the eigenvalue corrections to any order. These formulae are valid for self-adjoint, non-self-adjoint or even non-normal systems; therefore, they can be applied to a large range of problems, including fluid dynamics. Second, by analysing the expansion coefficients of the eigenvalue corrections as a function of a parameter of interest, we accurately estimate the radius of convergence of the power series. Third, we connect the existence of a finite radius of convergence to the existence of singularities in parameter space. We identify these singularities as exceptional points, which correspond to defective thermoacoustic eigenvalues, with infinite sensitivity to infinitesimal changes in the parameters. At an exceptional point, two eigenvalues and their associated eigenvectors coalesce. Close to an exceptional point, strong veering of the eigenvalue trajectories is observed. As demonstrated in recent work, exceptional points naturally arise in thermoacoustic systems due to the interaction between modes of acoustic and intrinsic origin. The role of exceptional points in thermoacoustic systems sheds new light on the physics and sensitivity of thermoacoustic stability, which can be leveraged for passive control by small design modifications

Algebraic geometric methods for the stabilizability and reliability of multivariable and of multimode systems

The extent to which feedback can alter the dynamic characteristics (e.g., instability, oscillations) of a control system, possibly operating in one or more modes (e.g., failure versus nonfailure of one or more components) is examined

Slow light in photonic crystals

The problem of slowing down light by orders of magnitude has been extensively discussed in the literature. Such a possibility can be useful in a variety of optical and microwave applications. Many qualitatively different approaches have been explored. Here we discuss how this goal can be achieved in linear dispersive media, such as photonic crystals. The existence of slowly propagating electromagnetic waves in photonic crystals is quite obvious and well known. The main problem, though, has been how to convert the input radiation into the slow mode without loosing a significant portion of the incident light energy to absorption, reflection, etc. We show that the so-called frozen mode regime offers a unique solution to the above problem. Under the frozen mode regime, the incident light enters the photonic crystal with little reflection and, subsequently, is completely converted into the frozen mode with huge amplitude and almost zero group velocity. The linearity of the above effect allows to slow light regardless of its intensity. An additional advantage of photonic crystals over other methods of slowing down light is that photonic crystals can preserve both time and space coherence of the input electromagnetic wave.Comment: 96 pages, 12 figure

Some Remarks on Puiseux Series for Multiple Imaginary Characteristic Roots of Linear Time-Delay Systems

International audienc

Some Remarks on Puiseux Series for Multiple Imaginary Characteristic Roots of Linear Time-Delay Systems

International audienc

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Spontane Teilchenerzeugung in zeitabhängigen überkritischen Feldern der QED

In the classical Dirac equation with strong potentials, called overcritical, a bound state reaches the negative continuum. In QED the presence of a static overcritical external electric field leads to a charged vacuum and indicates spontaneous particle creation when the overcritical field is switched on. The goal of this work is to clarify whether this effect exists, i.e. if it can be uniquely defined and proved, in time-dependent physical processes. Starting from a fundamental level of the theory we check all mathematical and interpretational steps from the algebra of fields to the very effect. In the first, theoretical part of this thesis we introduce the mathematical formulation of the classical and quantized Dirac theory with their most important results. Using this language we define rigorously the notion of spontaneous particle creation in overcritical fields. First, we give a rigorous definition of resonances as poles of the resolvent or the Green's function and show how eigenvalues become resonances under Hamiltonian perturbations. In particular, we consider essential for overcritical potentials perturbation of eigenvalues at the edge of the continuous spectrum. Next, we gather various adiabatic theorems and discuss well-posedness of the scattering in the adiabatic limit. Then, we construct Fock space representations of the field algebra, study their equivalence and give a unitary implementer of all Bogoliubov transformations induced by unitary transformations of the one-particle Hilbert space as well as by the projector (or vacuum vector) changes as long as they lead to unitarily equivalent Fock representations. We implement in Fock space self-adjoint and unitary operators from the one-particle space, discussing the charge, energy, evolution and scattering operators. Then we introduce the notion of particles and several particle interpretations for time-dependent processes with a different Fock space at every instant of time. We study how the charge, energy and number of particles change in consequence of a change of representation or in implemented evolution or scattering processes, what is especially interesting in presence of overcritical potentials. Using this language we define rigorously the notion of spontaneous particle creation. Then we look for physical processes which show the effect of vacuum decay and spontaneous particle creation exclusively due to the overcriticality of the potential. We consider several processes with static as well as suddenly switched on (and off) static overcritical potentials and conclude that they are unsatisfactory for observation of the spontaneous particle creation. Next, we consider properties of general time-dependent scattering processes with continuous switch on (and off) of an overcritical potential and show that they also fail to produce stable signatures of the particle creation due to overcriticality. Further, we study and successfully define the spontaneous particle creation in adiabatic processes, where the spontaneous antiparticle is created as a result of a resonance (wave packet) decay in the negative continuum. Unfortunately, they lead to physically questionable pair production as the adiabatic limit is approached. Finally, we consider extension of these ideas to non-adiabatic processes involving overcritical potentials and argue that they are the best candidate for showing the spontaneous pair creation in physical processes. Demanding creation of the spontaneous antiparticle in the state corresponding to the overcritical resonance rather quick than slow processes should be considered, with a possibly long frozen overcritical period. In the second part of this thesis we concentrate on a class of spherically symmetric square well potentials with a time-dependent depth. First, we solve the Dirac equation and analyze the structure and behaviour of bound states and appearance of overcriticality. Then, by analytic continuation we find and discuss the behaviour of resonances in overcritical potentials. Next, we derive and solve numerically (introducing a non-uniform continuum discretization for a consistent treatment of narrow peaks) a system of differential equations (coupled channel equations) to calculate particle and antiparticle production spectra for various time-dependent processes including sudden, quick, slow switch on and off of a sub- and overcritical potentials. We discuss in detail how and under which conditions an overcritical resonance decays during the evolution giving rise to the spontaneous production of an antiparticle. We compare the antiparticle production spectrum with the shape of the resonance in the overcritical potential. We study processes, where the overcritical potentials are switched on at different speed and are possibly frozen in the overcritical phase. We prove, in agreement with conclusions of the theoretical part, that the peak (wave packet) in the negative continuum representing a dived bound state partially follows the moving resonance and partially decays at every stage of its evolution. This continuous decay is more intensive in slow processes, while in quick processes the wave packet more precisely follows the resonance. In the adiabatic limit, the whole decay occurs already at the edge of the continuum, resulting in production of antiparticles with vanishing momentum. In contrast, in quick switch on processes with delay in the overcritical phase, the spectrum of the created antiparticles agrees best with the shape of the resonance. Finally, we address the question how much information about the time-dependent potential can be reconstructed from the scattering data, represented by the particle production spectrum. We propose a simple approximation method (master equation) basing on an exponential, decoherent decay of time-dependent resonances for prediction of particle creation spectra and obtain a good agreement with the results of full numerical calculations. Additionally, we discuss various sources of errors introduced by the numerical discretization, find estimations for them and prove convergence of the numerical schemes.In der klassischen Dirac Gleichung mit starken, so genannten überkritischen, Potentialen, ein gebundener Zustand erreicht das negative Kontinuum. In QED die Anwesenheit der statischen überkritischen externen elektrischen Feldern führt zu einem geladenen Vakuum und weist auf eine spontane Teilchenerzeugung hin, wenn das überkritische Feld eingeschaltet wird. Das Ziel dieser Arbeit ist es zu klären, ob der Effekt existiert, d.h. ob er in zeitabhängigen physikalischen Prozessen eindeutig definiert und bewiesen werden kann. Beginnend von dem fundamentalen Niveau der Theorie prüfen wir alle mathematischen und interpretationellen Schritte von der Algebra der Felder bis zu dem studierten Effekt. In dem ersten, theoretischen Teil der Arbeit führen wir die mathematische Formulierung der klassischen und quantisierten Dirac Theorie mit ihren wichtigsten Ergebnissen ein. Mit dieser Sprache definieren wir rigorös den Begriff der spontanen Teilchenerzeugung in überkritischen Feldern. Zuerst geben wir eine rigoröse Definition der Resonanzen als Pole der Resolvente oder der Greenschen Funktion und zeigen, wie Eigenwerte in Resonanzen unter Perturbationen des Hamiltonians übergehen. Speziell betrachten wir die für die überkritische Potentiale entscheidende Pertrubationen von Eigenwerten am Rande des kontinuierlichen Spektrum. Als nächstes sammeln wir verschiedene adiabatische Theoreme und diskutieren die Wohldefinierheit von Streuung im adiabatischen Limes. Dann konstruieren wir Fock Raum Darstellungen der Feld-Algebra, studieren deren äquivalenz und geben einen unitären Implementer aller Bogoliubov Transformationen, die durch unitäre Transformationen des Ein-Teilchen Hilbert Raumes genauso wie durch Projektor (oder Vakuum Vektor) änderung induziert werden, solange die zu unitär äquivalenten Fock Darstellungen führen. Wir implementieren im Fock Raum selbstadjungierte und unitäre Operatoren aus dem Ein-Teilchen Raum und diskutieren den Ladung-, Energie-, Evolution- und Streuoperator. Dann führen wir den Begriff der Teilchen und verschiedene Teilcheninterpretationen für zeitabhängige Prozesse mit unterschiedlichen Fock Räumen zu jedem Zeitpunkt ein. Wir studieren, wie sich die Ladung, Energie und Zahl der Teilchen als Folge der änderung der Darstellung oder in implementierten Evolution- oder Streu-Prozessen ändern, was in Präsenz von überkritischen Potentialen besonders interessant ist. Mit dieser Sprache definieren wir rigorös ten Begriff der spontanen Teilchenerzeugung. Weiterhin suchen wir nach physikalischen Prozessen, die den Effekt des Vakuumzerfalls und spontaner Teilchenerzeugung ausschließlich wegen der überkritikalität des Potentials zeigen. Wir betrachten verschiedene Prozesse sowohl mit statischen als auch mit plötzlich ein- (und aus-)geschalteten überkritischen Potentialen und folgern, dass sie nicht zufrieden stellend für die Beobachtung der spontanen Teilchenerzeugung sind. Als nächstes betrachten wir Eigenschaften von allgemeinen zeitabhängigen Streuprozessen mit kontinuierlichem Ein- (und Aus-)schalten des überkritischen Potentials und zeigen, dass sie auch ungeeignet sind, um stabile Anzeichen der Teilchenerzeugung wegen überkritikalität zu produzieren. Weiter studieren und definieren wir erfolgreich die spontane Teilchenerzeugung in adiabatischen Prozessen, wo das spontane Antiteilchen als Folge des Resonanz (Wellenpacket) Zerfalls im negativen Kontinuum erzeugt wird. Leider führen sie zu physikalisch fraglicher Paarerzeugung als der adiabatische Limes erreicht wird. Endlich betrachten wir eine Ausweitung dieser Ideen auf nicht adiabatische Prozesse mit überkritischen Potentialen und argumentieren, dass sie der beste Kandidat fürs Zeigen der spontanen Paarerzeugung in physikalischen Prozessen sind. Für das Verlangen der Erzeugung des spontanen Antiteilchens im Zustand, der einem überkritischen Resonanzen entspricht, sollen eher schnelle als langsame Prozesse, mit möglichst langer Verzögerung der überkritischen Phase, betrachtet werden. In dem zweiten Teil der Arbeit konzentrieren wir uns auf der Klasse der spherisch symmetrischen Potentialtöpfen mit zeitabhängiger Tiefe. Zuerst lösen wir die Dirac Gleichung und analysieren die Struktur und Verhalten der gebundenen Zuständen und das Auftreten von überkritikalität. Dann, durch analytische Fortsetzung, finden und diskutieren wir das Verhalten der Resonanzen in überkritischen Potentialen. Als nächstes leiten wir ab und lösen numerisch (mit Einführung von nicht gleichmäßiger Kontinuum-Diskretisierung für eine konsistente Behandlung von schmalen Peaks) ein System von Differentialgleichungen (gekoppelte Kanäle). Wir bekommen die Teilchen- und Antiteilchenproduktion Spektra für verschiedene zeitabhängige Prozesse wie plötzliches, schnelles, langsames Ein- und Ausschalten von unter- und überkritischen Potentialen. Wir diskutieren, wie und unter welchen Bedingungen ein überkritischer Resonanz während der Evolution zerfällt und zur spontanen Produktion eines Antiteilchens führt. Wir vergleichen das Antiteilchen Produktion Spektrum mit der Form des Resonanzen im überkritischen Potential. Wir studieren Prozesse, wo die überkritischen Potentiale mit unterschiedlichem Tempo Eingeschaltet werden und eventuell in der überkritischen Phase gefroren werden. Wir beweisen, in übereinstimmung mit den Folgerungen aus dem theoretischen Teil, dass der Peak (Wellenpacket) im negativen Kontinuum, der den eingetauchten gebundenen Zustand darstellt, teilweise den bewegenden Resonanz folgt und teilweise in jeder Phase seiner Evolution zerfällt. Der kontinuierliche Zerfall ist intensiver in langsamen Prozessen, während in schnellen Prozessen das Wellenpacket genauer den Resonanz folgt. Im adiabatischen Limes findet der gesamte Zerfall schon am Rande des Kontinuums statt, was zur Produktion von Antiteilchen mit verschwindendem Impuls führt. Dagegen, in schnellen Einschalt-Prozessen mit einer Verzögerung in der überkritischen Phase, stimmt das Spektrum der Erzeugten Antiteilchen mit der Form des Resonanzen am besten überein. Endlich betrachten wir die Frage, wieviel Information über das zeitabhängige Potential kann von den durch die Teilchenerzeugung Spektrum dargestellten Streudaten zurück gewonnen werden. Wir schlagen eine einfache Näherungsmethode vor, die auf dem exponentiellen, dekohärenten Zerfall der zeitabhängigen Resonanzen basiert, mit der wir die Teilchenerzeugung Spektren vorausberechnen und eine gute übereinstimmung mit den Ergebnissen der vollen numerischen Rechnungen bekommen. Zusätzlich diskutieren wir verschiedene Quellen von numerischen Fehlern, finden Abschätzungen für sie und beweisen Konvergenz der benutzten numerischen Methoden