Stability Exponent and Eigenvalue Abscissas by Way of the Imaginary Axis Eigenvalues

Existence Results for Differential Delay Equations, I

The reasons for and the effects of incorporating time lags into the differential equations of chemical kinetics are investigated for some simple model systems. Complex networks of first-order reactions may, in the limit of long times, be described by relatively simple systems of differential delay (differential difference) equations, in which the effects of intermediates are replaced by time lags. Explicit inclusion of time lags affords a more physically realistic description of nucleation in certain solid state reactions. An electrochemical cell with lagged feedback gives rise to a particularly simple and instructive delay equation. A two-chamber system in which molecules diffuse with a time lag through a membrane shows unexpected oscillatory, and seemingly nonconservative behavior, which is explained in terms of the number of molecules in passage through the membrane.

This study aims at seeking a fractional-order equation that is a good approximation for a delay equation. To this end, we consider a delay equation with simple initial and boundary conditions and obtain a fractional-order equation and an associated numerical method for approximating the solution of the delay equation. In order to determine the fractional-order equation that is a better approximation of the Delay equation, the Levenberg-Marquardt iterative method is employed to estimate the optimal parameters in the fractional-order equation. This obtained fractional-order equation is then tested and compared its solution with the solution of the delay equation. Results show that the fractional method is indeed a good approximation for the Delay equation.

Using known Cα-multiplier result, we give necessary and sufficient conditions for the second order delay equations: $$ u''(t) = Au(t) + Fu_t + Gu'_t + f(t), t \in \mathbb{R} $$ to have maximal regularity in Holder continuous function spaces Cα(ℝ, X), where X is a Banach space, A is a closed operator in X, F, G ∊ ℒ(C([−r, 0],X), X) are delay operators for some fixed r > 0.

We give necessary and sufficient conditions of L p -maximal regularity (resp. B -maximal regularity or F -maximal regularity) for the second order delay equations: u″(t) = Au(t) + Gu ′ + Fu t + f(t), t ∈ [0, 2π] with periodic boundary conditions u(0) = u(2π), u′(0) = u′(2π), where A is a closed operator in a Banach space X, F and G are delay operators on L p ([−2π, 0];X) (resp. B ([−2π, 0];X) or F ([−2π, 0];X)).

I Basic Theory.- Basic Theory for Linear Delay Equations.- II Stability and Robust Stability.- Complete Type Lyapunov-Krasovskii Functionals.- Robust Stability Conditio ns of Quasipolynomials by Frequency Sweeping.- Improvements on the Cluster Treatment of Characteristic Roots and the Case Studies.- From Lyapunov-Krasovskii Functionals for Delay-Independent Stability to LMI Conditions for -Ana1ysis.- III Control, Identification, and Observer Design.- Finite Eigenstructure Assignment for Input Delay Systems.- Control of Systems with Input Delay-An Elementary Approach.- On the Stabilization of Systems with Bounded and Delayed Input.- Identifiability and Identification of Linear Systems with Delays.- A Model Matching Solution of Robust Observer Design for Time-Delay Systems.- IV Computation, Software, and Implementation.- Adaptive Integration of Delay Differential Equations.- Software for Stability and Bifurcation Analysis of Delay Differential Equations and Applications to Stabilization.- Empirical Methods for Determining the Stability of Certain Linear Delay Systems.- Stability Exponent and Eigenvalue Abscissas by Way of the Imaginary Axis Eigenvalues.- The Effect of Approximating Distributed Delay Control Laws on Stability.- V Partial Differential Equations, Nonlinear and Neutral Systems.- Synchronization Through Boundary Interaction.- Output Regulation of Nonlinear Neutral Systems.- Robust Stability Analysis of Various Classes of Delay Systems.- On Strong Stability and Stabilizability of Linear Systems of Neutral Type.- Robust Delay Dependent Stability Analysis of Neutral Systems.- VI Applications.- On Delay-Based Linear Models and Robust Control of Cavity Flows.- Active-adaptive Control of Acoustic Resonances in Flows.- Robust Prediction-Dased Control for Unstable Delay Systems.- Robust Stability of Teleoperation Schemes SUbject to Constant and Time-Varying Communication Delays.- Bounded Control of Multiple-Delay Systems with Applications to ATM Networks.- Dynamic Time Delay Models for Load Balancing. Part I: Deterministic Models.- Dynamic Time Delay Models for lA>ad Balancing. Part II: A Stochastic Analysis of the Effect of Delay Uncertainty.- VII Miscellaneous Topics.- Asymptotic Properties of Stochastic Delay Systems.- Stability and Dissipativity Theory for Nonnegative and Compartmental Dynamical Systems with Time Delay.- List of Contributors.

We study the persistence of the asymptotic stability of delay equations both under linear and nonlinear perturbations. Namely, we consider nonautonomous linear delay equations v′ = L(t)vt with a nonuniform exponential contraction. Our main objective is to establish the persistence of the nonuniform exponential stability of the zero solution both under nonautonomous linear perturbations, i.e., for the equation v′ = (L(t) + M(t))vt, thus discussing the so-called robustness problem, and under a large class of nonlinear perturbations, namely for the equation v′ = L(t)vt + f(t, vt). In addition, we consider general contractions e−λρ(t) determined by an increasing function ρ that includes the usual exponential behavior with ρ(t) = t as a very special case. We also obtain corresponding results in the case of discrete time.

In the first part of this chapter we recall the notion of a characteristic matrix function for classes of operators as introduced in Kaashoek and Verduyn Lunel (2023). The characteristic matrix function completely describes the spectral properties of the corresponding operator. In the second part we show that the period map or monodromy operator associated with a periodic neutral delay equation has a characteristic matrix function. We end this chapter with a number of illustrative examples of periodic neutral delay equations for which we can compute the characteristic matrix function explicitly.

We prove a basic result which relates the structure of the spectrum to the interior of the numerical range.Using this result we derive corollaries concerning compact operators, quasinilpotents, and finite dimensional operators.In particular, we characterize finite dimensional convexoid operators.

We prove a basic result which relates the structure of the spectrum to the interior of the numerical range. Using this result we derive corollaries concerning compact operators, quasi- nilpotents, and finite dimensional operators. In particular, we characterize finite dimensional convexoid operators. 1. Introduction. In this paper will mean a bounded linear transformation of the complex Hilbert space 77 into itself. The thrust of the conclusions that we obtain here is to show that the numerical range of an operator T can be described provided that T—zI has closed range. For a general discussion of numerical range see Chapter 17 of (4). 2. Preliminaries. For an isolated eigenvalue z there are two different notions of eigenspace ; the geometric eigenspace is just the kernel of T—zI (which we simply write as T—z). The algebraic eigenspace associated with z is the range of an idempotent P defined by a contour integral according to the Banach space operational calculus (see pp. 178-181 of (5), for example). If the underlying Hilbert space is H then both F77 and (I—P)H are in- variant under F, and the restriction of T—z to F77 (which we denote T—zjPH) is quasi-nilpotent. We shall say that an eigenvalue is a normal eigenvalue if the corresponding geometric eigenspace reduces F; if a normal eigenvalue is an isolated eigenvalue and the geometric multi- plicity agrees with the algebraic multiplicity then we say that it is a normal-isolated eigenvalue. Clearly an isolated eigenvalue for a normal operator is a normal-isolated eigenvalue. It will be convenient to denote by W(T) the numerical range of F, i.e. {(Tfif): ||/|| = 1}, and the closure of the numerical range is denoted by W(T)~. Finally, an operator is convexoid provided that W(T)~ is the convex hull of the spectrum of T, denoted conv a(T). Note that it follows from Theorem 1.24 on p. 16 of (8) that every point of conv a(T) can be

From several points of view it is of advantage to know the properties of the collision operators in kinetic equations, e.g. in the well known Boltzmann equation, in particular for the purpose of solving eigenvalue problems. With regard to elastic, exciting and deexciting processes some attemps were recently made to investigate such operators in the Boltzmann equation describing the behaviour of electrons in weakly ionized plasmas. In the following we will prove that in the case of a finite dimensional inscattering operator the eigenvalues and the corresponding eigenfunctions can be represented explicitly. Finite dimensional operators were used successfully in special models to approximate the inscattering operators; they possess the property of compactness and are well suitable for analytical or numerical calculations. The representation has been obtained by solving an adequate linear equation system. The generalized eigenfunctions correspond to the normal solutions used by Case in the neutron transport theory. The regularization of the singular integrals which are necessary to obtain this solution will be given in detail. Further a velocity dependence of the collision frequency which need not be monotonous in the considered case and the dependence on the direction could be included.

Modelling Dependency of Volatility on Sampling Frequency via Delay Equations

Many times a scientist is choosing a programming language or a software for a specific purpose. For the field of scientific computing, the methods for solving differential equations are one of the important areas. What I would like to do is take the time to compare and contrast between the most popular offerings. This is a good way to reflect upon what's available and find out where there is room for improvement. I hope that by giving you the details for how each suite was put together (and the "why", as gathered from software publications) you can come to your own conclusion as to which suites are right for you.

A new non-oscillatory theory is presented for higher order delay and neutral equations. The results include both singular and nonsingular problems.

For control systems where the closed-loop system description is governed by linear delay-differential equations of neutral type, it is known that stability may be fragile, in the sense of sensitive to infinitesimal perturbations to parameters in the system model or arbitrarily small errors in the implementation of the controller. A natural approach to resolve this problem of ill-posedness and to break down the underlying instability mechanisms, rooted in characteristic roots moving from the left plane to the right one via the point at infinity, consists of including a low-pass filter in the control loop, provided the inclusion preserves stability. Independently of the particular control problem, the addition of a low-pass filter essentially boils down to a “regularization” of delay-difference equations and delay equations of neutral type in terms of parametrized delay equations of retarded type, where the parameter can be interpreted as the inverse of the filter's cut-off frequency. In this paper, the stability properties of these parametrized delay equations are analyzed in a general, multi-delay setting, with focus on the transition to the original delay-difference or neutral equations. It is illustrated that the spectral abscissa may not be continuous at the transition, which may impact stability. Hence, conditions for preservation of stability in terms of a robustified stability indicator called filtered spectral abscissa are presented, for which mathematical characterizations and a computationally tractable expression are provided. The application of a PD controller to a time-delay system with relative degree one is used to motivate the structure of the equations studied throughout the paper, and to explicate the implications of the presented results on control design, discussed in the last section.