Linear Delay Research Articles

A general decay criterion for a viscoelastic equation with nonlinear distributed delay and damping effects

In this paper, a viscoelastic equation with nonlinear distributed delay and nonlinear damping effects is considered. Under very general conditions that do not involve differential inequalities for the kernel function, a general decay criterion is established through the perturbed energy method and properties of convex functions. We extend the stability result of wave equation with linear delay and linear damping to the nonlinear case.

Relative Controllability of Impulsive Linear Discrete Delay Systems

Relative Controllability of Impulsive Linear Discrete Delay Systems

A numerical method for ℋ2$$ {\mathscr{H}}_2 $$ control of linear delay systems

SummaryThis paper is concerned with the output feedback stabilization of linear delay systems with exogenous disturbances. Based on the norm, an optimization problem is formulated for the suppression of exogenous disturbances in the linear delay systems. The effectiveness of the presented approach is demonstrated via several numerical examples.

Conditions of stability and guaranteed convergence rate for linear time-varying discrete-time delay systems

This work introduces novel results on the stability of linear time-varying discrete-time delay systems, addressing both internal stability and input-to-state stability. We start from the special case of positive delay systems and provide conditions of global exponential stability, both delay-dependent and delay-independent. Moreover, we formulate an explicit bound on the exponential decay rate of the state evolution, as a function of the maximum delay. We then show that the aforementioned stability conditions also prove input-to-state stability. The stability conditions consist of simple linear inequalities, albeit infinite-many due to the time-varying nature of the system. To overcome the computational burden, we also introduce a simple relaxation that allows to check the stability with a finite test, at the expense of conservatism. Finally, leveraging comparison systems and results from the theory of non-negative matrices, we generalize all the aforementioned results to systems with no sign constraint, for which we still provide stability conditions by means of linear inequalities and guaranteed convergence rate. Some consequences and a comparative example are provided for the well investigated special case of systems with constant matrices and time-varying delays.

The Lambert function method in qualitative analysis of fractional delay differential equations

We discuss an analytical method for qualitative investigations of linear fractional delay differential equations. This method originates from the Lambert function technique that is traditionally used in stability analysis of ordinary delay differential equations. Contrary to the existing results based on such a technique, we show that the method can result into fully explicit stability criteria for a linear fractional delay differential equation, supported by a precise description of its asymptotics. As a by-product of our investigations, we also state alternate proofs of some classical assertions that are given in a more lucid form compared to the existing proofs.

Oscillations analysis for two kinds of Nicholson models

Oscillations of two kinds of delay Nicholson models with density constraint are mainly considered. The nonlinear models are transformed into the corresponding linear delay differential equation by the theory of oscillation. The difference equations are obtained by using the linear θ−method. Oscillations conditions of the analytical solution and numerical solution of models are obtained. The properties of non-oscillatory solutions are investigated. In order to better explain the results, some numerical examples are given.

Stability Properties for the Delay Integro-Differential Equation

In this paper stability inequalities for the linear nonhomogeneous Volterra delay integro-differential equation (VDIDE) is being established. The particular problems are encountered to show the applicability of the method and to confirm the predicted theoretical analysis.

On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations

ABSTRACT The authors examine the oscillatory behavior of solutions to a class of third order half-linear delay differential equations. The results are obtained by a comparison with first-order linear delay differential equations whose oscillatory characters are known. An example is provided to illustrate the results.

Linear Quadratic Regulator for Delayed Systems Using the Hamiltonian Approach and Exact Closed-Loop Poles for First-Order Systems

Abstract We consider the linear quadratic regulator (LQR) for linear constant-coefficient delay differential equations (DDEs) with multiple delays. The Hamiltonian approach is used instead of an algebraic Riccati partial differential equation. Two coupled DDEs governing the state and control input are derived using the calculus of variations. This coupled system, with infinitely many roots in both left and right half-planes, defines a boundary value problem. Its left half-plane roots are the exact closed-loop poles of the controlled system. These closed-loop poles have not been used to compute the optimal feedback before. Here, the distributed delay kernel that yields exactly those poles is first computed using an eigenfunction expansion. Increasing the number of terms in the truncated expansion yields a highly oscillatory kernel. However, the oscillatory kernel's antiderivative converges to a piecewise smooth function on the delay interval plus a Dirac delta function at zero. Discontinuities in the kernel coincide with discrete delay values in the original DDE. Using this insight, a fitted piecewise polynomial kernel matches the exact closed-loop poles very well. The twofold contribution of the Hamiltonian approach is thus clarity on the form of the feedback kernel as well as the exact closed-loop poles. Subsequently, the fitted piecewise polynomial kernel can be used for a much simpler control calculation. The polynomial coefficients can be fitted by solving a few simultaneous linear equations. Two detailed numerical examples of the LQR for DDEs, one with two delays and one with three delays, show excellent results.

Stability and asymptotics for fractional delay diffusion‐wave equations

The asymptotic stability and long‐time decay rates of solutions to linear Caputo time‐fractional ordinary differential equations (FODEs) with order both and are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of FODEs decay only polynomially, leading to the so‐called Mittag–Leffler stability, which was already extended to linear fractional delay differential equations (FDDEs) with order . However, the study on the long‐time decay rates of the solutions for FDDEs with order is scarce. Hence, this paper is devoted to the asymptotical stability and long‐time decay rates for a class of FDDEs with , that is, fractional delay diffusion‐wave equations. We reformulate the fractional delay diffusion‐wave equations to infinite‐dimensional FDDEs by the eigenvalue decomposition method. Through a detailed analysis for the characteristic equations, a novel necessary and sufficient stability condition is established in a coefficient criterion. Moreover, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long‐time decay rates of the solutions for both linear FDDEs and fractional delay diffusion‐wave equations are . Finally, an example of an application model with feedback control is provided to show the effectiveness of our theoretical results.

A quadband polarization insensitive and angularly steady frequency Selective surface for gain enlargement of multiband antenna

A simply constructed bi-layered compact Frequency Selective Surface (FSS) is reported in this article to attain four distinct stop bands for multiband wireless applications. The compact FSS consists of complimentary oriented repeated cells in top and bottom layers with cell size of 0.135λL × 0.135λL and 0.12λL × 0.12λL respectively where λL refers the lower band edge wavelength. The FSS exhibits bi-polarized characteristic with angular steadiness till 450 in TE and 300 in TM polarization. The FSS, act as a sub-reflector to broaden the antenna gain, is placed beneath a simple co-planar waveguide (CPW) fed quadband antenna having three triangular patches with rectangular slots stacked together. This united ‘antenna-FSS’ structure attained S11 < -10 dB spectrums at 3.1–3.7 GHz (17.65 %), 5.3–6 GHz (12.34 %), 9.7–11.72 GHz (18.86 %) and 17.4–19.15 GHz (9.57 %) as per experimental evaluation. The ‘antenna-FSS’ offers maximum gain value 9.56 dBi with an enlargement of 4.9 dBi, 3.7 dBi, 4.3 dBi and 5 dBi in its four bands. Adequate radiation efficiency, linear transfer function response and smooth group delay contour are also conquered from ‘antenna-FSS’ in its four bands. The reported ‘antenna-FSS’ is appropriate for multiband wireless technologies as it covers WiMAX, WLAN, ISM, terrestrial and satellite spectrums.

Asymptotic Properties of a Class of Systems with Linear Delay

Sufficient conditions for the asymptotic stability of linear systems of differential equations with linear delay are obtained. On the basis of these conditions, some systems of linear differential equations are studied, and one of them is stabilized on an infinite time interval.

Nonlinear phase accumulation for a linear path delay in low coherence fourier transform spectral interferometry

A scheme to measure path delay in spectral interferometry by observing the phase accumulated by the superposed field is presented. Through the interference with an additional reference beam maintained at an out-of-phase condition near zero optical path delay with respect to the sample probe beam, a nonlinearity in measured phase is observed. A detailed simulation is presented, and its experimental proof-of concept demonstration is attempted using the classic low coherence spectral interferometry. The spectral phase is measured from the modulations in the recorded spectral interference using Fourier transform method of fringe analysis. A systematic calibration for setting up the experiment even with unknown parameters like amplitude ratio among the interfering fields, and location of the zero optical path delay is also provided.

Asymptotic Properties of a Class of Systems with Linear Delay

Asymptotic Properties of a Class of Systems with Linear Delay

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Helical Circular Array Configurations for Generation of Orbital Angular Momentum Beams

The vortex electromagnetic waves carrying the orbital angular momentum (OAM) are an increasing subject of study, and the design of a system capable of generating such waves is also of crucial importance. In this letter, a helical circular array (HCA) has been proposed to generate OAM radio beams. The proposed HCA has a more straightforward and more feasible scheme when compared to uniform circular arrays (UCAs) that require additional phase-shifting devices to form inter-element phases. The heights of the proposed HCA elements along the z-axis are exploited to generate the different OAM modes. For the high-order OAM modes, the height of the HCA is made more compact with the help of the proposed helical circular subarray (HCSA) configurations with spiral phase positions providing a linear phase delay with the azimuth angle. In addition, at high mode values, the HCSA structure creates a smoother OAM vortex beam than HCA. The full-wave simulation results show that the proposed HCA scheme can be used to generate vortex electromagnetic waves with a spiral phase wavefront structure.

Accurate Solution for the Pantograph Delay Differential Equation via Laplace Transform

The Pantograph equation is a fundamental mathematical model in the field of delay differential equations. A special case of the Pantograph equation is well known as the Ambartsumian delay equation which has a particular application in Astrophysics. In this paper, the Laplace transform is successfully applied to solve the Pantograph delay equation. The solution is obtained in a closed series form in terms of exponential functions. This closed form reduces to the corresponding solution in the relevant literature for the Ambartsumian delay equation as a special case. In addition, the convergence of the obtained series is proved theoretically and validated graphically. Furthermore, the accuracy of the numerical results are estimated through several computations of the residual errors. It is shown that such residuals tend to zero, even in a huge domain. The obtained results reveal that the Laplace transform is a powerful approach to solve linear delay differential equations, including the Pantograph model.

Vibration resonance and fork bifurcation of under-damped Duffing system with fractional and linear delay terms

Vibration resonance and fork bifurcation of under-damped Duffing system with fractional and linear delay terms

Magnus integrators for linear and quasilinear delay differential equations

A procedure to numerically integrate non-autonomous linear delay differential equations is presented. It is based on the use of a spectral discretization of the delayed part to transform the original problem into a matrix linear ordinary differential equation which is subsequently solved with numerical integrators obtained from the Magnus expansion. The algorithm can be used in the periodic case to get both accurate approximations of the characteristic multipliers and the solution itself. In addition, it can be extended to deal with certain quasilinear delay equations.

The Cauchy Problem for the Delay Differential Equation with Dzhrbashyan – Nersesyan Fractional Derivative

Последние десятилетия количество работ, посвященных исследованию задач для дифференциальных уравнений дробного порядка, заметно растет. Интерес исследователей вызван тем, что количество областей науки, в которых используются уравнения, содержащие дробные производные, варьируется от биологии и медицины до теории управления, инженерии, финансов, а также оптики, физики и так далее. Включение запаздывания в уравнение дробного порядка существенно влияет на ход процесса, описываемого этим уравнением, так как неизвестная функция задается при различных значениях аргумента, что вносит эффект предыстории в уравнение. Поэтому, математические модели, содержащие дробный оператор и запаздывающий аргумент, более точны, чем модели, содержащие производные целого порядка. В данной работе исследуется задача Коши для линейного обыкновенного дифференциального уравнения с запаздывающим аргументом c оператором дробного дифференцирования Джрбашяна – Нерсесяна, обобщающим известные дробные операторы Римана – Лиувилля и Герасимова – Капуто. Результаты работы получены с использованием методов теории целого и дробного исчислений, методов теории дифференциальных уравнений с запаздывающим аргументом, метода специальных функций. В работе доказывается теорема о справедливости аналога формулы Лагранжа. Также доказано, что специальная функция Wγm​​(t), которая, в свою очередь, определяется через обобщенную функцию Миттаг – Леффлера (или функция Прабхакара), удовлетворяет уравнению и условиям, сопряженным исследуемому, и является фундаментальным решением рассматриваемого уравнения. Сформулирована и доказана теорема существования и единственности решения начальной задачи. Решение поставленной задачи выписано в терминах специальной функции Wν​(t). In recent, the number of works devoted to the study of problems for fractional order differential equations is growing noticeably. The interest of researchers is due to the fact that the number of areas of science in which equations containing fractional derivatives are used varies from biology and medicine to control theory, engineering, finance, as well as optics, physics, and so on. The inclusion of delay in the fractional order equation significantly affects the course of the process described by this equation, since the unknown function is given for different values of the argument, which includes a history effect into the equation. Therefore, mathematical models containing a fractional operator and a delay argument are more accurate than models containing integer derivatives. In this paper, we study the Cauchy problem for a linear ordinary delay differential equation with the Dzhrbashyan – Nersesyan fractional differentiation operator, which is generalizing the Riemann – Liouville and Gerasimov – Caputo fractional operators. The results of the work are obtained using the methods of the theory of integer and fractional calculus, methods of the theory of delay differential equations, the method of special functions. In this paper proves a theorem on the validity of an analogue of the Lagrange formula. It is also proved that the special function Wγm​​(t), which is defined in terms of the generalized Mittag-Leffler function (or the Prabhakar function), satisfies the equation and conditions associated with the one under study, and is the fundamental solution of the considered equation. The main result is that the existence and uniqueness theorem to the initial value problem is proved. The solution to the problem is written out in terms of the special function Wν​(t).