Relevant bibliographies by topics / Delay difference equation

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Delay difference equation'

Author: Grafiati
Published: 4 June 2021
Last updated: 2 February 2022

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Journal articles on the topic "Delay difference equation"

1

Tang, X. H., and J. S. Yu. "Oscillation of delay difference equation." Computers & Mathematics with Applications 37, no. 7 (April 1999): 11–20. http://dx.doi.org/10.1016/s0898-1221(99)00083-8.

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Davies, Roy O., and A. J. Ostaszewski. "On a Difference-Delay Equation." Journal of Mathematical Analysis and Applications 247, no. 2 (July 2000): 608–26. http://dx.doi.org/10.1006/jmaa.2000.6893.

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Wang, Chao, Ravi P. Agarwal, and Donal O’Regan. "δ-Almost Periodic Functions and Applications to Dynamic Equations." Mathematics 7, no. 6 (June 9, 2019): 525. http://dx.doi.org/10.3390/math7060525.

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In this paper, by employing matched spaces for time scales, we introduce a δ -almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of admitting exponential dichotomy for the homogeneous equation, the expression of a δ -almost periodic solution for a type of nonhomogeneous dynamic equation is obtained and the existence of δ -almost periodic solutions for new delay dynamic equations is considered. The results in this paper are valid for delay q-difference equations and delay dynamic equations whose delays may be completely separated from the time scale T .
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Tang, X. H. "Oscillation for nonlinear delay difference equations." Tamkang Journal of Mathematics 32, no. 4 (December 31, 2001): 275–80. http://dx.doi.org/10.5556/j.tkjm.32.2001.342.

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The oscillatory behavior of the first order nonlinear delay difference equation of the form $$ x_{n+1} - x_n + p_n x_{n-k}^{\alpha} = 0, ~~~ n = 0, 1, 2, \ldots ~~~~~~~ \eqno{(*)} $$ is investigated. A necessary and sufficient condition of oscillation for sublinear equation (*) ($ 0 < \alpha < 1 $) and an almost sharp sufficient condition of oscillation for superlinear equation (*) ($ \alpha > 1 $) are obtained.
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Ding, Xiaohua. "Exponential stability of a kind of stochastic delay difference equations." Discrete Dynamics in Nature and Society 2006 (2006): 1–9. http://dx.doi.org/10.1155/ddns/2006/94656.

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We present a Razumilchin-type theorem for stochastic delay difference equation, and use it to investigate the mean square exponential stability of a kind of nonautonomous stochastic difference equation which may also be viewed as an approximation of a nonautonomous stochastic delay integrodifferential equations (SDIDEs), and of a difference equation arises from some of the earliest mathematical models of the macroeconomic “trade cycle” with the environmental noise.
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Dai, Binxiang, and Na Zhang. "Stability and global attractivity for a class of nonlinear delay difference equations." Discrete Dynamics in Nature and Society 2005, no. 3 (2005): 227–34. http://dx.doi.org/10.1155/ddns.2005.227.

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A class of nonlinear delay difference equations are considered and some sufficient conditions for global attractivity of solutions of the equation are obtained. It is shown that the stability properties, both local and global, of the equilibrium of the delay equation can be derived from those of an associated nondelay equation.
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Wei, Zhijian. "Periodicity in a Class of Systems of Delay Difference Equations." Journal of Applied Mathematics 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/735825.

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We study a system of delay difference equations modeling four-dimensional discrete-time delayed neural networks with no internal decay. Such a discrete-time system can be regarded as the discrete analog of a differential equation with piecewise constant argument. By using semicycle analysis method, it is shown that every bounded solution of this discrete-time system is eventually periodic. The obtained results are new, and they complement previously known results.
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Győri, I., G. Ladas, and P. N. Vlahos. "Global attractivity in a delay difference equation." Nonlinear Analysis: Theory, Methods & Applications 17, no. 5 (January 1991): 473–79. http://dx.doi.org/10.1016/0362-546x(91)90142-n.

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Sun, Taixiang, Hongjian Xi, and Mingde Xie. "Global stability for a delay difference equation." Journal of Applied Mathematics and Computing 29, no. 1-2 (September 3, 2008): 367–72. http://dx.doi.org/10.1007/s12190-008-0137-1.

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Ashyralyev, A., K. Turk, and D. Agirseven. "On the stable difference scheme for the time delay telegraph equation." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 99, no. 3 (September 30, 2020): 105–19. http://dx.doi.org/10.31489/2020m3/105-119.

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The stable difference scheme for the approximate solution of the initial boundary value problem for the telegraph equation with time delay in a Hilbert space is presented. The main theorem on stability of the difference scheme is established. In applications, stability estimates for the solution of difference schemes for the two type of the time delay telegraph equations are obtained. As a test problem, one-dimensional delay telegraph equation with nonlocal boundary conditions is considered. Numerical results are provided.
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Dissertations / Theses on the topic "Delay difference equation"

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Jánský, Jiří. "Delay Difference Equations and Their Applications." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2010. http://www.nusl.cz/ntk/nusl-233892.

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Disertační práce se zabývá vyšetřováním kvalitativních vlastností diferenčních rovnic se zpožděním, které vznikly diskretizací příslušných diferenciálních rovnic se zpožděním pomocí tzv. $\Theta$-metody. Cílem je analyzovat asymptotické vlastnosti numerického řešení těchto rovnic a formulovat jeho horní odhady. Studována je rovněž stabilita vybraných numerických diskretizací. Práce obsahuje také srovnání s dosud známými výsledky a několik příkladů ilustrujících hlavní dosažené výsledky.
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Dvořáková, Stanislava. "The Qualitative and Numerical Analysis of Nonlinear Delay Differential Equations." Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2011. http://www.nusl.cz/ntk/nusl-233952.

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Disertační práce formuluje asymptotické odhady řešení tzv. sublineárních a superlineárních diferenciálních rovnic se zpožděním. V těchto odhadech vystupuje řešení pomocných funkcionálních rovnic a nerovností. Dále práce pojednává o kvalitativních vlastnostech diferenčních rovnic se zpožděním, které vznikly diskretizací studovaných diferenciálních rovnic. Pozornost je věnována souvislostem asympotického chování řešení rovnic ve spojitém a diskrétním tvaru, a to v obecném i v konkrétních případech. Studována je rovněž stabilita numerické diskretizace vycházející z $\theta$-metody. Práce obsahuje několik příkladů ilustrujících dosažené výsledky.
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Morávková, Blanka. "Reprezentace řešení lineárních diskrétních systémů se zpožděním." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2014. http://www.nusl.cz/ntk/nusl-233649.

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Disertační práce se zabývá lineárními diskrétními systémy s konstantními maticemi a s jedním nebo dvěma zpožděními. Hlavním cílem je odvodit vzorce analyticky popisující řešení počátečních úloh. K tomu jsou definovány speciální maticové funkce zvané diskrétní maticové zpožděné exponenciály a je dokázána jejich základní vlastnost. Tyto speciální maticové funkce jsou základem analytických vzorců reprezentujících řešení počáteční úlohy. Nejprve je uvažována počáteční úloha s impulsy, které působí na řešení v některých předepsaných bodech, a jsou odvozeny vzorce popisující řešení této úlohy. V další části disertační práce jsou definovány dvě různé diskrétní maticové zpožděné exponenciály pro dvě zpoždění a jsou dokázány jejich základní vlastnosti. Tyto diskrétní maticové zpožděné exponenciály nám dávají možnost najít reprezentaci řešení lineárních systémů se dvěma zpožděními. Tato řešení jsou konstruována v poslední kapitole disertační práce, kde je řešení tohoto problému dáno pomocí dvou různých vzorců.
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Bou, Saba David. "Analyse et commande modulaires de réseaux de lois de bilan en dimension infinie." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSEI084/document.

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Les réseaux de lois de bilan sont définis par l'interconnexion, via des conditions aux bords, de modules élémentaires individuellement caractérisés par la conservation de certaines quantités. Des applications industrielles se trouvent dans les réseaux de lignes de transmission électriques (réseaux HVDC), hydrauliques et pneumatiques (réseaux de distribution du gaz, de l'eau et du fuel). La thèse se concentre sur l'analyse modulaire et la commande au bord d'une ligne élémentaire représentée par un système de lois de bilan en dimension infinie, où la dynamique de la ligne est prise en considération au moyen d'équations aux dérivées partielles hyperboliques linéaires du premier ordre et couplées deux à deux. Cette dynamique permet de modéliser d'une manière rigoureuse les phénomènes de transport et les vitesses finies de propagation, aspects normalement négligés dans le régime transitoire. Les développements de ces travaux sont des outils d'analyse qui testent la stabilité du système, et de commande au bord pour la stabilisation autour d'un point d'équilibre. Dans la partie analyse, nous considérons un système de lois de bilan avec des couplages statiques aux bords et anti-diagonaux à l’intérieur du domaine. Nous proposons des conditions suffisantes de stabilité, tant explicites en termes des coefficients du système, que numériques par la construction d'un algorithme. La méthode se base sur la reformulation du problème en une analyse, dans le domaine fréquentiel, d'un système à retard obtenu en appliquant une transformation backstepping au système de départ. Dans le travail de stabilisation, un couplage avec des dynamiques décrites par des équations différentielles ordinaires (EDO) aux deux bords des EDP est considéré. Nous développons une transformation backstepping (bornée et inversible) et une loi de commande qui, à la fois stabilise les EDP à l'intérieur du domaine et la dynamique des EDO, et élimine les couplages qui peuvent potentiellement mener à l’instabilité. L'efficacité de la loi de commande est illustrée par une simulation numérique
Networks of balance laws are defined by the interconnection, via boundary conditions, of elementary modules individually characterized by the conservation of physical quantities. Industrial applications of such networks can be found in electric (HVDC networks), hydraulic and pneumatic (gas, water and oil distribution) transmission lines. The thesis is focused on modular analysis and boundary control of an elementary line represented by a system of balance laws in infinite dimension, where the dynamics of the line is taken into consideration by means of first order two by two coupled linear hyperbolic partial differential equations. This representation allows to rigorously model the transport phenomena and finite propagation speed, aspects usually neglected in transient regime. The developments of this work are analysis tools that test the stability, as well as boundary control for the stabilization around an equilibrium point. In the analysis section, we consider a system of balance laws with static boundary conditions and anti-diagonal in-domain couplings. We propose sufficient stability conditions, explicit in terms of the system coefficients, and numerical by constructing an algorithm. The method is based on reformulating the analysis problem as an analysis of a delay system in the frequency domain, obtained by applying a backstepping transform to the original system. In the stabilization work, couplings with dynamic boundary conditions, described by ordinary differential equations (ODE), at both boundaries of the PDEs are considered. We develop a backstepping (bounded and invertible) transform and a control law that at the same time, stabilizes the PDEs inside the domain and the ODE dynamics, and eliminates the couplings that are a potential source of instability. The effectiveness of the control law is illustrated by a numerical simulation
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Mensour, Boualem. "Dynamical invariants, multistability, controllability and synchronization in delay-differential and difference equations." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ28360.pdf.

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Smith, Dale T. "Expotential decay of resolvents of banded matrices and asymptotics of solutions of linear difference equations." Diss., Georgia Institute of Technology, 1990. http://hdl.handle.net/1853/29218.

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Foley, Dawn Christine. "Applications of State space realization of nonlinear input/output difference equations." Thesis, Georgia Institute of Technology, 1999. http://hdl.handle.net/1853/16818.

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Kemajou, Elisabeth. "A Stochastic Delay Model for Pricing Corporate Liabilities." OpenSIUC, 2012. https://opensiuc.lib.siu.edu/dissertations/547.

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We suppose that the price of a firm follows a nonlinear stochastic delay differential equation. We also assume that any claim whose value depends on firm value and time follows a nonlinear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a random partial differential equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt. We then analyze the risk structure of a levered firm. We also evaluate loan guarantees in the presence of more than one debt. Furthermore, we perform numerical simulations for specific companies and compare our results with existing models.
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Thai, Son Doan. "Lyapunov Exponents for Random Dynamical Systems." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-25314.

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In this thesis the Lyapunov exponents of random dynamical systems are presented and investigated. The main results are: 1. In the space of all unbounded linear cocycles satisfying a certain integrability condition, we construct an open set of linear cocycles have simple Lyapunov spectrum and no exponential separation. Thus, unlike the bounded case, the exponential separation property is nongeneric in the space of unbounded cocycles. 2. The multiplicative ergodic theorem is established for random difference equations as well as random differential equations with random delay. 3. We provide a computational method for computing an invariant measure for infinite iterated functions systems as well as the Lyapunov exponents of products of random matrices
In den vorliegenden Arbeit werden Lyapunov-Exponented für zufällige dynamische Systeme untersucht. Die Hauptresultate sind: 1. Im Raum aller unbeschränkten linearen Kozyklen, die eine gewisse Integrabilitätsbedingung erfüllen, konstruieren wir eine offene Menge linearer Kyzyklen, die einfaches Lyapunov-Spektrum besitzen und nicht exponentiell separiert sind. Im Gegensatz zum beschränkten Fall ist die Eingenschaft der exponentiellen Separiertheit nicht generisch in Raum der unbeschränkten Kozyklen. 2. Sowohl für zufällige Differenzengleichungen, als auch für zufällige Differentialgleichungen, mit zufälligem Delay wird ein multiplikatives Ergodentheorem bewiesen. 3.Eine algorithmisch implementierbare Methode wird entwickelt zur Berechnung von invarianten Maßen für unendliche iterierte Funktionensysteme und zur Berechnung von Lyapunov-Exponenten für Produkte von zufälligen Matrizen
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Stankovic, Nikola. "Set-based control methods for systems affected by time-varying delay." Thesis, Supélec, 2013. http://www.theses.fr/2013SUPL0025/document.

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On considère la synthèse de la commande basée sur un asservissement affecté par des retards. L’approche utilisée repose sur des méthodes ensemblistes. Une partie de cette thèse est consacrée à une conception de commande active pour la compensation des retards qui apparaissent dans des canaux de communication entre le capteur et correcteur. Ce problème est considéré dans une perspective générale du cadre de commande tolérante aux défauts où des retards variés sont vus comme un mode particulier de dégradation du capteur. Le cas avec transmission de mesure retardée pour des systèmes avec des capteurs redondants est également examiné. Par conséquent, un cadre unifié est proposé afin de régler le problème de commande basé sur la transmission des mesures avec retard qui peuvent également être fournies par des capteurs qui sont affectés par des défauts soudains.Dans la deuxième partie le concept d’invariance positive pour des systèmes linéaires à retard à temps discret est exposé. En ce qui concerne l’invariance pour cette classe des systèmes dynamiques, il existe deux idées principales. La première approche repose sur la réécriture d’un tel système dans l’espace d’état augmenté et de le considérer comme un système linéaire. D’autre part, la seconde approche considère l’invariance dans l’espace d’état initial. Cependant, la caractérisation d’un tel ensemble invariant est encore une question ouverte, même pour le cas linéaire. Par conséquent, l’objectif de cette thèse est d’introduire une notion générale d’invariance positive pour des systèmes linéaires à retard à temps discret. Également, certains nouveaux éclairages sur l’existence et la construction pour les ensembles invariants positifs robustes sont détaillés. En outre, les nouveaux concepts d’invariance alternatives sont décrits
We considered the process regulation which is based on feedback affected by varying delays. Proposed approach relies on set-based control methods. One part of the thesis examines active control design for compensation of delays in sensor-to controller communication channel. This problem is regarded in a general perspective of the fault tolerant control where delays are considered as a particular degradation mode of the sensor. Obtained results are also adapted to the systems with redundant sensing elements that are prone to abrupt faults. In this sense, an unified framework is proposed in order to address the control design with outdated measurements provided by unreliable sensors.Positive invariance for linear discrete-time systems with delays is outlined in the second part of the thesis. Concerning this class of dynamics, there are two main approaches which define positive invariance. The first one relies on rewriting a delay-difference equation in the augmented state-space and applying standard analysis and control design tools for the linear systems. The second approach considers invariance in the initial state-space. However, the initial state-space characterization is still an open problem even for the linear case and it represents our main subject of interest. As a contribution, we provide new insights on the existence of the positively invariant sets in the initial state-space. Moreover, a construction algorithm for the minimal robust D-invariant set is outlined. Additionally, alternative invariance concepts are discussed
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Books on the topic "Delay difference equation"

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Zhang, B. G. Qualitative analysis of delay partial difference equations. Cairo, Egypt: Hindawi Pub. Co., 2007.

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Hartung, Ferenc, and Mihály Pituk, eds. Recent Advances in Delay Differential and Difference Equations. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08251-6.

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Oscillation Theory of Delay Differential and Difference Equations: Second and Third Orders. Saarbrücken: VDM Verlag Dr. Müller, 2010.

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Recent Advances in Delay Differential and Difference Equations. Springer, 2014.

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Hartung, Ferenc, and Mihály Pituk. Recent Advances in Delay Differential and Difference Equations. Springer, 2016.

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Zhu, Yang, and Miroslav Krstic. Delay-Adaptive Linear Control. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691202549.001.0001.

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Actuator and sensor delays are among the most common dynamic phenomena in engineering practice, and when disregarded, they render controlled systems unstable. Over the past sixty years, predictor feedback has been a key tool for compensating such delays, but conventional predictor feedback algorithms assume that the delays and other parameters of a given system are known. When incorrect parameter values are used in the predictor, the resulting controller may be as destabilizing as without the delay compensation. This book develops adaptive predictor feedback algorithms equipped with online estimators of unknown delays and other parameters. Such estimators are designed as nonlinear differential equations, which dynamically adjust the parameters of the predictor. The design and analysis of the adaptive predictors involves a Lyapunov stability study of systems whose dimension is infinite, because of the delays, and nonlinear, because of the parameter estimators. This book solves adaptive delay compensation problems for systems with single and multiple inputs/outputs, unknown and distinct delays in different input channels, unknown delay kernels, unknown plant parameters, unmeasurable finite-dimensional plant states, and unmeasurable infinite-dimensional actuator states. Presenting breakthroughs in adaptive control and control of delay systems, the book offers powerful new tools for the control engineer and the mathematician.
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Book chapters on the topic "Delay difference equation"

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Kashchenko, Ilia. "Asymptotics of an Equation with Large State-Dependent Delay." In Differential and Difference Equations with Applications, 339–46. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56323-3_26.

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Wei, Gengping. "On a Linear Delay Partial Difference Equation with Impulses." In Difference Equations, Discrete Dynamical Systems and Applications, 145–52. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-24747-2_11.

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Tashirova, Ekaterina. "The Numerical Solution of Wave Equation with Delay for the Case of Variable Velocity Coefficient." In Differential and Difference Equations with Applications, 181–93. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56323-3_15.

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Lekomtsev, Andrei. "The Method of Fractional Steps for the Numerical Solution of a Multidimensional Heat Conduction Equation with Delay for the Case of Variable Coefficient of Heat Conductivity." In Differential and Difference Equations with Applications, 105–21. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-56323-3_9.

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Li, Hongfei, and Keqin Gu. "Lyapunov-Krasovskii Functional Approach for Coupled Differential-Difference Equations with Multiple Delays." In Delay Differential Equations, 1–30. Boston, MA: Springer US, 2009. http://dx.doi.org/10.1007/978-0-387-85595-0_1.

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Ashyralyev, Allaberen, and Pavel E. Sobolevskii. "Appendix: Delay Parabolic Differential Equations." In New Difference Schemes for Partial Differential Equations, 393–410. Basel: Birkhäuser Basel, 2004. http://dx.doi.org/10.1007/978-3-0348-7922-4_8.

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Agarwal, Ravi P., and Patricia J. Y. Wong. "Oscillation for Second Order Neutral Delay Difference Equations." In Advanced Topics in Difference Equations, 219–27. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8899-7_20.

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Agarwal, Ravi P., and Patricia J. Y. Wong. "Oscillation for Higher Order Neutral Delay Difference Equations." In Advanced Topics in Difference Equations, 233–41. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8899-7_22.

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Agarwal, Ravi P., Donal O’Regan, and Patricia J. Y. Wong. "Delay Boundary Value Problems." In Positive Solutions of Differential, Difference and Integral Equations, 110–18. Dordrecht: Springer Netherlands, 1999. http://dx.doi.org/10.1007/978-94-015-9171-3_10.

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Agarwal, Ravi P., and Patricia J. Y. Wong. "Oscillation for Second Order Neutral Delay Difference Equations (Contd.)." In Advanced Topics in Difference Equations, 227–33. Dordrecht: Springer Netherlands, 1997. http://dx.doi.org/10.1007/978-94-015-8899-7_21.

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Conference papers on the topic "Delay difference equation"

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Hamaya, Yoshihiro. "Asymptotic Constancy to the Neutral Delay Difference Equation." In Proceedings of the Third International Conference on Difference Equations. Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742: CRC Press, 2017. http://dx.doi.org/10.4324/9780203745854-16.

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Li, Zongcheng, Di Liang, and Qingli Zhao. "Chaotic Behavior in a Delay Difference Equation." In 2012 5th International Workshop on Chaos-Fractals Theories and Applications (IWCFTA). IEEE, 2012. http://dx.doi.org/10.1109/iwcfta.2012.23.

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Gorbova, Tatiana, and Svyatoslav Solodushkin. "Nonlinear difference scheme for fractional equation with functional delay." In PROCEEDINGS OF THE X ALL-RUSSIAN CONFERENCE “Actual Problems of Applied Mathematics and Mechanics” with International Participation, Dedicated to the Memory of Academician A.F. Sidorov and 100th Anniversary of UrFU: AFSID-2020. AIP Publishing, 2020. http://dx.doi.org/10.1063/5.0035580.

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Ratchagit, Manlika, Benchawan Wiwatanapataphee, and Darfiana Nur. "On Parameter Estimation of Stochastic Delay Difference Equation using the Two $m$-delay Autoregressive Coefficients." In 2020 3rd International Seminar on Research of Information Technology and Intelligent Systems (ISRITI). IEEE, 2020. http://dx.doi.org/10.1109/isriti51436.2020.9315414.

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Fangfang Jiang and Jun Yang. "Frequent oscillatory solutions of a neutral partial difference equation with variable delay." In 2011 International Conference on Multimedia Technology (ICMT). IEEE, 2011. http://dx.doi.org/10.1109/icmt.2011.6002567.

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6

Kalma´r-Nagy, Tama´s. "A New Look at the Stability Analysis of Delay-Differential Equations." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84740.

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It is shown that the method of steps for linear delay-differential equations combined with the Laplace-transform can be used to determine the stability of the equation. The result of the method is an infinite dimensional difference equation whose stability corresponds to that of the transcendental characteristic equation. Truncations of this difference equation are used to construct numerical stability charts. The method is demonstrated on a first and second order delay equation. Correspondence between the transcendental characteristic equation and the difference equation is proved for the first order case.
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Liang, Jinsong, Weiwei Zhang, YangQuan Chen, and Igor Podlubny. "Robustness of Boundary Control of Fractional Wave Equations With Delayed Boundary Measurement Using Fractional Order Controller and the Smith Predictor." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85299.

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In this paper, we analyze the robustness of the fractional wave equation with a fractional order boundary controller subject to delayed boundary measurement. Conditions are given to guarantee stability when the delay is small. For large delays, the Smith predictor is applied to solve the instability problem and the scheme is proved to be robust against a small difference between the assumed delay and the actual delay. The analysis shows that fractional order controllers are better than integer order controllers in the robustness against delays in the boundary measurement.
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Li, Hongfei, and Keqin Gu. "Discretized Lyapunov-Krasovskii Functional for Systems With Multiple Delay Channels." In ASME 2008 Dynamic Systems and Control Conference. ASMEDC, 2008. http://dx.doi.org/10.1115/dscc2008-2282.

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Many practical systems have a large number of state variables but only a few components have time delays. These delay components are often scalar or low dimensional, and involve single time delay in each component. A coupled differential-difference equation is well suited to formulate such systems. It is known that such a formulation is very general. Systems with multiple related or independent delays can be transformed into this standard form. Similar to regular time-delay systems, the existence of a quadratic Lyapunov-Krasovkii functional is necessary and sufficient for stability. This article discusses the discretization of such a quadratic Lyapunov-Krasovskii functional. Even for time-delay systems of retarded type, the formulation has significant advantage over the traditional formulation, as the size of the resulting linear matrix inequalities are drastically reduced for such systems. Indeed, the computational effort needed for checking stability of such a large system with a few low dimensional delays is quite reasonable.
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Butcher, Eric A., and Oleg A. Bobrenkov. "The Chebyshev Spectral Continuous Time Approximation for Periodic Delay Differential Equations." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86641.

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In this paper, the approximation technique proposed in [1] for converting a system of constant-coefficient delay differential equations (DDEs) into a system of ordinary differential equations (ODEs) using pseudospectral differencing is applied to both constant and periodic systems of DDEs. Specifically, the use of Chebyshev spectral collocation is proposed in order to obtain the “spectral accuracy” convergence behavior shown in [1]. The proposed technique is used to study the stability of first and second order constant coefficient DDEs with one or two fixed delays with or without cubic nonlinearity and parametric sinusoidal excitation, as well as of the delayed Mathieu’s equation. In all the examples, the results of the approximation by the proposed method show good agreement with either analytical results, or the results obtained before by other reliable approximation methods. In particular, the greater accuracy and convergence properties of this method compared to the finite difference-based continuous time approximation proposed recently in [2] is shown.
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Esteban Villegas, Helio S., Carlos Borrás Pinilla, and Nejat Olgac. "Delay Scheduling of a LQR and PID Controlled Pendubot Using CTCR Method." In ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/imece2020-24273.

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Abstract Delays are a common physic effect that is present in a huge quantity of industrial systems with feedback control. Sometimes the impact of the delay presence in a system is neglected without any difference in the performance of the controller. But in some cases, the Delay quantity reaches levels that increase the overshoot significantly or destabilizes the system. For this system the CTCR method can be used to design a “delayed scheduled” controller able to reject the delays effect. To be able to apply this method it is necessary for both the system and the controller to be Linear or linearized. In this article the study case is an articulated inverted pendulum or also called “Pendubot”. This configuration of pendulum was selected because, in spite of being a simple pendulum, it has four equilibrium points, where two equilibrium points were selected to be controlled, the most unstable which has the maximum potential energy for each link and the most stable with the lowest potential energy. The purpose of work with those equilibrium points is compare the difference between the delay rejection in each point with similar setting times and overshoots from controllers. In order to get an accurate model, the state of the current is added along with the viscous friction terms for each joint in the pendulum. To tune the linear and non-linear model an experimental validation was carry on the physical prototype from the Universidad Industrial de Santander (VIE-5373 UIS). The control laws applied were a classical PID and LQR control. Due the controller sample frequency is extremely high in comparison to the states response, all the controllers and linear models were implemented in continuous space. In the first tuning process it was observed that the PID control gets a significantly lowest performance than the LQR control in the unstable equilibrium point, for that reason the comparison between points only was carried on with the LQR equation of feedback system, this can be done in different ways, but for the PID control the Mason theorems was applied and for LQR control only with matrix operation the equation can be obtained. After applying the “Delay scheduling” it was observed that the tuned LQR get a highest delay rejection that PID. An observer fact was that although the controllers for each point have a similar performance the delay pockets have completely different values due the final poles locations for each point. Also was observed that the system only gets one stability pocket, this could because only one delay in the actuator was induced.
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