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Journal articles on the topic 'Second order delay system'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Polyanin, Andrei, and Alexei Zhurov. "Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions." Mathematics 10, no. 9 (2022): 1529. http://dx.doi.org/10.3390/math10091529.

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The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schr

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2

Chen, Yipeng, Yicheng Liu, and Xiao Wang. "The critical delay of the consensus for a class of multi-agent system involving task strategies." Networks and Heterogeneous Media 18, no. 2 (2023): 513–31. http://dx.doi.org/10.3934/nhm.2023021.

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<abstract><p>The time delay may induce oscillatory behaviour in multi-agent systems, which may destroy the consensus of the system. Therefore, the critical delay that is the maximum value of the delay to guarantee the consensus of the system, is an important performance index of multi-agent systems. This paper studies the influence of the processing delay on the consensus for a class of multi-agent system involving task strategies. The first-order system with a single delay and the second-order system with two different delays are investigated respectively. A critical delay indepen

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3

Guo, Gao Fei, Shun Xiang Wu, and Da Cao. "Transient Responses of Second-Order System." Applied Mechanics and Materials 55-57 (May 2011): 224–28. http://dx.doi.org/10.4028/www.scientific.net/amm.55-57.224.

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This paper analyses the transient response of second-order system through time domain analysis, root locus and frequency domain analysis, meanwhile, studies the influence exerted to the system by the second-order system damping ratio and the coefficient ratio as well as the research and damping ratio associated with the relevant parameters, like delay time, rise time, peak time, overshoot, time regulation, basing on the unit step response, and the stability of the system is studied by root locus. Finally, graphics are built through the application of Matlab in order to have an intuitive unders

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4

Fridman, Leonid, Emilia Fridman, and Eugenii Shustin. "Oscillations in a second-order discontinuous system with delay." Discrete and Continuous Dynamical Systems 9, no. 2 (2002): 339–58. http://dx.doi.org/10.3934/dcds.2003.9.339.

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5

Yu, Yuanhong. "Stability criteria for linear second order delay differential system." Acta Mathematicae Applicatae Sinica 4, no. 2 (1988): 109–12. http://dx.doi.org/10.1007/bf02006058.

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6

Elmadssia, Sami, Karim Saadaoui, Atef Zaguia, Tahar Ezzedine, and Qing-Guo Wang. "Stabilization Domains for Second Order Delay Systems." IEEE Access 9 (2021): 53518–29. http://dx.doi.org/10.1109/access.2021.3070430.

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7

Bae, Sang-Wook, Jeonghyeon Yang, Jaesung Kwon, Sungwoong Choi, and Beomsoo Kim. "Design of stabilizing controller for second-order time-delay system." Journal of the Korean Society of Marine Engineering 43, no. 7 (2019): 545–51. http://dx.doi.org/10.5916/jkosme.2019.43.7.545.

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8

Tunç, Cemil, and Osman Tunç. "On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order." Mathematics 10, no. 22 (2022): 4235. http://dx.doi.org/10.3390/math10224235.

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In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-di

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9

Zhao, Qiwen, Ying He, Yanfang Yang, Huifang Zhang, and Yi Xu. "Second-Order Sidebands and Group Delays in Coupled Optomechanical Cavity System with a Cubic Nonlinear Harmonic Oscillator." Photonics 11, no. 3 (2024): 256. http://dx.doi.org/10.3390/photonics11030256.

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The generation of second-order sidebands and its associated group delay is an important subject in optical storage and switch. In this work, the efficiency of second-order sideband generation in a coupled optomechanical cavity system with a cubic nonlinear harmonic oscillator is theoretically investigated. It is found that the efficiency of second-order sideband generation can be effectively enhanced with the decrease in decay rate of optomechanical cavity, the increase in coupling strength between two cavities and the power of probe field. The slow light effect (i.e., positive group delay) is

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10

Abuasbeh, Kinda, Nazim I. Mahmudov, and Muath Awadalla. "Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems." Mathematics 11, no. 4 (2023): 806. http://dx.doi.org/10.3390/math11040806.

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We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and Ω. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-dela

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11

Liu, Bo, Guangming Xie, Yanping Gao, Jiaxi Wu, Jianguo Zhang, and Wenguang Luo. "Consensus Analysis of Second-Order Multiagent Systems with General Topology and Time Delay." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/359750.

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This paper addresses the consensus of second-order multiagent systems with general topology and time delay based on the nearest neighbor rule. By using the Laplace transform technique, it is proved that the second-order multi-agent system in the presence of time-delay can reach consensus if the network topology contains a globally reachable node and time delay is bounded. The bound of time-delay only depends on eigenvalues of the Laplacian matrix of the system. The main contribution of this paper is that the accurate state of the consensus center and the upper bound of the communication delay

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12

Farkh, Rihem, Kaouther Laabidi, and Mekki Ksouri. "Computation of All Stabilizing PID Gain for Second-Order Delay System." Mathematical Problems in Engineering 2009 (2009): 1–17. http://dx.doi.org/10.1155/2009/212053.

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The problem of stabilizing a second-order delay system using classical proportional-integral-derivative (PID) controller is considered. An extension of the Hermite-Biehler theorem, which is applicable to quasipolynomials, is used to seek the set of complete stabilizing PID parameters. The range of admissible proportional gains is determined in closed form. For each proportional gain, the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid or a triangle.

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13

Kumar, Ankit, Ramesh K. Vats, and Avadhesh Kumar. "Approximate Controllability of Second-order Non-autonomous System with Finite Delay." Journal of Dynamical and Control Systems 26, no. 4 (2020): 611–27. http://dx.doi.org/10.1007/s10883-019-09475-0.

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14

Gang Liu, A. Zinober, and Y. B. Shtessel. "Second-Order SM Approach to SISO Time-Delay System Output Tracking." IEEE Transactions on Industrial Electronics 56, no. 9 (2009): 3638–45. http://dx.doi.org/10.1109/tie.2009.2017089.

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15

Manel, Abdelli, and Bouzettouta Lamine. "Energy decay of one-dimensional thermoelastic bresse system with distributed neutral delay and a second sound." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 1 (2024): 2932–56. http://dx.doi.org/10.54021/seesv5n1-147.

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In this paper we study the exponential stability of a one-dimensional thermoelastic Bresse system, with distributed neutral delay and a second sound, under suitable assumptions on the kernel of neutral delay term. In 1859, Bresse developed the circular arc problem which consists of three wave equations where the main variables, represent the vertical, longitudinal and shear angle displacements respectively. The Bresse system reduces to the well-known Timoshenko system where the longitudinal displacement w is not considered (l = 0), on the other hand, the energy associated with this system rema

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16

Arief Muhammad Luthfi Yanuar, Fuqaha Asnan Said, Reihan Diaz Pramudya, Jilan Ma’rifat Al Faiz, and Tatyantoro Andrasto. "Aplikasi Metode Root-Locus dengan Aproksimasi Padé untuk Analisis Pengaruh Time Delay pada Kestabilan Sistem Kendali." JURAL RISET RUMPUN ILMU TEKNIK 4, no. 2 (2025): 28–40. https://doi.org/10.55606/jurritek.v4i2.5467.

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Control systems with time delays introduce system stability problems because time delays cause exponential effects to the system response. Conventional root-locus methods cannot be used directly on systems with delays due to irrational mathematical forms. This study analyzes the shifting effect of time delay on the stability of linear control systems by using the first-order Padé approach to enable the application of the root-locus method. The system model used is a second-order linear system with a transfer function of , and is analyzed under conditions without delay and with delays of 0.5, 1

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17

Zhou, Yingxia, and Mengmeng Li. "Existence and Uniqueness of Second-Order Impulsive Delay Differential Systems." Axioms 13, no. 12 (2024): 834. http://dx.doi.org/10.3390/axioms13120834.

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In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results.

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18

Moraru, Dumitru. "Tuning method of automatic controllers to object models with second order advance-delay and dead time." Acta et commentationes Ştiinţe Exacte şi ale Naturii 16, no. 2 (2023): 78–88. http://dx.doi.org/10.36120/2587-3644.v16i2.78-88.

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This article explores the application of mathematical models in the design and analysis of automatic control system. By integrating mathematical concepts such as linear algebra, mathematical analysis, the performance and reliability of automatic control systems can be optimized. In the paper, an efficient procedure has been developed for tuning the standardized P, PI, PD, and PID control algorithms to mathematical models of second-order advance-delay with dead time control objects with known parameters, using the maximal stability degree method with iterations. The advantages of the maximum st

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19

Qi, Bin, Xuyang Lou, and Baotong Cui. "Containment control of second-order multi-agent systems with directed topology and time-delays." Kybernetes 43, no. 8 (2014): 1248–61. http://dx.doi.org/10.1108/k-06-2013-0113.

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Purpose – The purpose of this paper is to discuss the impacts of the communication time-delays to the distributed containment control of the second-order multi-agent systems with directed topology. Design/methodology/approach – A basic theoretical analysis is first carried out for the containment control of the second-order multi-agent systems under directed topology without communication time-delay and a sufficient condition is proposed for the achievement of containment control. Based on the above result and frequency-domain analysis method, a sufficient condition is also derived for the ach

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20

Gözen, Melek. "Boundedness of Vector Linéard Equation with Multiple Variable Delays." Mathematics 12, no. 5 (2024): 769. http://dx.doi.org/10.3390/math12050769.

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In this article, we consider a system of ordinary differential equations (ODEs) of second order with two variable time delays. We obtain new conditions for uniform ultimate bounded (UUB) solutions of the considered system. The technique of the proof is based on the Lyapunov–Krasovskii functional (LKF) method using a new LKF. The main result of this article extends and improves a recent result for ODEs of second order with a constant delay to a more general system of ODEs of second order with two variable time delays. In this particular case, we also give a numerical example to verify the appli

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21

Li, Xue, Huai Wu, and Yikang Yang. "Consensus of Second-Order Multiagent Systems with Fixed Topology and Time-Delay." Mathematical Problems in Engineering 2015 (2015): 1–6. http://dx.doi.org/10.1155/2015/138276.

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We are concerned with the consensus problems for networks of second-order agents, where each agent can only access the relative position information from its neighbours. We aim to find the largest tolerable input delay such that the system consensus can be reached. We introduce a protocol with time-delay and fixed topology. A sufficient and necessary condition is given to guarantee the consensus. By using eigenvector-eigenvalue method and frequency domain method, it is proved that the largest tolerable time-delay is only related to the eigenvalues of the graph Laplacian. And simulation results

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22

Karthikeyan, Kulandhivel, Dhatchinamoorthy Tamizharasan, and Dimplekumar N. Chalishajar. "Analysis on Controllability Results for Wellposedness of Impulsive Functional Abstract Second-Order Differential Equation with State-Dependent Delay." Axioms 10, no. 3 (2021): 188. http://dx.doi.org/10.3390/axioms10030188.

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The functional abstract second order impulsive differential equation with state dependent delay is studied in this paper. First, we consider a second order system and use a control to determine the controllability result. Then, using Sadovskii’s fixed point theorem, we get sufficient conditions for the controllability of the proposed system in a Banach space. The major goal of this study is to demonstrate the controllability of an abstract second-order impulsive differential system with a state dependent delay mechanism. The wellposed condition is then defined. Next, we studied whether the def

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23

Xiao, Zhi-Hua, and Yao-Lin Jiang. "Multi-order Arnoldi-based model order reduction of second-order time-delay systems." International Journal of Systems Science 47, no. 12 (2015): 2925–34. http://dx.doi.org/10.1080/00207721.2015.1042087.

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24

Sun, Li, and Dan Ma. "Stabilization of Unstable Second-Order Delay Plants under PID Control: A Nyquist Curve Analysis." Actuators 10, no. 9 (2021): 227. http://dx.doi.org/10.3390/act10090227.

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Time delays arise in various components of control systems, including actuators, sensors, control algorithms, and communication links. If not properly taken into consideration, time delays will degrade the closed-loop performance and may even result in instability. This paper studies the stabilization problem of the second-order delay plants with two unstable real poles. Stabilization conditions under PD and PID control are derived using the Nyquist stability criterion. Algorithms for computing feasible PD and PID parameter regions are proposed. In some special cases, the maximal range of dela

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25

Kandala, Shanti S., Surya Samukham, Thomas K. Uchida, and C. P. Vyasarayani. "Spurious roots of delay differential equations using Galerkin approximations." Journal of Vibration and Control 26, no. 15-16 (2020): 1178–84. http://dx.doi.org/10.1177/1077546319894172.

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The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerki

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26

Sadalla, Talar, Dariusz Horla, Wojciech Giernacki, and Piotr Kozierski. "Influence of time delay on fractional-order PI-controlled system for a second-order oscillatory plant model with time delay." Archives of Electrical Engineering 66, no. 4 (2017): 693–704. http://dx.doi.org/10.1515/aee-2017-0052.

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Abstract The paper aims at presenting the influence of an open-loop time delay on the stability and tracking performance of a second-order open-loop system and continuoustime fractional-order PI controller. The tuning method of this controller is based on Hermite- Biehler and Pontryagin theorems, and the tracking performance is evaluated on the basis of two integral performance indices, namely IAE and ISE. The paper extends the results and methodology presented in previous work of the authors to analysis of the influence of time delay on the closed-loop system taking its destabilizing properti

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27

Wang, Zhi, Xiangtao Ran, Bin Zhao, and Jie Zhao. "GA Tuning PID Controller Based on Second-order Time-delay Industrial System." MATEC Web of Conferences 288 (2019): 01009. http://dx.doi.org/10.1051/matecconf/201928801009.

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For a typical second-order time-delay system model, an intelligent genetic algorithm is used to initially optimize the initial parameters of the PID controller, and a step response curve of the system is obtained, and the performance index is compared with performance indexes obtained by other optimization methods. The results show that the GA optimization method is more robust.

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Levaggi, Laura, and Elisabetta Punta. "Stabilization of a second-order variable-structure system with unknown input delay." IFAC Proceedings Volumes 36, no. 19 (2003): 105–10. http://dx.doi.org/10.1016/s1474-6670(17)33310-4.

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29

Farkh, Rihem, Kaouther Laabidi, and Mekki Ksouri. "Stabilizing Sets of PI/PID Controllers for Unstable Second Order Delay System." International Journal of Automation and Computing 11, no. 2 (2014): 210–22. http://dx.doi.org/10.1007/s11633-014-0783-8.

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30

Qi-lin, Peng. "Qualitative analysis for a class of second-order nonlinear system with delay." Applied Mathematics and Mechanics 22, no. 7 (2001): 842–45. http://dx.doi.org/10.1007/bf02438229.

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31

Tang, Man, Zhi-Hua Xiao, and Umair Zulfiqar. "Structure-Preserving Low-Rank Model Reduction for Second-Order Time-Delay Systems." Mathematics 13, no. 3 (2025): 474. https://doi.org/10.3390/math13030474.

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This paper introduces two model order-reduction techniques for second-order time-delay systems. The first method involves converting the second-order system into a first-order form, along with a set of related structure-preserving algorithms. The second method avoids converting the original model into a first-order form and uses direct projection to produce the reduced system, which can also retain the structure of the original one. The key idea of the proposed methods is to utilize low-rank Gramian approximations to construct reduced-order models. The time-delay Gramians are decomposed into l

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32

Jerzy, Klamka. "Constrained Controllability of Second Order Dynamical Systems with Delay." IFAC Proceedings Volumes 42, no. 13 (2009): 15–20. http://dx.doi.org/10.3182/20090819-3-pl-3002.00004.

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33

Weng, Aizhi, and Jitao Sun. "Impulsive stabilization of second-order nonlinear delay differential systems." Applied Mathematics and Computation 214, no. 1 (2009): 95–101. http://dx.doi.org/10.1016/j.amc.2009.03.071.

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34

Mechee, M., F. Ismail, N. Senu, and Z. Siri. "Directly Solving Special Second Order Delay Differential Equations Using Runge-Kutta-Nyström Method." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/830317.

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Runge-Kutta-Nyström (RKN) method is adapted for solving the special second order delay differential equations (DDEs). The stability polynomial is obtained when this method is used for solving linear second order delay differential equation. A standard set of test problems is solved using the method together with a cubic interpolation for evaluating the delay terms. The same set of problems is reduced to a system of first order delay differential equations and then solved using the existing Runge-Kutta (RK) method. Numerical results show that the RKN method is more efficient in terms of accurac

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35

Yang, Panpan, Ye Tang, Maode Yan, and Lei Zuo. "Leader-Follower Consensus of Second-Order Multiagent Systems with Absent Velocity Measurement and Time Delay." Mathematical Problems in Engineering 2018 (October 8, 2018): 1–11. http://dx.doi.org/10.1155/2018/3079868.

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The leader-follower consensus problem of second-order multiagent systems with both absent velocity measurement and time delay is considered. First of all, the consensus protocol is designed by introducing an auxiliary system to compensate for the unavailability of the velocity information. Then, time delay is incorporated into the consensus protocol and two cases with, respectively, constant time delay and time-varying delay are investigated. For the case of constant time delay, Lyapunov-Razumikhin theorem is deployed to obtain the sufficient conditions that guarantee the stability of the cons

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36

Bayili, Gilbert, Akram Ben Aissa, and Serge Nicaise. "Same decay rate of second order evolution equations with or without delay." Systems & Control Letters 141 (July 2020): 104700. http://dx.doi.org/10.1016/j.sysconle.2020.104700.

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37

Wang, Lihui. "Automatic control of constant temperature and humidity in building air conditioning systems based on frequency domain analysis." International Journal of Renewable Energy Development 13, no. 4 (2024): 618–29. http://dx.doi.org/10.61435/ijred.2024.60102.

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How to solve their automatic control of constant temperature and humidity gradually becomes a research hotspot as the continuous upgrading of air conditioning systems. This study aims to optimize the traditional proportional-integral-differential controller for improvement to solve the time-delay instability phenomenon in temperature and humidity control. The objective of this study is to optimize existing proportional-integral-differential controllers to improve the time-delay instability problem that is common in temperature and humidity control. Firstly, it treats the controlled object as a

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Kiss, Gábor, and Bernd Krauskopf. "Stability implications of delay distribution for first-order and second-order systems." Discrete & Continuous Dynamical Systems - B 13, no. 2 (2010): 327–45. http://dx.doi.org/10.3934/dcdsb.2010.13.327.

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Pei, Huiqin, Qiang Lai, and Guang Li. "Consensus of Second-Order Multiagent Systems with Directed Signed Networks and Communication Delays." Complexity 2020 (September 29, 2020): 1–10. http://dx.doi.org/10.1155/2020/1712643.

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This paper is concerned with the consensus problem for second-order multiagent systems (MASs) with directed signed networks and local communication delays. First, under a strongly connected directed graph, some properties for signed digraph and the model of second-order MASs are provided. A distributed nonlinear consensus control protocol is proposed for implementing consensus of second-order MASs with signed digraph and communication delay. Then, for structurally balanced and unbalanced networks, the consensus of second-order MASs with directed signed networks and local delays is proved via t

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40

Kim, Beomsoo, Jaesung Kwon, Sungwoong Choi, and Jeonghyeon Yang. "Feedback Stabilization of First Order Neutral Delay Systems Using the Lambert W Function." Applied Sciences 9, no. 17 (2019): 3539. http://dx.doi.org/10.3390/app9173539.

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This paper presents a new approach to stabilize the first order neutral delay differential systems with two time delays. First, we provided a few oscillation and non-oscillation criteria for the neutral delay differential equations using spectrum analysis and the Lambert W function. These conditions were explicit and the real roots were analytically expressed in terms of the Lambert W function in the case of non-oscillation. Second, we designed a stabilizing state feedback controller for the neutral delay differential systems with two time delays, wherein the proportional and derivative gains

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41

Fan, Lingling, and Chengyan Wu. "Distributed Finite-Time Consensus Control of Second-Order Multiagent Systems Subject to Communication Time Delay." Journal of Control Science and Engineering 2021 (December 21, 2021): 1–12. http://dx.doi.org/10.1155/2021/3786530.

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This paper studies the consensus problem of a second-order multiagent system (MAS) with fixed communication delay under the structure of leaderless and leader-following systems. By using graph theory and finite-time control scheme, a distributed control protocol is proposed for each agent to reach consensus in a finite time. In practical application, the time delay of states is unavoidable, and for this, the consensus method is supposed to be extended to solve the time-delay problem. Thus, a finite-time consensus protocol with communication time delay is proposed in this paper. Compared with t

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42

Diblík, Josef, Denis Khusainov, Oleksandra Kukharenko, and Zdeněk Svoboda. "Solution of the First Boundary-Value Problem for a System of Autonomous Second-Order Linear Partial Differential Equations of Parabolic Type with a Single Delay." Abstract and Applied Analysis 2012 (2012): 1–27. http://dx.doi.org/10.1155/2012/219040.

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The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier

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43

R., Priyanka, and Ramadevi S. "A Hyperbolic PDE ODE System with Delay Robust Stabilization." International Journal of Trend in Scientific Research and Development 2, no. 5 (2018): 1988–90. https://doi.org/10.31142/ijtsrd17157.

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This paper is concerned with a new development of a delay robust stabilizing feedback control law for linear ordinary differential equation coupled with two linear first order hyperbolic equations in the actuation path. A second change of variables that reduces the stabilization problem of the PDE ODE system to that of a time delay system for which a forecaster can be constructed. Hence, by choosing the pole placement on the ODE when constructing the forecaster, enabling a trade off between convergence rate and delay robustness. The proposed feedback law is finally proved to be robust to small

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44

Wilkey, Andrew, Yogesh N. Joglekar, and Gautam Vemuri. "Exceptional Points in a Non-Markovian Anti-Parity-Time Symmetric System." Photonics 10, no. 12 (2023): 1299. http://dx.doi.org/10.3390/photonics10121299.

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By studying the eigenvalues and eigenvectors of a non-Markovian anti parity-time (APT) symmetric system, we investigate the possibility of exceptional points (EPs) that may arise within it. Our work is motivated by a recently studied APT-symmetric experimental configuration consisting of a pair of time-delay coupled semiconductor lasers (SCLs). In such a system, a single time-delay represents the memory. The time-delayed coupling makes the system’s effective Hamiltonian infinite-dimensional and leads to novel features in the corresponding eigenvalues and eigenvectors. In particular, we demonst

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Farahani, Hamed Shirinabadi, Heidar Ali Talebi, and Mohammad Baghermenhaj. "An adaptive control scheme for hyperbolic partial differential equation system (drilling system) with unknown coefficient." Archives of Control Sciences 27, no. 1 (2017): 63–76. http://dx.doi.org/10.1515/acsc-2017-0004.

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Abstract The adaptive boundary stabilization is investigated for a class of systems described by second-order hyperbolic PDEs with unknown coefficient. The proposed control scheme only utilizes measurement on top boundary and assume anti-damping dynamics on the opposite boundary which is the main feature of our work. To cope with the lack of full state measurements, we introduce Riemann variables which allow us reformulate the second-order in time hyperbolic PDE as a system with linear input-delay dynamics. Then, the infinite-dimensional time-delay tools are employed to design the controller.

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Prasolov, A. V. "On the construction of a periodic solution of a second-order delay system." Differential Equations 36, no. 4 (2000): 524–28. http://dx.doi.org/10.1007/bf02754247.

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Tunç, Cemil, and Osman Tunç. "Qualitative analysis for a variable delay system of differential equations of second order." Journal of Taibah University for Science 13, no. 1 (2019): 468–77. http://dx.doi.org/10.1080/16583655.2019.1595359.

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Berezansky, Leonid, Elena Braverman, and Lev Idels. "Stabilisation of second-order nonlinear equations with variable delay." International Journal of Control 88, no. 8 (2015): 1533–39. http://dx.doi.org/10.1080/00207179.2015.1009948.

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Duan, Yameng, Wieslaw Krawcewicz, and Huafeng Xiao. "Periodic solutions in reversible systems in second order systems with distributed delays." AIMS Mathematics 9, no. 4 (2024): 8461–75. http://dx.doi.org/10.3934/math.2024411.

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<abstract><p>In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.</p></abstract>

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Sakthivel, R., E. R. Anandhi, and N. I. Mahmudov. "Approximate Controllability of Second-Order Systems with State-Dependent Delay." Numerical Functional Analysis and Optimization 29, no. 11-12 (2008): 1347–62. http://dx.doi.org/10.1080/01630560802580901.

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