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Author: Grafiati

Published: 9 June 2022

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1

Di Cintio, Pierfrancesco, and Lapo Casetti. "Discreteness effects, N-body chaos and the onset of radial-orbit instability." Monthly Notices of the Royal Astronomical Society 494, no. 1 (March 20, 2020): 1027–34. http://dx.doi.org/10.1093/mnras/staa741.

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ABSTRACT We study the stability of a family of spherical equilibrium models of self-gravitating systems, the so-called γ models with Osipkov–Merritt velocity anisotropy, by means of N-body simulations. In particular, we analyse the effect of self-consistent N-body chaos on the onset of radial-orbit instability. We find that degree of chaoticity of the system associated with its largest Lyapunov exponent Λmax has no appreciable relation with the stability of the model for fixed density profile and different values of radial velocity anisotropy. However, by studying the distribution of the Lyapunov exponents λm of the individual particles in the single-particle phase space, we find that more anisotropic systems have a larger fraction of orbits with larger λm.

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2

Xu, Kexin, Xianqing Wu, Miao Ma, and Yibo Zhang. "Energy-based output feedback control of the underactuated 2DTORA system with saturated inputs." Transactions of the Institute of Measurement and Control 42, no. 14 (July 2, 2020): 2822–29. http://dx.doi.org/10.1177/0142331220933475.

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In this paper, we consider the control issues of the two-dimensional translational oscillator with rotational actuator (2DTORA) system, which has two translational carts and one rotational rotor. An output feedback controller for the 2DTORA system is proposed, which can prevent the unwinding behaviour. In addition, the velocity signal unavailability and actuator saturation are taken into account, simultaneously. In particular, the dynamics of the 2DTORA system are given first. On the basis of the passivity and control objectives of the 2DTORA system, an elaborate Lyapunov function is constructed. Then, based on the introduced Lyapunov function, a novel output feedback control method is proposed straightforwardly for the 2DTORA system. Lyapunov theory and LaSalle’s invariance principle are utilized to analyse the stability of the closed-loop system and the convergence of the states. Finally, simulation results are provided to illustrate the excellent control performance of the proposed controller in comparison with the existing method.

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3

Ge, Z.-M., C.-S. Chen, H.-H. Chen, and S.-C. Lee. "Regular and chaotic dynamics of a simplified fly-ball governor." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 213, no. 5 (May 1, 1999): 461–75. http://dx.doi.org/10.1243/0954406991522707.

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The dynamics of a simplified model of a fly-ball speed governor undergoing a harmonic variation about its rotational speed is studied in this paper. This system is a non-linear damped system subjected to parametric excitation. The harmonic balance method is applied to analyse the stability of period attractors and the behaviour of bifurcations. The time evolutions of the response of the non-linear dynamic system are described by time history, phase portraits and Poincaré maps. The regular and chaotic behaviour is observed by various numerical techniques such as power spectra, Lyapunov exponents and Lyapunov dimension. Finally, the domains of attraction of periodic and stranger attractors of the system are located by applying the interpolated cell mapping (ICM) method.

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4

Greulich, Philip, Ben D. MacArthur, Cristina Parigini, and Rubén J. Sánchez-García. "Stability and steady state of complex cooperative systems: a diakoptic approach." Royal Society Open Science 6, no. 12 (December 2019): 191090. http://dx.doi.org/10.1098/rsos.191090.

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Cooperative dynamics are common in ecology and population dynamics. However, their commonly high degree of complexity with a large number of coupled degrees of freedom renders them difficult to analyse. Here, we present a graph-theoretical criterion, via a diakoptic approach (divide-and-conquer) to determine a cooperative system’s stability by decomposing the system’s dependence graph into its strongly connected components (SCCs). In particular, we show that a linear cooperative system is Lyapunov stable if the SCCs of the associated dependence graph all have non-positive dominant eigenvalues, and if no SCCs which have dominant eigenvalue zero are connected by a path.

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Chanthorn, Pharunyou, Grienggrai Rajchakit, Jenjira Thipcha, Chanikan Emharuethai, Ramalingam Sriraman, Chee Peng Lim, and Raja Ramachandran. "Robust Stability of Complex-Valued Stochastic Neural Networks with Time-Varying Delays and Parameter Uncertainties." Mathematics 8, no. 5 (May 8, 2020): 742. http://dx.doi.org/10.3390/math8050742.

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In practical applications, stochastic effects are normally viewed as the major sources that lead to the system’s unwilling behaviours when modelling real neural systems. As such, the research on network models with stochastic effects is significant. In view of this, in this paper, we analyse the issue of robust stability for a class of uncertain complex-valued stochastic neural networks (UCVSNNs) with time-varying delays. Based on the real-imaginary separate-type activation function, the original UCVSNN model is analysed using an equivalent representation consisting of two real-valued neural networks. By constructing the proper Lyapunov–Krasovskii functional and applying Jensen’s inequality, a number of sufficient conditions can be derived by utilizing It o ^ ’s formula, the homeomorphism principle, the linear matrix inequality, and other analytic techniques. As a result, new sufficient conditions to ensure robust, globally asymptotic stability in the mean square for the considered UCVSNN models are derived. Numerical simulations are presented to illustrate the merit of the obtained results.

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6

Richard, Quentin. "Global stability in a competitive infection-age structured model." Mathematical Modelling of Natural Phenomena 15 (2020): 54. http://dx.doi.org/10.1051/mmnp/2020007.

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We study a competitive infection-age structured SI model between two diseases. The well-posedness of the system is handled by using integrated semigroups theory, while the existence and the stability of disease-free or endemic equilibria are ensured, depending on the basic reproduction number R0x and R0y of each strain. We then exhibit Lyapunov functionals to analyse the global stability and we prove that the disease-free equilibrium is globally asymptotically stable whenever max{R0x, R0y} ≤ 1. With respect to explicit basin of attraction, the competitive exclusion principle occurs in the case where R0x ≠ R0y and max{R0x, R0y} > 1, meaning that the strain with the largest R0 persists and eliminates the other strain. In the limit case R0x = Ry0 > 1, an infinite number of endemic equilibria exists and constitute a globally attractive set.

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7

Song, Yi, Zehang Song, and Xiaodong Yao. "Permanent Magnet Synchronous Motor Control Based on New Sliding Mode Observer." Journal of Physics: Conference Series 2218, no. 1 (March 1, 2022): 012058. http://dx.doi.org/10.1088/1742-6596/2218/1/012058.

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Abstract Aiming at the problems of large chattering and slow response time existing in the preexisting sliding mode observer permanent magnet synchronous motor control, this paper proposes a new sliding mode observer control way. The model of the permanent magnet synchronous motor was established, and design a new function to take the place of the preexisting transforming function. And use Lyapunov stability criterion to analyse the system stability. At the same time, Use Matlab/Simulink simulation software to carry out simulation experiments, build simulation experiment models, and obtain the control result waveforms of the integrated sliding mode observer control system and the newly designed sliding mode observer control system. Compared to pre-existing sliding mode observers, it can be known that, constrasted with the pre-existing sliding film observer, better performance and stronger stability are possessed by the new sliding mode observer proposed in this artical.

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8

Ibrahim, M. O., A. A. Ayoade, O. J. Peter, and F. A. Oguntolu. "ON THE GLOBAL STABILITY OF CHOLERA MODEL WITH PREVENTION AND CONTROL." MALAYSIAN JOURNAL OF COMPUTING 3, no. 1 (June 29, 2018): 28. http://dx.doi.org/10.24191/mjoc.v3i1.4812.

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In this study, a system of first order ordinary differential equations is used to analyse the dynamics of cholera disease via a mathematical model extended from Fung (2014) cholera model. The global stability analysis is conducted for the extended model by suitable Lyapunov function and LaSalle’s invariance principle. It is shown that the disease free equilibrium (DFE) for the extended model is globally asymptotically stable if 𝑅0 𝑞 < 1 and the disease eventually disappears in the population with time while there exists a unique endemic equilibrium that is globally asymptotically stable whenever 𝑅0 𝑞 > 1 for the extended model or 𝑅0 > 1 for the original model and the disease persists at a positive level though with mild waves (i.e few cases of cholera) in the case of𝑅0 𝑞 > 1. Numerical simulations for strong, weak, and no prevention and control measures are carried out to verify the analytical results and Maple 18 is used to carry out the computations.

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9

Kruthika, H. A., Arun D. Mahindrakar, and Ramkrishna Pasumarthy. "Stability Analysis of Nonlinear Time–Delayed Systems with Application to Biological Models." International Journal of Applied Mathematics and Computer Science 27, no. 1 (March 28, 2017): 91–103. http://dx.doi.org/10.1515/amcs-2017-0007.

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Abstract In this paper, we analyse the local stability of a gene-regulatory network and immunotherapy for cancer modelled as nonlinear time-delay systems. A numerically generated kernel, using the sum-of-squares decomposition of multivariate polynomials, is used in the construction of an appropriate Lyapunov–Krasovskii functional for stability analysis of the networks around an equilibrium point. This analysis translates to verifying equivalent LMI conditions. A delay-independent asymptotic stability of a second-order model of a gene regulatory network, taking into consideration multiple commensurate delays, is established. In the case of cancer immunotherapy, a predator–prey type model is adopted to describe the dynamics with cancer cells and immune cells contributing to the predator–prey population, respectively. A delay-dependent asymptotic stability of the cancer-free equilibrium point is proved. Apart from the system and control point of view, in the case of gene-regulatory networks such stability analysis of dynamics aids mimicking gene networks synthetically using integrated circuits like neurochips learnt from biological neural networks, and in the case of cancer immunotherapy it helps determine the long-term outcome of therapy and thus aids oncologists in deciding upon the right approach.

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10

Hanel, Rudolf, Manfred Pöchacker, and Stefan Thurner. "Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1933 (December 28, 2010): 5583–96. http://dx.doi.org/10.1098/rsta.2010.0267.

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Linearized catalytic reaction equations (modelling, for example, the dynamics of genetic regulatory networks), under the constraint that expression levels, i.e. molecular concentrations of nucleic material, are positive, exhibit non-trivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems, an inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity, which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems , their basic properties allow us to understand the fundamental dynamical properties of complex biological reaction networks. We analyse the Lyapunov spectrum, determine the probability of finding stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network , and study how the frequency distributions of oscillatory modes of such a system depend on the average connectivity.

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11

Li, Yihong, Jinxiao Pan, and Zhen Jin. "Dynamic Modeling and Analysis of the Email Virus Propagation." Discrete Dynamics in Nature and Society 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/472072.

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A novel deterministic SEIS model for the transmission of email viruses in growing communication networks is formulated. Interestingly, the model is different from classical SEIS models not only in the form, but also in the dynamical features. We study the equilibria and their stability and analyse the bifurcation dynamics of the model. In particular, we find that the virus-free equilibrium is locally asymptotically stable for any parameter values, which may attribute to the absence of the basic reproduction number. It is shown that the model undergoes a saddle-node bifurcation and admits the bistable phenomenon. Moreover, on the basis of the Lyapunov function, the domains of attraction of equilibria are estimated by solving an LMI optimization problem. Based on the above theoretical results, some effective strategies are also provided to control the propagation of the email viruses. Additionally, our results are confirmed by numerical simulations.

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12

Taheri Andani, Majid, Zahra Ramezani, Saeed Moazami, Jinde Cao, Mohammad Mehdi Arefi, and Hassan Zargarzadeh. "Observer-Based Sliding Mode Control for Path Tracking of a Spherical Robot." Complexity 2018 (October 16, 2018): 1–15. http://dx.doi.org/10.1155/2018/3129398.

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Due to their complicated dynamics and underactuated nature, spherical robots require advanced control methods to reveal all their manoeuvrability features. This paper considers the path tracking control problem of a spherical robot equipped with a 2-DOF pendulum. The pendulum has two input torques that allow it to take angles about the robot’s transverse and longitudinal axes. Due to mechanical technicalities, it is assumed that these angles are immeasurable. First, a neural network observer is designed to estimate the pendulum angles. Then a modified sliding mode controller is proposed for the robot’s tracking control in the presence of uncertainties. Next, the Lyapunov theorem is utilized to analyse the overall stability of the proposed scheme, including the convergence of the observer estimation and the trajectory tracking errors. Finally, simulation results are provided to indicate the effectiveness of the proposed method in comparison with the other available control approaches.

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13

Dong, Zaopeng, Yang Liu, Hao Wang, and Tao Qin. "Method of Cooperative Formation Control for Underactuated USVS Based on Nonlinear Backstepping and Cascade System Theory." Polish Maritime Research 28, no. 1 (March 1, 2021): 149–62. http://dx.doi.org/10.2478/pomr-2021-0014.

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Abstract This paper presents a method for the cooperative formation control of a group of underactuated USVs. The problem of formation control is first converted to one of stabilisation control of the tracking errors of the follower USVs using system state transformation design. The followers must keep a fixed distance from the leader USV and a specific heading angle in order to maintain a certain type of formation. A global differential homeomorphism transformation is then designed to create a tracking error system for the follower USVs, in order to simplify the description of the control system. This makes the complex formation control system easy to analyse, and allows it to be decomposed into a cascaded system. In addition, several intermediate state variables and virtual control laws are designed based on nonlinear backstepping, and actual control algorithms for the follower USVs to control the surge force and yaw moment are presented. A global system that can ensure uniform asymptotic stability of the USVs’ cooperative formation control is achieved by combining Lyapunov stability theory and cascade system theory. Finally, several simulation experiments are carried out to verify the validity, stability and reliability of our cooperative formation control method.

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14

George, Sara Mohan, Selvi S.S., and Raol J.R. "Aircraft Parameter Estimation Using Gaussian Sum Information Filter with Lyapunov Stability Analysis." Webology 19, no. 1 (January 20, 2022): 3275–90. http://dx.doi.org/10.14704/web/v19i1/web19216.

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Accurate estimation of aerodynamic parameters of an aircraft is a matter of great importance in aviation industry as well in aircraft design and development programs. In this paper, Gaussian sum extended information filter (GSEIF) is studied for aircraft parameter estimation. The convergence condition for the GSEIF has been obtained using the normalized Lyapunov energy functional by proposing a nonlinear observer for the filter and studying its asymptotic behavior, this is a novel approach for deriving stability results for corresponding nonlinear filter. The performance of extended information filter (EIF) and GSEIF is evaluated for aircraft parameter estimation using simulated flight data generated using the MATLAB implementations. The estimation results using GSEIF are better than that obtained using EIF, especially when the initial state is not known (well), which shows that the proposed technique is really feasible for aerodynamic parameter estimation.

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15

Zhao, Rongyong, Ping Jia, Yan Wang, Cuiling Li, Yunlong Ma, and Zhishu Zhang. "Acceleration-critical density time-delay model for crowd stability analysis based on Lyapunov theory." MATEC Web of Conferences 355 (2022): 03019. http://dx.doi.org/10.1051/matecconf/202235503019.

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Crowd stability analysis is one of research hotspots to alleviate the severe situation of stampede accidents worldwide. Different from the conventional analysis models for crowd stability based on pedestrian density, this study analyses the characteristics of external disturbances and internal obstacle disturbance based on Lyapunov's theory. The critical range of crowd acceleration in crowd evacuation is obtained, a crowd merging acceleration-critical density time delay model is established, and a stability criterion of acceleration vector based on Lyapunov is obtained based on Lyapunov stability analysis. This provides new information for ensuring the stability of crowd movement in public places, assessing the stability of the crowd in the area, and taking reasonable protection and guidance measures prior to instability of a crowd flow.

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16

Zhang, Yang, Duanwei Shi, Tong Xiao, Ji Zhou, and Xionghao Cheng. "Pitch Stability Analysis for Mechanical-Hydraulic-Structural-Fluid Coupling System of High-Lift Hoist Vertical Shiplift." Strojniški vestnik – Journal of Mechanical Engineering 66, no. 4 (April 15, 2020): 256–75. http://dx.doi.org/10.5545/sv-jme.2019.6467.

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Pitch stability of the high-lift wire rope hoist vertical shiplift under dynamic hydraulic levelling has always been an issue of concern. It not only affects working efficiency but also brings significant challenges to operational safety. A new mechanical-hydraulic-structural-fluid (MHSF) coupling dynamics model and a developed semi-analytical method are presented for stable property analysis. The models of the hydraulic levelling subsystem, shallow water sloshing subsystem, the main hoist mechanical subsystem, and the shiplift chamber structure subsystem are built using a closed-loop transfer function, multi-modal theory, and an second-type Lagrangian equation, respectively. Then, a core twenty-one order state matrix of the MHSF coupling system is established using the state-space method. Subsequently, the Lyapunov motion stability theory and Eigen-analysis method are used in combination to judge the pitch stability and analyse the characteristics of the subsystems. Taking four typical high-lift hoist vertical shiplifts as examples, the rationality of the proposed model and method is validated. The results indicate that although the pitch stability safety factor under hydraulic dynamic levelling is reduced by about 15 % to 44 % with respect to hydraulic static levelling, hydraulic dynamic levelling still can meet stability requirements. Furthermore, for the designed 200 m level hoist vertical shiplift, the preliminary design parameters can ensure the pitch stability safety factor under dynamic hydraulic levelling of not less than 1.1. The element most prone to instability is the shallow water sloshing subsystem; increasing the synchronous shaft stiffness or the water boundary layer damping ratio can effectively enhance the pitch stability.

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17

Barreira, Luis, and Claudia Valls. "Stability theory and Lyapunov regularity." Journal of Differential Equations 232, no. 2 (January 2007): 675–701. http://dx.doi.org/10.1016/j.jde.2006.09.021.

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18

Clarke, F. H., Yu S. Ledyaev, and R. J. Stern. "Asymptotic Stability and Smooth Lyapunov Functions." Journal of Differential Equations 149, no. 1 (October 1998): 69–114. http://dx.doi.org/10.1006/jdeq.1998.3476.

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19

Michalak, Anna. "Dual approach to Lyapunov stability." Nonlinear Analysis: Theory, Methods & Applications 85 (July 2013): 174–79. http://dx.doi.org/10.1016/j.na.2013.02.007.

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20

Kalitin, B. S. "Lyapunov Stability and Orbital Stability of Dynamical Systems." Differential Equations 40, no. 8 (August 2004): 1096–105. http://dx.doi.org/10.1023/b:dieq.0000049826.73745.c1.

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21

Cong, Xin-rong, and Long-suo Li. "Analysis of Robust Stability for a Class of Stochastic Systems via Output Feedback: The LMI Approach." Journal of Function Spaces and Applications 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/873578.

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This paper investigates the robust stability for a class of stochastic systems with both state and control inputs. The problem of the robust stability is solved via static output feedback, and we convert the problem to a constrained convex optimization problem involving linear matrix inequality (LMI). We show how the proposed linear matrix inequality framework can be used to select a quadratic Lyapunov function. The control laws can be produced by assuming the stability of the systems. We verify that all controllers can robustly stabilize the corresponding system. Further, the numerical simulation results verify the theoretical analysis results.

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22

Rodrigues, Hildebrando M., and J. Solà-Morales. "An example on Lyapunov stability and linearization." Journal of Differential Equations 269, no. 2 (July 2020): 1349–59. http://dx.doi.org/10.1016/j.jde.2020.01.027.

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23

Souza, Josiney A., and Luana H. Takamoto. "Lyapunov stability for impulsive control affine systems." Journal of Differential Equations 266, no. 7 (March 2019): 4232–67. http://dx.doi.org/10.1016/j.jde.2018.09.033.

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24

Petrov, Alexey A., and Sergei Yu Pilyugin. "Lyapunov functions, shadowing and topological stability." Topological Methods in Nonlinear Analysis 43, no. 1 (April 12, 2016): 231. http://dx.doi.org/10.12775/tmna.2014.013.

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25

Yao, Fengqi, Feiqi Deng, and Pei Cheng. "Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/548712.

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A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.

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26

Brooks, M. L. "On Lyapunov stability analyses of the switching regulator/filter system." IEEE Transactions on Aerospace and Electronic Systems 26, no. 2 (March 1990): 421–23. http://dx.doi.org/10.1109/7.53452.

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27

Freitas, Pedro, and Carlos Rocha. "Lyapunov Functionals and Stability for FitzHugh–Nagumo Systems." Journal of Differential Equations 169, no. 1 (January 2001): 208–27. http://dx.doi.org/10.1006/jdeq.2000.3901.

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28

Mukdasai, Kanit, and Piyapong Niamsup. "An LMI Approach to Stability for Linear Time-Varying System with Nonlinear Perturbation on Time Scales." Abstract and Applied Analysis 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/860506.

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We consider Lyapunov stability theory of linear time-varying system and derive sufficient conditions for uniform stability, uniform exponential stability, -uniform stability, andh-stability for linear time-varying system with nonlinear perturbation on time scales. We construct appropriate Lyapunov functions and derive several stability conditions. Numerical examples are presented to illustrate the effectiveness of the theoretical results.

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29

Pereira, F. L., and G. N. Silva. "Lyapunov Stability of Measure Driven Impulsive Systems." Differential Equations 40, no. 8 (August 2004): 1122–30. http://dx.doi.org/10.1023/b:dieq.0000049829.32675.ca.

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30

Koksal, Semen. "Nonuniform stability properties and perturbing lyapunov functions." Applicable Analysis 43, no. 1-2 (January 1992): 99–107. http://dx.doi.org/10.1080/00036819208840054.

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Barreira, Luis, and Claudia Valls. "Stability of delay equations via Lyapunov functions." Journal of Mathematical Analysis and Applications 365, no. 2 (May 2010): 797–805. http://dx.doi.org/10.1016/j.jmaa.2009.12.018.

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Yang, Ying, and Guopei Chen. "Finite Time Stability of Stochastic Hybrid Systems." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/867189.

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This paper considers the finite time stability of stochastic hybrid systems, which has both Markovian switching and impulsive effect. First, the concept of finite time stability is extended to stochastic hybrid systems. Then, by using common Lyapunov function and multiple Lyapunov functions theory, two sufficient conditions for finite time stability of stochastic hybrid systems are presented. Furthermore, a new notion called stochastic minimum dwell time is proposed and then, combining it with the method of multiple Lyapunov functions, a sufficient condition for finite time stability of stochastic hybrid systems is given. Finally, a numerical example is provided to illustrate the theoretical results.

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33

You, Jin, and Shurong Sun. "Practical stability for fractional impulsive control systems with noninstantaneous impulses on networks." Nonlinear Analysis: Modelling and Control 27, no. 1 (January 1, 2022): 102–20. http://dx.doi.org/10.15388/namc.2022.27.25204.

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This paper investigates practical stability for a class of fractional-order impulsive control coupled systems with noninstantaneous impulses on networks. Using graph theory and Lyapunov method, new criteria for practical stability, uniform practical stability as well as practical asymptotic stability are established. In this paper, we extend graph theory to fractional-order system via piecewise Lyapunov-like functions in each vertex system to construct global Lyapunov-like functions. Our results are generalization of some known results of practical stability in the literature and provide a new method of impulsive control law for impulsive control systems with noninstantaneous impulses. Examples are given to illustrate the effectiveness of our results

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El Allaoui, Abdelati, Said Melliani, and Lalla Saadia Chadli. "Stability of Fuzzy Dynamical Systems via Lyapunov Functions." International Journal of Differential Equations 2020 (August 1, 2020): 1–7. http://dx.doi.org/10.1155/2020/6218424.

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The purpose of this paper is to introduce the concept of fuzzy Lyapunov functions to study the notion of stability of equilibrium points for fuzzy dynamical systems associated with fuzzy initial value problems, through the principle of Zadeh. Our contribution consists in a qualitative characterization of stability by a study of the trajectories of fuzzy dynamical systems, using auxiliary functions, and they will be called fuzzy Lyapunov functions. And, among the main results that have been proven is that the existence of fuzzy Lyapunov functions is a necessary and sufficient condition for stability. Some examples are given to illustrate the obtained results.

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35

Tunç, Osman, Özkan Atan, Cemil Tunç, and Jen-Chih Yao. "Qualitative Analyses of Integro-Fractional Differential Equations with Caputo Derivatives and Retardations via the Lyapunov–Razumikhin Method." Axioms 10, no. 2 (April 9, 2021): 58. http://dx.doi.org/10.3390/axioms10020058.

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The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided.

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36

Ortega, Juan-Pablo, Víctor Planas-Bielsa, and Tudor S. Ratiu. "Asymptotic and Lyapunov stability of constrained and Poisson equilibria." Journal of Differential Equations 214, no. 1 (July 2005): 92–127. http://dx.doi.org/10.1016/j.jde.2004.09.016.

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37

Shaw, M. "Generalized Stability of Motion and Matrix Lyapunov Functions." Journal of Mathematical Analysis and Applications 189, no. 1 (January 1995): 104–14. http://dx.doi.org/10.1006/jmaa.1995.1006.

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38

Yin, Li Zi. "Stability Analyses of Neural Networks with Unbounded Time-Varying Delays and Nonlinear Perturbations." Applied Mechanics and Materials 278-280 (January 2013): 1247–50. http://dx.doi.org/10.4028/www.scientific.net/amm.278-280.1247.

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In this paper, we consider the problem of mu-stability of unbounded time-varying delays neural systems with nonlinear perturbations. Some mu-stability criterias are derived by using Lyapunov- Krasovski functional method . Those criteria are expressed in the form of linear matrix inequalities (LMIs).

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Li, Huijuan, and Qingxia Ma. "Finite-Time Lyapunov Functions and Impulsive Control Design." Complexity 2020 (October 27, 2020): 1–9. http://dx.doi.org/10.1155/2020/5179752.

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In this paper, we introduce finite-time Lyapunov functions for impulsive systems. The relaxed sufficient conditions for asymptotic stability of an equilibrium of an impulsive system are given via finite-time Lyapunov functions. A converse finite-time Lyapunov theorem for controlling the impulsive system is proposed. Three examples are presented to show how to analyze the stability of an equilibrium of the considered impulsive system via finite-time Lyapunov functions. Furthermore, according to the results, we design an impulsive controller for a chaotic system modified from the Lorenz system.

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Wu, Min, Zhengfeng Yang, and Wang Lin. "Exact Asymptotic Stability Analysis and Region-of-Attraction Estimation for Nonlinear Systems." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/146137.

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We address the problem of asymptotic stability and region-of-attraction analysis of nonlinear dynamical systems. A hybrid symbolic-numeric method is presented to compute exact Lyapunov functions and exact estimates of regions of attraction of nonlinear systems efficiently. A numerical Lyapunov function and an estimate of region of attraction can be obtained by solving an (bilinear) SOS programming via BMI solver, then the modified Newton refinement and rational vector recovery techniques are applied to obtain exact Lyapunov functions and verified estimates of regions of attraction with rational coefficients. Experiments on some benchmarks are given to illustrate the efficiency of our algorithm.

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Akeju, T. A. I. "Torsional Buckling of Columns by the Lyapunov Method." Journal of Ship Research 29, no. 03 (September 1, 1985): 189–93. http://dx.doi.org/10.5957/jsr.1985.29.3.189.

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The paper presents an application of Lyapunov's direct method to torsional and torsional-flexural buckling of columns. A metric space and a Lyapunov functional are proposed for each of the problems. Making use of Zubov's stability theorem and appropriate eigenvalue inequalities, the functionals yield the expressions for the buckling loads for simple and fixed supports. Of particular interest is the relative ease with which the expressions are derived, especially in the torsional-flexural buckling case, where it has not been necessary to seek the roots of the traditional cubic equation which governs the critical loads of the member.

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Fitzgibbon, W. B., S. L. Hollis, and J. J. Morgan. "Stability and Lyapunov Functions for Reaction-Diffusion Systems." SIAM Journal on Mathematical Analysis 28, no. 3 (May 1997): 595–610. http://dx.doi.org/10.1137/s0036141094272241.

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Cañada, Antonio, and Salvador Villegas. "Stability, resonance and Lyapunov inequalities for periodic conservative systems." Nonlinear Analysis: Theory, Methods & Applications 74, no. 5 (March 2011): 1913–25. http://dx.doi.org/10.1016/j.na.2010.10.061.

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Chiu, Chuang-Hsiung. "Lyapunov Functions for the Global Stability of Competing Predators." Journal of Mathematical Analysis and Applications 230, no. 1 (February 1999): 232–41. http://dx.doi.org/10.1006/jmaa.1998.6198.

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Xu, Bugong. "Stability of Retarded Dynamical Systems: A Lyapunov Function Approach." Journal of Mathematical Analysis and Applications 253, no. 2 (January 2001): 590–615. http://dx.doi.org/10.1006/jmaa.2000.7167.

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Liu, Xinzhi. "Quasi stability via lyapunov functions for impulsive differential systems." Applicable Analysis 31, no. 3 (January 1988): 201–13. http://dx.doi.org/10.1080/00036818808839824.

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Zeng, Caibin, Qigui Yang, and YangQuan Chen. "Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/292653.

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Little seems to be known about evaluating the stochastic stability of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) via stochastic Lyapunov technique. The objective of this paper is to work with stochastic stability criterions for such systems. By defining a new derivative operator and constructing some suitable stochastic Lyapunov function, we establish some sufficient conditions for two types of stability, that is, stability in probability and moment exponential stability of a class of nonlinear SDEs driven by fBm. We will also give an example to illustrate our theory. Specifically, the obtained results open a possible way to stochastic stabilization and destabilization problem associated with nonlinear SDEs driven by fBm.

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Alkhalifa, Loay, and Khaled Zennir. "New Estimates of Solution to Coupled System of Damped Wave Equations with Logarithmic External Forces." Journal of Function Spaces 2021 (April 9, 2021): 1–7. http://dx.doi.org/10.1155/2021/9924504.

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In the paper, we consider new stability results of solution to class of coupled damped wave equations with logarithmic sources in ℝ n . We prove a new scenario of stability estimates by introducing a suitable Lyapunov functional combined with some estimates.

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Liu, Zixin, Shu Lü, Shouming Zhong, and Mao Ye. "Improved Robust Stability Criteria of Uncertain Neutral Systems with Mixed Delays." Abstract and Applied Analysis 2009 (2009): 1–18. http://dx.doi.org/10.1155/2009/294845.

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The problem of robust stability for a class of neutral control systems with mixed delays is investigated. Based on Lyapunov stable theory, by constructing a new Lyapunov-Krasovskii function, some new stable criteria are obtained. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.

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Yin, Lizi, and Yungang Liu. "Exponential Stability Analysis for Genetic Regulatory Networks with Both Time-Varying and Continuous Distributed Delays." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/897280.

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The global exponential stability is investigated for genetic regulatory networks with time-varying delays and continuous distributed delays. By choosing an appropriate Lyapunov-Krasovskii functional, new conditions of delay-dependent stability are obtained in the form of linear matrix inequality (LMI). The lower bound of derivatives of time-varying delay is first taken into account in genetic networks stability analysis, and the main results with less conservatism are established by interactive convex combination method to estimate the upper bound of derivative function of the Lyapunov-Krasovskii functional. In addition, two numerical examples are provided to illustrate the effectiveness of the theoretical results.

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