Journal articles: 'Lyapunov Functions'

1

Stawiska, Małgorzata. "Plurisubharmonic Lyapunov functions." Michigan Mathematical Journal 52, no. 1 (April 2004): 131–40. http://dx.doi.org/10.1307/mmj/1080837739.

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2

Olas, Andrzej. "Recursive Lyapunov Functions." Journal of Dynamic Systems, Measurement, and Control 111, no. 4 (December 1, 1989): 641–45. http://dx.doi.org/10.1115/1.3153107.

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The paper presents the concept of recursive Lyapunov function. The concept is applied to investigation of asymptotic stability problem of autonomous systems. The sequence of functions {Uα(i)} and corresponding performance measures λ(i) are introduced. It is proven that λ(i+1) ≤ λ(i) and in most cases the inequality is a strong one. This fact leads to a concept of a recursive Lyapunov function. For the very important applications case of exponential stability the procedure is effective under very weak conditions imposed on the function V = U(0). The procedure may be particularly applicable for the systems dependent on parameters, when the Lyapunov function determined from one set of parameters may be employed at the first step of the procedure.

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3

Avallone, Anna. "Lyapunov modular functions." Rendiconti del Circolo Matematico di Palermo 53, no. 2 (June 2004): 195–204. http://dx.doi.org/10.1007/bf02872871.

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4

Sassano, Mario, and Alessandro Astolfi. "Dynamic Lyapunov functions." Automatica 49, no. 4 (April 2013): 1058–67. http://dx.doi.org/10.1016/j.automatica.2013.01.027.

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5

Fathi, Albert, and Pierre Pageault. "Smoothing Lyapunov functions." Transactions of the American Mathematical Society 371, no. 3 (September 10, 2018): 1677–700. http://dx.doi.org/10.1090/tran/7329.

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Sassano, M., and A. Astolfi. "Dynamic Lyapunov Functions." IFAC Proceedings Volumes 44, no. 1 (January 2011): 3409–14. http://dx.doi.org/10.3182/20110828-6-it-1002.03522.

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7

Levit, M. "Optimum Lyapunov functions." Dynamics and Control 5, no. 2 (April 1995): 163–90. http://dx.doi.org/10.1007/bf01973947.

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8

Pappalardo, Massimo, and Mauro Passacantando. "Gap Functions and Lyapunov Functions." Journal of Global Optimization 28, no. 3/4 (April 2004): 379–85. http://dx.doi.org/10.1023/b:jogo.0000026455.72523.ed.

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9

Köksal, S., and V. Lakshmikantham. "Higher derivatives of Lyapunov functions and cone-valued Lyapunov functions." Nonlinear Analysis: Theory, Methods & Applications 26, no. 9 (May 1996): 1555–64. http://dx.doi.org/10.1016/0362-546x(94)00002-y.

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10

Ates, Muzaffer, and Nezir Kadah. "Novel stability and passivity analysis for three types of nonlinear LRC circuits." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 11, no. 2 (July 31, 2021): 227–37. http://dx.doi.org/10.11121/ijocta.01.2021.001073.

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In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

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11

Kellett, Christopher M., and Andrew R. Teel. "Weak Converse Lyapunov Theorems and Control-Lyapunov Functions." SIAM Journal on Control and Optimization 42, no. 6 (January 2004): 1934–59. http://dx.doi.org/10.1137/s0363012901398186.

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12

Reĭzinš, Linard Eduardovič. "Nonlocal Lyapunov-Krasovski functions." Časopis pro pěstování matematiky 111, no. 3 (1986): 280–93. http://dx.doi.org/10.21136/cpm.1986.108156.

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13

Raković, Saša V. "Control Minkowski–Lyapunov functions." Automatica 128 (June 2021): 109598. http://dx.doi.org/10.1016/j.automatica.2021.109598.

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14

Daizhan Cheng, Lei Guo, and Jie Huang. "On quadratic lyapunov functions." IEEE Transactions on Automatic Control 48, no. 5 (May 2003): 885–90. http://dx.doi.org/10.1109/tac.2003.811274.

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15

Barreira, Luis, and Claudia Valls. "Robustness via Lyapunov functions." Journal of Differential Equations 246, no. 7 (April 2009): 2891–907. http://dx.doi.org/10.1016/j.jde.2008.11.010.

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16

Raković, Saša V. "Robust Minkowski–Lyapunov functions." Automatica 120 (October 2020): 109168. http://dx.doi.org/10.1016/j.automatica.2020.109168.

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17

Thang, VIET NGUYEN, Takehiro MORI, Yasuaki KUROE, and Yoshihiro MORI. "Relations between Common Quadratic Lyapunov Functions and Common Infinity-Norm Lyapunov Functions." Transactions of the Society of Instrument and Control Engineers 40, no. 10 (2004): 1067–69. http://dx.doi.org/10.9746/sicetr1965.40.1067.

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18

Martynyuk, Anatoly A. "Analysis of stability problems via matrix Lyapunov functions." Journal of Applied Mathematics and Stochastic Analysis 3, no. 4 (January 1, 1990): 209–26. http://dx.doi.org/10.1155/s104895339000020x.

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The stability of nonlinear systems is analyzed by the direct Lyapunov's method in terms of Lyapunov matrix functions. The given paper surveys the main theorems on stability, asymptotic stability and nonstability. They are applied to systems of nonlinear equations, singularly-perturbed systems and hybrid systems. The results are demonstrated by an example of a two-component system.

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19

Tsygankov, A. A. "Higher Derivatives of Lyapunov Functions." Differential Equations 40, no. 2 (February 2004): 298–301. http://dx.doi.org/10.1023/b:dieq.0000033722.83393.58.

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20

Clarke, F. H., Yu S. Ledyaev, L. Rifford, and R. J. Stern. "Feedback Stabilization and Lyapunov Functions." SIAM Journal on Control and Optimization 39, no. 1 (January 2000): 25–48. http://dx.doi.org/10.1137/s0363012999352297.

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21

Rapisarda, P., and C. Kojima. "Stabilization, Lyapunov functions, and dissipation." Systems & Control Letters 59, no. 12 (December 2010): 806–11. http://dx.doi.org/10.1016/j.sysconle.2010.09.008.

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22

Iggidr, Abderrahman, Boris Kalitine, and Gauthier Sallet. "Lyapunov theorems with semidefinite functions." IFAC Proceedings Volumes 32, no. 2 (July 1999): 1802–7. http://dx.doi.org/10.1016/s1474-6670(17)56306-5.

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23

Barreira, Luis, and Claudia Valls. "Nonuniform Dichotomies via Lyapunov Functions." Milan Journal of Mathematics 82, no. 2 (May 23, 2014): 243–71. http://dx.doi.org/10.1007/s00032-014-0221-y.

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24

Barreira, Luis, Davor Dragičević, and Claudia Valls. "Lyapunov Functions and Cone Families." Journal of Statistical Physics 148, no. 1 (June 20, 2012): 137–63. http://dx.doi.org/10.1007/s10955-012-0524-8.

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25

Calvo, M., M. P. Laburta, J. I. Montijano, and L. Rández. "Projection methods preserving Lyapunov functions." BIT Numerical Mathematics 50, no. 2 (March 5, 2010): 223–41. http://dx.doi.org/10.1007/s10543-010-0259-3.

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26

Fall, A., A. Iggidr, G. Sallet, and J. J. Tewa. "Epidemiological Models and Lyapunov Functions." Mathematical Modelling of Natural Phenomena 2, no. 1 (2007): 62–83. http://dx.doi.org/10.1051/mmnp:2008011.

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27

Raković, Saša V. "Robust control Minkowski–Lyapunov functions." Automatica 125 (March 2021): 109437. http://dx.doi.org/10.1016/j.automatica.2020.109437.

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28

Hirschorn, Ronald. "Lower Bounded Control-Lyapunov Functions." Communications in Information and Systems 8, no. 4 (2008): 399–412. http://dx.doi.org/10.4310/cis.2008.v8.n4.a3.

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29

Balestrino, A., A. Caiti, and E. Crisostomi. "Logical composition of Lyapunov functions." International Journal of Control 84, no. 3 (March 2011): 563–73. http://dx.doi.org/10.1080/00207179.2011.562549.

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30

Barreira, Luis, and Claudia Valls. "Lyapunov functions versus exponential contractions." Mathematische Zeitschrift 268, no. 1-2 (January 21, 2010): 187–96. http://dx.doi.org/10.1007/s00209-010-0665-x.

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31

Goebel, Rafal, Christophe Prieur, and Andrew R. Teel. "Smooth patchy control Lyapunov functions." Automatica 45, no. 3 (March 2009): 675–83. http://dx.doi.org/10.1016/j.automatica.2008.10.023.

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32

Hafstein, Sigurdur Freyr Hafstein. "algorithm for constructing Lyapunov functions." Electronic Journal of Differential Equations 1, Mon. 01-09 (August 15, 2007): 08. http://dx.doi.org/10.58997/ejde.mon.08.

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In this monograph we develop an algorithm for constructing Lyapunov functions for arbitrary switched dynamical systems \(\dot{\mathbf{x}} = \mathbf{f}_\sigma(t,\mathbf{x})\), possessing a uniformly asymptotically stable equilibrium. Let \(\dot{\mathbf{x}}=\mathbf{f}_p(t,\mathbf{x})\), \(p\in \mathcal{P}\), be the collection of the ODEs, to which the switched system corresponds. The number of the vector fields \(\mathbf{f}_p\) on the right-hand side of the differential equation is assumed to be finite and we assume that their components \(f_{p,i}\) are \(\mathcal{C}^2\) functions and that we can give some bounds, not necessarily close, on their second-order partial derivatives. The inputs of the algorithm are solely a finite number of the function values of the vector fields \(\mathbf{f}_p\) and these bounds. The domain of the Lyapunov function constructed by the algorithm is only limited by the size of the equilibrium's region of attraction. Note, that the concept of a Lyapunov function for the arbitrary switched system \(\dot{\mathbf{x}} = \mathbf{f}_\sigma(t,\mathbf{x})\) is equivalent to the concept of a common Lyapunov function for the systems \(\dot{\mathbf{x}}=\mathbf{f}_p(t,\mathbf{x})\), \(p\in\mathcal{P}\), and that if \(\mathcal{P}\) contains exactly one element, then the switched system is just a usual ODE \(\dot{\mathbf{x}}=\mathbf{f}(t,\mathbf{x})\). We give numerous examples of Lyapunov functions constructed by our method at the end of this monograph.

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33

Hurley, Mike. "Weak attractors from Lyapunov functions." Topology and its Applications 109, no. 2 (January 2001): 201–10. http://dx.doi.org/10.1016/s0166-8641(99)00158-3.

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34

Beisenbi, Мamyrbek, and Samal Kaliyeva. "Approach to the synthesis of an aperiodic robust automatic control system based on the gradient-speed method of Lyapunov vector functions." Eastern-European Journal of Enterprise Technologies 1, no. 3 (121) (February 28, 2023): 6–14. http://dx.doi.org/10.15587/1729-4061.2023.274063.

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One of the actual problems for the theory and practice of control of dynamic objects is the development of methods for research and synthesis of control systems of multidimensional objects. The paper proposes a universal approach to construct Lyapunov vector functions directly from the equation of state of control system and a new gradient-speed method of Lyapunov vector functions to study aperiodic robust stability of linear control system with m inputs and n outputs. The study of aperiodic robust stability of automatic control systems is based on the construction of Lyapunov vector functions and gradient-speed dynamic control systems. The basic statements of Lyapunov's theorem about asymptotic stability and notions of stability of dynamic systems are used. The representation of control systems as gradient systems and Lyapunov functions as potential functions of gradient systems from the catastrophe theory allow to construct the full-time derivative of Lyapunov vector functions always as a sign-negative function equal to the scalar product of the velocity vector on the gradient vector. The conditions of aperiodic robust stability are obtained as a system of inequalities on the uncertain parameters of the automatic control system, which are a condition for the existence of the Lyapunov vector-function. A numerical example of synthesis of aperiodic robustness of a multidimensional control object is given. The example shows the main stages of the developed synthesis method, the study of the system stability at different values of the coefficients k, confirming the consistency of the proposed method. Transients in the system satisfy all requirements

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35

NAKAMURA, Hisakazu, Yuh YAMASHITA, and Hirokazu NISHITANI. "Lyapunov Functions for Homogeneous Differential Inclusions." Transactions of the Society of Instrument and Control Engineers 39, no. 4 (2003): 365–74. http://dx.doi.org/10.9746/sicetr1965.39.365.

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36

DÍAZ-SIERRA, R., and V. FAIRÉN. "NEW METHOD FOR THE ESTIMATION OF DOMAINS OF ATTRACTION OF FIXED POINTS FROM LYAPUNOV FUNCTIONS." International Journal of Bifurcation and Chaos 12, no. 11 (November 2002): 2467–77. http://dx.doi.org/10.1142/s0218127402005984.

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The estimation of the domain of stability of fixed points of nonlinear differential systems constitutes a practical problem of much interest in engineering. The procedures based on Lyapunov's second method configures an alternative worth consideration. It has the appeal of reducing calculation complexity and is time-saving with respect to the direct, computer crunching approach which requires a systematic numerical integration of the evolution equations from a gridlike pattern of initial conditions. However, it is not devoid of problems inasmuch as the Lyapunov function itself is problem-dependent and relies too much on presumptions. Additionally, the evaluation of its corresponding domain is produced in terms of a nonlinear programming problem with inequality constraints the resolution of which may sometimes require a large investment in computer time. These problems are in part avoided by restricting to quadratic Lyapunov functions, with the possible obvious consequence of limiting the estimation of the domain. In order to simplify the estimation of domains we exploit here a novel formulation of the issue of stability of invariant surfaces within Lyapunov's direct method [Díaz-Sierra et al., 2001]. The resulting method addresses directly the optimization problem associated to the evaluation of the stability domain. The problem is recast in a new, simpler form by playing both on the Lyapunov function itself and on the constraints. The gains from the procedure permit to conceive increased returns in the application of Lyapunov's direct method once it is realized that it is not prohibitive from a computational point of view to depart from the limited quadratic Lyapunov functions.

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37

Voßwinkel, Rick, and Klaus Röbenack. "Systematic Analysis and Design of Control Systems Based on Lyapunov’s Direct Method." Algorithms 16, no. 8 (August 14, 2023): 389. http://dx.doi.org/10.3390/a16080389.

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This paper deals with systematic approaches for the analysis of stability properties and controller design for nonlinear dynamical systems. Numerical methods based on sum-of-squares decomposition or algebraic methods based on quantifier elimination are used. Starting from Lyapunov’s direct method, these methods can be used to derive conditions for the automatic verification of Lyapunov functions as well as for the structural determination of control laws. This contribution describes methods for the automatic verification of (control) Lyapunov functions as well as for the constructive determination of control laws.

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38

Li, Haitao, and Yuzhen Wang. "Lyapunov-Based Stability and Construction of Lyapunov Functions for Boolean Networks." SIAM Journal on Control and Optimization 55, no. 6 (January 2017): 3437–57. http://dx.doi.org/10.1137/16m1092581.

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39

McCluskey, C. Connell. "Using Lyapunov Functions to Construct Lyapunov Functionals for Delay Differential Equations." SIAM Journal on Applied Dynamical Systems 14, no. 1 (January 2015): 1–24. http://dx.doi.org/10.1137/140971683.

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40

Hafstein, Sigurdur, and Stefan Suhr. "Smooth complete Lyapunov functions for ODEs." Journal of Mathematical Analysis and Applications 499, no. 1 (July 2021): 125003. http://dx.doi.org/10.1016/j.jmaa.2021.125003.

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41

Li, Lina, and Lina Li. "From Lyapunov functions to Sobolev inequalities." Teoriya Veroyatnostei i ee Primeneniya 59, no. 4 (2014): 808–14. http://dx.doi.org/10.4213/tvp4599.

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42

Leong, Yoke Peng, Matanya B. Horowitz, and Joel W. Burdick. "Linearly Solvable Stochastic Control Lyapunov Functions." SIAM Journal on Control and Optimization 54, no. 6 (January 2016): 3106–25. http://dx.doi.org/10.1137/16m105767x.

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43

Barreira, Luis, Davor Dragičević, and Claudia Valls. "Nonuniform exponential dichotomies and Lyapunov functions." Regular and Chaotic Dynamics 22, no. 3 (May 2017): 197–209. http://dx.doi.org/10.1134/s1560354717030017.

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44

Liberzon, D., and R. Tempo. "Common Lyapunov Functions and Gradient Algorithms." IEEE Transactions on Automatic Control 49, no. 6 (June 2004): 990–94. http://dx.doi.org/10.1109/tac.2004.829632.

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45

Giesl, Peter, and Sigurdur Hafstein. "Computation and Verification of Lyapunov Functions." SIAM Journal on Applied Dynamical Systems 14, no. 4 (January 2015): 1663–98. http://dx.doi.org/10.1137/140988802.

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46

QU, ZHIHUA, and JOHN DORSEY. "Robust control by two Lyapunov functions." International Journal of Control 55, no. 6 (June 1992): 1335–50. http://dx.doi.org/10.1080/00207179208934288.

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47

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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48

Badii, R., K. Heinzelmann, P. F. Meier, and A. Politi. "Correlation functions and generalized Lyapunov exponents." Physical Review A 37, no. 4 (February 1, 1988): 1323–28. http://dx.doi.org/10.1103/physreva.37.1323.

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49

Li, L. "From Lyapunov Functions to Sobolev Inequalities." Theory of Probability & Its Applications 59, no. 4 (January 2015): 693–99. http://dx.doi.org/10.1137/s0040585x97t987405.

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50

Yu, Jie, and Athanasios Sideris. "H∞ control with parametric Lyapunov functions." Systems & Control Letters 30, no. 2-3 (April 1997): 57–69. http://dx.doi.org/10.1016/s0167-6911(96)00075-8.

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Journal articles on the topic 'Lyapunov Functions'

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