353 results on '"Liapunov functions"'
Search Results
2. Global Attractivity For a Volterra Difference Equation
- Author
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Saito, Kaori, Baigent, Steve, editor, Bohner, Martin, editor, and Elaydi, Saber, editor
- Published
- 2020
- Full Text
- View/download PDF
3. Bounded and Periodic Solutions of Integral Equations
- Author
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T. A Burton and Bo Zhang
- Subjects
Integral Equations ,Boundedness ,Periodic Solutions ,Liapunov Functions ,Mathematics ,QA1-939 - Abstract
In this paper we introduce a new method for obtaining boundedness of solutions of integral equations. From the integral equation we form an integrodifferential equation by computing x' + kx to which we apply a Liapunov functional. This can be far more effective than the usual technique of differentiating the equation. The qualitative properties derived from that equation are then transferred to a majorizing function for the integral equation. Schaefer's fixed point theorem is used to conclude that there is a periodic solution. Three kinds of integral equations are studied and they are shown to be related through two examples.En este artículo presentamos un nuevo método para obtener acotación de soluciones de ecuaciones integrales. A partir de la ecuación integral, formamos una ecuación integrales diferencial calculando x' + kx mediante la aplicación de un funcional de Liapunov. Ello puede resultar bastante más efectivo que la técnica usual de diferenciación de la ecuación. Las propiedades cualitativas derivadas de la ecuación son entonces transferidas a la función mayorante para la ecuación integral. El teorema del punto fijo de Schaefer es usado para concluir que hay una solución periódica. Se estudia tres tipos de ecuaciones integrales y se muestra que ellas están relacionadas a través de dos ejemplos.
- Published
- 2012
4. Stability properties of differential systems under constantly acting perturbations
- Author
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Giancarlo Cantarelli and Giuseppe Zappala
- Subjects
Stability ,persistent disturbance ,two measures ,Liapunov functions ,Mathematics ,QA1-939 - Abstract
In this article, we find stability criteria for perturbed differential systems, in terms of two measures. Our main tool is a definition of total stability based on two classes of perturbations.
- Published
- 2010
5. Non-asymptotic stability and integral stability trough a reduction principle.
- Author
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Salvadori, Luigi and Visentin, Francesca
- Abstract
This paper concerns the analysis of transferring stability properties from an invariant manifold to the whole space for an ordinary differential system. In previous papers we already treated this problem in the case of asymptotic and total stability. Here we deal with the case of non-asymptotic stability. We generalize to differential systems depending on time a reduction principle (Kelley in J Math Anal Appl 18:336-344, ; Pliss in Izv Akad Nauk SSSR Mat Ser 28:1297-1324, ) relative to autonomous systems. Our procedure is very different from the fixed point theorem argument used in Kelley (J Math Anal Appl 18:336-344, ), and it is based on the use of a suitable Liapunov function. Some results concerning integral stability are also given. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
6. Chaotic monetary dynamics with confidence
- Author
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Serletis, Apostolos and Shintani, Mototsugu
- Subjects
Chaos theory ,Liapunov functions ,Business ,Economics - Abstract
This paper uses tools from dynamical systems theory to investigate the properties of Canadian and US money and velocity measures. In doing so, we follow the recent contribution by Whang and Linton [Whang, Y.-J., Linton, O., 1999. The asymptotic distribution of nonparametric estimates of the Lyapunov exponent for stochastic time series. Journal of Econometrics 91, 1-42] and construct the standard error for the Nychka et al. [Nychka, D.W., Ellner, S., Gallant, R.A., McCaffrey, D., 1992. Finding chaos in noisy systems. Journal of the Royal Statistical Society B 54, 399-426] dominant Lyapunov exponent. Comparisons are made among simple-sum, Divisia, and currency equivalent aggregates at different levels of monetary aggregation. We find statistically significant evidence against low-dimensional chaos and point to the use of stochastic models and statistical inference in the modeling of these variables. JEL classification: E40; E50; C32 Keywords: Divisia; Chaos; Lyapunov exponent
- Published
- 2006
7. Parametric optimization of digitally controlled nonlinear reactor dynamics using Zubov-like functional equations
- Author
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Huynh, Nguyen and Kazantzis, Nikolaos
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Functional equations ,Functions ,Liapunov functions ,Mathematics ,Nonlinear theories ,Mathematics - Published
- 2005
8. An algorithmic approach to chain recurrence
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Kalies, W.D., Mischaikow, K., and VanderVorst, R.C.A.M.
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Algorithms ,Computational physics ,Liapunov functions ,Mathematical physics ,Algorithm ,Mathematics - Abstract
Dedicated to Steve Smale on His 75th Birthday Abstract. In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. [...]
- Published
- 2005
9. Global analysis of competition for perfectly substitutable resources with linear response
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Ballyk, Mary M., McCluskey, C. Connell, and Wolkowicz, Gail S.K.
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Bifurcation theory ,Chemical reactions ,Competition (Biology) ,Liapunov functions ,Mathematics - Published
- 2005
10. The energy function of a general multimachine system with a unified power flow controller
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Azbe, Valentin, Gabrijel, Uros, Povh, Dusan, and Mihalic, Rafael
- Subjects
Liapunov functions ,Electric power systems ,Differential equations ,Business ,Electronics ,Electronics and electrical industries - Abstract
In order to be able to successfully apply direct methods or other energy-function-based calculations in power systems, which include flexible AC transmission systems (FACTS) devices, the influence of those devices should be properly considered. This is currently not always possible in power systems incorporating unified power flow controllers (UPFCs) because the present energy functions for electric power systems do not involve proper UPFC actions. This paper presents a way of incorporating the transient-stability augmentation action of the most versatile FACTS device, i.e., the UPFC, into an energy function for multimachine systems. After making some assumptions, a new term of the structure-preserving energy function (SPEF) that represents a UPFC's energy function was constructed. This term can simply be added to any existing SPEF. The extended SPEF was tested for one UPFC in a longitudinal test system and for one and two UPFCs in an IEEE nine-bus machine test system. A comparison between the critical clearing times (CCTs) acquired directly with the use of the newly constructed SPEF and those obtained with time-simulation results shows that the proposed UPFC's energy function is suitable because the correct CCTs were obtained. Index Terms--FACTS devices, Lyapunov methods, power system control, power system transient stability.
- Published
- 2005
11. Stability of discontinuous retarded functional differential equations with applications
- Author
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Sun, Ye, Michel, Anthony N., and Zhai, Guisheng
- Subjects
Functions, Discontinuous ,Differential equations ,Liapunov functions ,Functions, Exponential - Abstract
We present Lyapunov stability results, including converse theorems, for a class of discontinuous dynamical systems (DDS) determined by the solutions of retarded functional differential equations. We demonstrate the applicability of these results in the analysis of several specific important classes of DDS determined by functional differential equations and differential-difference equations. Index Terms--Asymptotic stability, differential-difference equations, discontinuous dynamical systems (DDS), exponential stability, hybrid systems, Lyapunov stability, retarded functional differential equations, switched systems, systems with delays.
- Published
- 2005
12. A delayed neural network for solving linear projection equations and its analysis
- Author
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Liu, Qingshan, Cao, Jinde, and Xia, Youshen
- Subjects
Quadratic programming ,Nonlinear programming ,Liapunov functions ,Neural networks ,Neural network ,Business ,Computers ,Electronics ,Electronics and electrical industries - Abstract
In this paper, we present a delayed neural network approach to solve linear projection equations. The Lyapunov-Krasovskii theory for functional differential equations and the linear matrix inequality (LMI) approach are employed to analyze the global asymptotic stability and global exponential stability of the delayed neural network. Compared with the existing linear projection neural network, theoretical results and illustrative examples show that the delayed neural network can effectively solve a class of linear projection equations and some quadratic programming problems. Index Terms--Asymptotical stability, delayed neural networks, exponential stability, linear matrix inequality (LMI), Lyapunov-Krasovskii functional, quadratic programming.
- Published
- 2005
13. Remarks on the [L.sup.p]-input converging-state property
- Author
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Ryan, E.P.
- Subjects
Liapunov functions ,Differential equations - Abstract
Let X C [R.sup.N] and consider a system x = f (x, u), f : X X [R.sup.M] [right arrow] [R.sup.N], with the property that the associated autonomous system x = f (x, 0) has an asymptotically stable compactum C with region of attraction A. Assume that x is a solution of the former, defined on [0, [infinity]), corresponding to an input function u. Assume further that, for each compact K C X, there exists k > 0 such that | f (z, v)--f (z, 0) | [less than or equal to] k | v | for all (z, v) [member of] K X [R.sup.M]. A simple proof is given of the following [L.sup.p]-input converging-state property: if u [member of] [L.sup.p] for some p [member of] [1, [infinity]) and x has an [omega]-limit point in A, then x approaches C. Index Terms--Asymptotic stability, converse Lyapunov theory, domain of attraction.
- Published
- 2005
14. A unifying proof of global asymptotical stability of neural networks with delay
- Author
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Huang, Ying Sue and Wu, Chai Wah
- Subjects
Differential equations -- Delay equations ,Liapunov functions ,Asymptotic expansions ,Neural networks ,Neural network ,Business ,Computers and office automation industries ,Electronics ,Electronics and electrical industries - Abstract
We present some new global stability results of neural networks with delay and show that these results generalize recently published stability results. In particular, several different stability conditions in the literature which were proved using different Lyapunov functionals are generalized and unified by proving them using the same Lyapunov functional. We also show that under certain conditions, reversing the directions of the coupling between neurons preserves the global asymptotical stability of the neural network. Index Terms--Asymptotical stability, delay equations, Lyapunov functional, neural networks.
- Published
- 2005
15. Delay-dependent/delay-independent stability of linear systems with multiple time-varying delays
- Author
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Xu, Bugong and Liu, Yun-Hui
- Subjects
Linear systems ,Liapunov functions ,Stability - Abstract
Delay-dependent/delay-independent uniform asymptotic stability and uniform stability criteria for linear systems with multiple time-varying delays are established respectively in this note. The results are derived based on a new-type stability theorem for retarded dynamical systems and a new analysis technique for estimating the derivative of a Lyapunov function along the solution of a system at certain specific instants. Four remarks together with an illustrative example are given to compare the obtained results with and to show their superiority to those in the literature. Index Terms--Linear systems, Lyapunov methods, stability, time delay.
- Published
- 2003
16. Quadratic characterization and use of output stabilizable subspaces
- Author
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Castelan, Eugenio B., Hennet, Jean-Claude, and Llanos Villarreal, Elmer R.
- Subjects
Convex programming ,Liapunov functions ,Equations, Quadratic - Abstract
This note treats the problem of stabilization of linear systems by static output feedback using the concept of (C, A, B)-invariant subspaces. The work provides a new characterization of output stabilizable (C, A, B)-invariant subspaces through two coupled quadratic stabilization conditions. An equivalence is shown between the existence of a solution to this set of conditions and the possibility to stabilize the system by static output feedback. An algorithm is provided and numerical examples are reported to illustrate the approach. Index Terms--Convex programming, geometric approach, Lyapunov equations, output feedback, quadratic stabilizability.
- Published
- 2003
17. An improved LMI condition for robust D-stability of uncertain polytopic systems
- Author
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Leite, Valter J.S. and Peres, Pedro L.D.
- Subjects
Matrices ,Liapunov functions ,Linear systems ,Stability - Abstract
A sufficient condition for robust D-stability of linear systems with polytope type uncertainties is proposed. The result is based on a linear parameter-dependent Lyapunov function obtained from the feasibility test of a set of linear matrix inequalities (LMIs) defined at the vertices of the polytope. This improved LMI condition encompasses previous results based on additional inequalities, as well as results based on extra variables. Index Terms--Linear matrix inequalities (LMIs), parameter-dependent Lyapunov functions, parametric uncertainty, polytopic uncertainty, robust stability.
- Published
- 2003
18. Composite quadratic Lyapunov functions for constrained control systems
- Author
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Hu, Tingshu and Lin, Zongli
- Subjects
Invariants ,Liapunov functions ,Equations, Quadratic - Abstract
A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant. Index Terms--Constrained control, invariant set, quadratic functions.
- Published
- 2003
19. Algebraic criteria for global exponential stability of cellular neural networks with multiple time delays
- Author
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Liao, Xiao-xin and Wang, Jun
- Subjects
Neural networks -- Research ,Liapunov functions ,Neural network ,Business ,Computers and office automation industries ,Electronics ,Electronics and electrical industries - Abstract
This brief presents three sufficient conditions for the global exponential stability of cellular neural networks with time delays. The new stability results provide algebraic criteria for stability verifications and improve upon exisfing ones with stronger conditions. To demonstrate the differences and features of the new stability criteria, several examples are discussed to compare the present results with the existing ones. Index Terms--Cellular neural networks (CNNs), global exponential stability, Lyapunov function, time delays.
- Published
- 2003
20. Synchronization in coupled arrays of chaotic oscillators with nonreciprocal coupling
- Author
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Wu, Chai Wah
- Subjects
Chaos theory -- Research ,Liapunov functions ,Oscillators (Electronics) ,Nonlinear programming ,Business ,Computers and office automation industries ,Electronics ,Electronics and electrical industries - Abstract
There are, in general, two classes of results regarding the synchronization of chaos in an array of coupled identical chaotic systems. The first class of results relies on Lyapunov's direct method and gives analytical criteria for global or local synchronization. The second class of results relies on linearization around the synchronization manifold and the computation of Lyapunov exponents. The computation of Lyapunov exponents is mainly done via numerical experiments and can only show local synchronization in the neighborhood of the synchronization manifold. On the other hand, Lyapunov's direct method is more rigorous and can give global results. The coupling topology is generally expressed in matrix form and the first class of methods mainly deals with symmetric matrices whereas the second class of methods can work with all diagonalizable matrices. The purpose of this brief is to bridge the gap in the applicability of the two classes of methods by considering the nonsymmetric case for the first class of methods. We derive a synchronization criterion for nonreciprocal coupling related to a numerical quantity that depends on the coupling topology and we present methods for computing this quantity. Index Terms--Chaos, convex programming, coupled oscillators, Lyapunov exponents, Lyapunov functions, nonlinear programming, nonreciprocal coupling, synchronization.
- Published
- 2003
21. STABILITY PROPERTIES OF DIFFERENTIAL SYSTEMS UNDER CONSTANTLY ACTING PERTURBATIONS.
- Author
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CANTARELLI, GIANCARLO and ZAPPALÁ, GIUSEPPE
- Subjects
- *
PERTURBATION theory , *EXTERIOR differential systems , *EUCLIDEAN algorithm , *CAUCHY problem , *INITIAL value problems - Abstract
In this article, we find stability criteria for perturbed differential systems, in terms of two measures. Our main tool is a definition of total stability based on two classes of perturbations. [ABSTRACT FROM AUTHOR]
- Published
- 2010
22. ISLES OF EDEN AND THE ZUK THEOREM IN ℝd.
- Author
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GARAY, BARNABAS M. and CHUA, LEON O.
- Subjects
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CELLULAR automata , *MACHINE theory , *TOPOLOGICAL dynamics , *MATHEMATICAL symmetry , *INVARIANT sets , *LYAPUNOV functions - Abstract
The concept of isles of Eden when generalized from finite cellular automata to classical dynamical systems on ℝd calls the attention to the ZUK Theorem — Z for [Zubov, 1957], U and K for [Ura & Kimura, 1960] — a fifty years old and almost forgotten result in topological dynamics, a classification theorem for compact isolated invariant sets. We provide a simple proof and discuss various consequences including a new result on pointwise Liapunov functions. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
23. Global exponential stability of delayed BAM network on time scale
- Author
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Chen, Anping and Du, Dejun
- Subjects
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ASSOCIATIVE storage , *ARTIFICIAL neural networks , *LYAPUNOV functions , *EXPONENTIAL families (Statistics) , *CALCULUS , *COMPUTER networks - Abstract
Abstract: Some sufficient conditions are derived to ensure the global exponential stability of delayed bi-directional associative memory (BAM) neural network on time scale, using the time scale calculus theory and the Liapunov functional method. The conditions possess highly important significance and can be easily checked in practice by simple algebraic methods. This is the first time applying the time scale calculus theory to unify and improve discrete-time and continuous-time bi-directional associate memory neural network under the same framework. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
24. ISLES OF EDEN AND THE ZUK THEOREM IN ℝd.
- Author
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GARAY, BARNABAS M. and CHUA, LEON O.
- Subjects
CELLULAR automata ,MACHINE theory ,TOPOLOGICAL dynamics ,MATHEMATICAL symmetry ,INVARIANT sets ,LYAPUNOV functions - Abstract
The concept of isles of Eden when generalized from finite cellular automata to classical dynamical systems on ℝ
d calls the attention to the ZUK Theorem — Z for [Zubov, 1957], U and K for [Ura & Kimura, 1960] — a fifty years old and almost forgotten result in topological dynamics, a classification theorem for compact isolated invariant sets. We provide a simple proof and discuss various consequences including a new result on pointwise Liapunov functions. [ABSTRACT FROM AUTHOR]- Published
- 2008
- Full Text
- View/download PDF
25. Liapunov functions for closed relations.
- Author
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Wiandt, T.
- Subjects
- *
HAUSDORFF compactifications , *ORBIT determination , *LYAPUNOV functions , *HAUSDORFF measures , *ORBIT method - Abstract
We create real valued continuous functions which decrease on orbits of relations. This will generalize the notion of Liapunov functions to the more abstract setting of closed relations on compact Hausdorff spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
26. Integral equations, large and small forcing functions: Periodicity
- Author
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Burton, T.A.
- Subjects
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INTEGRAL equations , *FUNCTIONAL equations , *FUNCTIONAL analysis , *OPERATIONAL calculus - Abstract
Abstract: The defining property of an integral equation with resolvent is the relation between and for functions in a given vector space. We study the behaviour of a solution of an integral equation: when is periodic, , while is typified by with . There is a resolvent, , so that We show that the integral so closely approximates that the only trace of that large function, , in the solution is an -function, . In short, that large function has essentially no long-term effect on the solution which turns out to be the sum of a periodic function, a function tending to zero, and an -function. The noteworthy property here is that with great precision the integral can duplicate vector spaces of functions both large and small, both monotone and oscillatory; however, it cannot duplicate a given nontrivial periodic function other than where is constant. The integral is an approximation to for , but contraction mappings show us that precisely at that approximation fails and approaches a nontrivial periodic function. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
27. ON PERIODIC SOLUTIONS OF A TWO-NEURON NETWORK SYSTEM WITH SIGMOIDAL ACTIVATION FUNCTIONS.
- Author
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CHENG-HSIUNG HSU, SUH-YUH YANG, TING-HUI YANG, and TZI-SHENG YANG
- Subjects
- *
ARTIFICIAL neural networks , *ARTIFICIAL intelligence , *NEURONS , *DIFFERENTIAL equations , *CALCULUS - Abstract
In this paper we study the existence, uniqueness and stability of periodic solutions for a two-neuron network system with or without external inputs. The system consists of two identical neurons, each possessing nonlinear feedback and connected to the other neuron via a nonlinear sigmoidal activation function. In the absence of external inputs but with appropriate conditions on the feedback and connection strengths, we prove the existence, uniqueness and stability of periodic solutions by using the Poincaré–Bendixson theorem together with Dulac's criterion. On the other hand, for the system with periodic external inputs, combining the techniques of the Liapunov function with the contraction mapping theorem, we propose some sufficient conditions for establishing the existence, uniqueness and exponential stability of the periodic solutions. Some numerical results are also provided to demonstrate the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
28. A new approach to stability of impulsive functional differential equations
- Author
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Xing, Yepeng and Han, Maoan
- Subjects
- *
DIFFERENTIAL equations , *STABILITY (Mechanics) , *MATHEMATICS , *BESSEL functions - Abstract
In this work, a new approach to stability theory of impulsive functional differential equations is proposed. Instead of putting all components of the state variable
x in one Liapunov function, several functions of partial components ofx , which can be much easier constructed, are used so that the conditions ensuring that stability are simpler and less restrictive. Also, an example is given to illustrate the advantages of the obtained results. [Copyright &y& Elsevier]- Published
- 2004
- Full Text
- View/download PDF
29. Generalized neural networks for spectral analysis: dynamics and Liapunov functions
- Author
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Vegas, José M. and Zufiria, Pedro J.
- Subjects
- *
ARTIFICIAL neural networks , *PRINCIPAL components analysis , *LYAPUNOV stability , *LYAPUNOV functions - Abstract
This paper analyzes local and global behavior of several dynamical systems which generalize some artificial neural network (ANN) semilinear models originally designed for principal component analysis (PCA) in the characterization of random vectors. These systems implicitly performed the spectral analysis of correlation (i.e. symmetric positive definite) matrices. Here, the proposed generalizations cover both nonsymmetric matrices as well as fully nonlinear models. Local stability analysis is performed via linearization and global behavior is analyzed by constructing several Liapunov functions. [Copyright &y& Elsevier]
- Published
- 2004
- Full Text
- View/download PDF
30. Localizing attractors via a generalized La Salle principle
- Author
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Sabatini, M. and Trussoni, L.G.
- Subjects
- *
LYAPUNOV functions , *POLYNOMIALS - Abstract
We are concerned with plane differential systems of the form
x˙ = P(x,y), y˙ = Q(x,y) , withP, Q analytic. We propose a formal-numeric method to localize the attractors and the repellers of the system. Such a method consists of looking for a power series solution to a PDE of the typePSHAPE="BUILT" ALIGN="C" STYLE="S"> + Q∂V ∂x SHAPE="BUILT" ALIGN="C" STYLE="S"> , with μ is an arbitrary analytic function. When∂V ∂y = μ(V)μ(V) = ρ(V − V2) ,ρ > 0 , the attractors are contained in the setV = 1 , the repellers in the setV = 0 . [Copyright &y& Elsevier]- Published
- 2002
- Full Text
- View/download PDF
31. Liapunov and lagrange stability: Inverse theorems for discontinuous systems.
- Author
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Bacciotti, Andrea and Rosier, Lionel
- Abstract
The main result of this paper is a converse Liapunov theorem which applies to systems of ordinary differential equations with a discontinuous righthand side. We treat both the problem of local stability of an equilibrium position and the problem of boundedness of solutions. In particular, we show that in order to achieve a necessary and sufficient condition in terms of continuous Liapunov functions, the classical definitions need to be strengthened in a convenient way. This work was motivated by the recently renewed interest in stabilization by discontinuous feedback and analysis of the state evolution with respect to bounded inputs. To achieve a more general treatment, the exposition is developed in the framework of differential inclusions theory. [ABSTRACT FROM AUTHOR]
- Published
- 1998
- Full Text
- View/download PDF
32. Robust Regulator for Flexible-Joint Robots Using Integrator Backstepping.
- Author
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Abouelsoud, A.
- Abstract
A robust regulator for flexible-joint robots is proposed, which yields constant torque disturbance rejection acting on the links. The design uses the integrator backstepping technique [4,5] to cancel nonlinearities and disturbance not in the range space of the control. Stability of the closed loop system is shown using iterative Liapunov functions. [ABSTRACT FROM AUTHOR]
- Published
- 1998
- Full Text
- View/download PDF
33. The Dynamics of Nonlinear Relaxation Labeling Processes.
- Author
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Pelillo, Marcello
- Abstract
We present some new results which definitively explain thebehavior of the classical, heuristic nonlinear relaxation labelingalgorithm of Rosenfeld, Hummel, and Zucker in terms of theHummel-Zucker consistency theory and dynamical systems theory. Inparticular, it is shown that, when a certain symmetry condition is met,the algorithm possesses a Liapunov function which turns out to be (thenegative of) a well-known consistency measure. This follows almostimmediately from a powerful result of Baum and Eagon developed in thecontext of Markov chain theory. Moreover, it is seen that most of theessential dynamical properties of the algorithm are retained when thesymmetry restriction is relaxed. These properties are also shown tonaturally generalize to higher-order relaxation schemes. Someapplications and implications of the presented results are finallyoutlined. [ABSTRACT FROM AUTHOR]
- Published
- 1997
- Full Text
- View/download PDF
34. Remarks on Dwell Time Solutions and Stability of Families of Nonlinear Vector Fields.
- Author
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Bacciotti, Andrea and Mazzi, Luisa
- Subjects
- *
VECTOR analysis , *LYAPUNOV functions , *TIME series analysis , *STABILITY (Mechanics) , *DIFFERENTIAL equations - Abstract
In this note, we discuss the problem of stability of (finite or infinite) families of continuous vector fields, all of them asymptotically stable but, in general, not exponentially stable. Under a multiple Liapunov function condition and an average dwell time constraint, we prove that the system possesses a form of stability, weaker than the standard one. The advantage of our results is that the conditions imposed on the Liapunov functions can be verified a priori, with no previous knowledge of the integral curves of the family. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
35. Comments on 'chaotic monetary dynamics with confidence'
- Author
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Barnett, William A.
- Subjects
Chaos theory ,Liapunov functions ,Business ,Economics - Abstract
This comment on the Serletis and Shintani paper in this journal issue emphasizes misunderstandings and oversimplifications that exist within the literature, regarding the relevancy and importance of chaos in economics. The paper by Serletis and Shintani provides substantial insights into that frequently misunderstood literature. JEL classification: E40; E50; C32 Keywords: Divisia; Chaos; Lyapunov exponent
- Published
- 2006
36. Stabilnost sustava diferencijskih jednadžbi
- Author
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Jurković, Antonija and Starčević, Maja
- Subjects
Liapunov functions ,PRIRODNE ZNANOSTI. Matematika ,sustav diferencijskih jednadžbi ,equilibrium points ,Ljapunovljeve funkcije ,system of difference equations ,Liapunov theorem ,Ljapunovljev teorem ,NATURAL SCIENCES. Mathematics ,stabilnost točaka ravnoteže - Abstract
U ovome se radu primarno proučava kvalitativna teorija sustava diferencijskih jednadžbi s posebnim naglaskom na stabilnost točaka ravnoteže. Sustavi diferencijskih jednadžbi su iznimno važni jer opisuju mnoge procese u prirodi te promjene i odnose u znanostima poput ekonomije, strojarstva, građevine te općenito u tehničkim znanostima. U radu je posebna pozornost posvećena stabilnosti točaka ravnoteže sustava jednadžbi te smo detaljno obradili različite kriterije stabilnosti. U prvom smo poglavlju dali osnovne teoreme stabilnosti te smo riješili nekoliko primjera diferencijskih jednadžbi. Drugo poglavlje daje osnovnu teoriju sustava diferencijskih jednadžbi. U trećem poglavlju smo analizirali faznu ravninu, definirali Ljapunovljevu funkciju te primijenili Ljapunovljev teorem. In this graduate work we studied qualitative theory of systems of difference equations with special accent on equilibrium points. Systems of difference equations are extremely important because they describe numerous models in natural sciences, economy, engineering, etc. In this work we especially observed stability of equilibrium points of systems of equations. We also thoroughly observed different criteria of stability. In the first chapter we gave some basic stability theorems and we solved several examples of difference equations. The second chapter gives basic theory of systems of difference equations. In the third chapter we did the phase space analysis and we also defined Liapunov functions and applied the Liapunov theorem.
- Published
- 2015
37. Stability conditions and Liapunov functions for quasi-polynomial systems
- Author
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Benito Hernández-Bermejo
- Subjects
Lyapunov function ,Quasi-polynomial systems ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Quasi-polynomial ,Stability (probability) ,symbols.namesake ,Lotka-Volterra systems ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Quantitative Biology::Populations and Evolution ,Physics - Biological Physics ,Mathematics - Dynamical Systems ,Mathematical Physics ,Mathematics ,Equilibrium point ,Applied Mathematics ,Mathematical analysis ,Ode ,Mathematical Physics (math-ph) ,Liapunov functions ,Stability conditions ,Liapunov function ,Biological Physics (physics.bio-ph) ,Mathematics - Classical Analysis and ODEs ,symbols ,Stability - Abstract
The stability of equilibrium points of quasi-polynomial systems of ODEs is considered. The criteria and Liapunov functions found generalize those traditionally known for Lotka-Volterra equations, that now appear as a particular case.
- Published
- 2002
- Full Text
- View/download PDF
38. A stability theory for perturbed differential equations
- Author
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Sheldon P. Gordon
- Subjects
Liapunov functions ,asymptotic behavior of solutions ,asymptotic equivalence. ,Mathematics ,QA1-939 - Abstract
The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is X′=f(t,X) and the associated perturbed differential equation is Y′=f(t,Y)+g(t,Y).
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- 1979
- Full Text
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39. Control and filtering of time-varying linear systems via parameter dependent Lyapunov functions
- Author
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Renato Alves Borges, Peres, Pedro Luis Dias, 1960, Trofino Neto, Alexandre, Palhares, Reinaldo Martinez, Santos, Juan Francisco Camino dos, Val, João Bosco Ribeiro do, Universidade Estadual de Campinas. Faculdade de Engenharia Elétrica e de Computação, Programa de Pós-Graduação em Engenharia Elétrica, and UNIVERSIDADE ESTADUAL DE CAMPINAS
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Funções de Lyapunov ,Liapunov functions ,Control theory ,Teoria do controle ,Mathematical optimization ,Linear time invariant systems ,Stability ,Sistemas incertos variantes no tempo ,Estabilidade ,Otimização matemática - Abstract
Orientador: Pedro Luis Dias Peres Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação Resumo: A principal contribuição desta tese é a proposta de condições de sintese de filtros e controladores lineares, tanto robustos quanto dependentes de parametros, para sistemas discretos variantes no tempo.Os controladores, ou filtros, são obtidos solucionando problemas de otimização formulados em termos de desigualdades matriciais bililineares, por meio de um metodo que se baseia na alternancia de problemas convexos descritos por desigualdades matriciais lineares. Para obtenção das condiçoes de sintese foram utilizadas tanto funções de Lyapunov afins nos parâmetros quanto ametros, alem de variáveis multi-afins em diferentes instantes de tempo dos parâmetros, alem de variaveis extras introduzidas pelo lema de Finsler. Nesse contexto, sao tratados problemas de sintese com custo garantido H, assegurando robustez em relação a incertezas não estruturadas. Simulaçoes numéricas ilustram a eficiencia dos metodos propostos em termos de desempenho H quando comparados com outros metodos da literatura Abstract: The main contribution of this dissertation is to propose conditions for linear filter and controller design, considering both robust and parameter dependent structures, for discrete time-varying systems. The controllers, or filters, are obtained through the solution of optimization problems, formulated in terms of bilinear matrix inequalities, using a method that alternates convex optimization problems described in terms of linear matrix inequalities. Both affine and multi-affine in different instants of time (path dependent)Lyapunov functions were usedto obtain the design conditions, as wellas extra variables introduced bythe Finsler's lemma.Design problems that take into account an H guaranteed cost were investigated, providing robustness with respect to unstructured uncertainties. Numerical simulations show the effciency of the proposed methods in terms of H performance when compared with other strategies from the literature Doutorado Automação Doutor em Engenharia Elétrica
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- 2009
40. Discussions on : 'Global output stability for systems described by retarded functional differential equations : Lyapunov characterizations' and on : 'Input-to-output stability for systems described by retarded functional differential equations'
- Author
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Frédéric Mazenc, Water Resource Modeling (MERE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de la Recherche Agronomique (INRA), Analyse des Systèmes et Biométrie (ASB), Institut National de la Recherche Agronomique (INRA), and Rapaport, Alain
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Lyapunov function ,0209 industrial biotechnology ,Differential equation ,[INFO.INFO-OH]Computer Science [cs]/Other [cs.OH] ,INPUT TO OUTPUT STABILITY ,AUTOMATIQUE ,02 engineering and technology ,01 natural sciences ,Stability (probability) ,symbols.namesake ,020901 industrial engineering & automation ,Distributed parameter system ,Stability theory ,TIME VARYING SYSTEM ,STABILITE ENTREE SORTIE ,[INFO]Computer Science [cs] ,0101 mathematics ,[MATH]Mathematics [math] ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,LIAPUNOV FUNCTIONS ,SYSTEME PARAMETRE VARIABLE ,010102 general mathematics ,Mathematical analysis ,General Engineering ,DIFFERENTIAL EQUATION ,[INFO.INFO-OH] Computer Science [cs]/Other [cs.OH] ,Stochastic partial differential equation ,DELAY TIME ,symbols ,TEMPS RETARD ,LIAPUNOV FUNCTIONS INPUT TO OUTPUT STABILITY DIFFERENTIAL EQUATION DELAY TIME TIME VARYING SYSTEM AUTOMATIQUE FONCTION LIAPUNOV TEMPS RETARD STABILITE ENTREE SORTIE SYSTEME PARAMETRE VARIABLE ,FONCTION LIAPUNOV ,Differential algebraic equation - Abstract
aeres : ACL; International audience; no abstract
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- 2008
41. New upper estimates for the solution of the continuous algebraic Lyapunov equation
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Savov, Svetoslav and Popchev, Ivan
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Control engineering -- Models ,Control engineering -- Methods ,Liapunov functions - Abstract
A new result for bounding the summations for solution eigen-values of the algebraic Lyapunov equation is presented. This makes possible to generate the best known upper scalar solution estimates. Index Terms--Lyapunov equation, matrix measure, solution bounds, trace.
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- 2004
42. The neuron physiology and Hodgkin-Huxley and FitzHugh-Nagumo models
- Author
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Peterson Taylor C. Barbosa, Saa, Alberto Vazquez, 1966, Yang, Hyun Mo, Maia, Leonardo Paulo, Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica, Programa de Pós-Graduação em Matemática Aplicada, and UNIVERSIDADE ESTADUAL DE CAMPINAS
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Funções de Lyapunov ,Liapunov functions ,Dinâmica simbólica ,Physiology - Mathematics ,Dynamics, Symbolic ,Fisiologia - Matemática - Abstract
Orientador: Alberto Vazquez Saa Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica Resumo: Apresentaremos neste trabalho descrições detalhadas da modelagem e conceitos fisiológicos associados aos modelos de Hodgkin-Huxley e Fitzhugh-Nagumo, para em seguida compararmos suas características. Tentaremos justificar a opção pelo modelo de Fitzhugh-Nagumo como objeto de estudo viável pra interações entre neurônios, e em seguida introduziremos um tipo de interação na qual o comportamento do sistema para dois neurônios apresenta fenômenos de comportamento irregular. Para isso, mostraremos alguns resultados referentes à esta interação, chamada acoplamento repulsivo, descrito por Yanagita et al. (2005), e analisaremos fatos relativos às características qualitativas e quantitativas das equações envolvidas, como os Expoentes de Liapunov e o Intervalo entre Disparos, a partir de alguns resultados numéricos pertinentes Abstract: We present in this work some detailed descriptions concerning modeling and physiological concepts on Hodgkin-Huxley and Fitzhugh-Nagumo models, and after that to compare its main features. We will try to justify the choice for the Fitzhugh-Nagumo model as a viable study object for neuronal interactions, and than we introduce a particular kind of interaction in which the reaction of a two-neuron system shows irregular behavior phenomena. In order to achieve this, we also present some results associated to this interaction, called repulsive coupling, described by Yanagita et al. (2005), and afterward, we analyze some facts related to quantitative and qualitative characteristics of the involved equations, such as Liapunov Exponents and the Interspike Interval, based on numerical simulations of interest Mestrado Biomatemática Mestre em Matemática Aplicada
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- 2007
43. Noise-induced global asymptotic stability
- Author
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Mackey, Michael C., Longtin, André, and Lasota, Andrzej
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- 1990
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44. Stability and Control of Mobile Communications Systems With Time Varying Channels
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Consider the forward link of a mobile communications system with a single transmitter and rather arbitrary randomly time varying channels connecting the base to the mobiles. Data arrives at the base in some random way (and might have a burst character) and is queued according to the destination until transmitted. The main issues are the allocation of transmitter power and time to the various queues in a queue and channel-state dependent way to assure stability and good operation. The control decisions are made at the beginning of the (small) scheduling intervals. Stability methods are used to allocate time and power. Many schemes of current interest can be handled: For example, CDMA with control over the bit interval and power per bit, TDMA with control over the time allocated, power per bit, and bit interval, as well as arbitrary combinations. There might be random errors in transmission which require retransmission. The channel-state process might be known or only partially known. The details of the scheme are not directly involved; all essential factors are incorporated into a rate and error function. The system and channel process are scaled by speed. Under a stability assumption on a model obtained from the mean drift, and some other natural conditions, it is shown that the scaled physical system can be controlled to be stable, uniformly in the speed, for fast enough speeds. Owing to the non-Markov nature of the problem, we use the perturbed Liapunov function method, which is very useful for the analysis of non-Markovian systems. Finally, the stability method is used to actually choose the power and time allocations. The allocation will depend on the Liapunov function. But each such function corresponds loosely to an optimization problem for some performance criterion. Since there is a choice of Liapunov functions, various performance criteria can be taken into account in the allocations. The resulting controls are quite reasonable.
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- 2001
45. Stability robustness analysis of linear systems
- Author
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Karan, Mehmet, Sezer, Erol, Sezer, M. Erol, and Elektrik-Elektronik Mühendisliği Anabilim Dalı
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Robust Stability ,Additive perturbations ,Elektrik ve Elektronik Mühendisliği ,Liapunov stability ,Linear Systems ,Additive Functions ,System Analysis ,TK153 .K18 1990 ,Interconnected systems ,Discrete-time systems ,Liapunov Functions ,Electric Networks ,Electrical Engineering-Mathematics ,Stability ,Electrical and Electronics Engineering - Abstract
ÖZET DO?RUSAL SİSTEMLERİN KARARLILI?ININ GÜRBÜZLÜK AÇISINDAN İNCELENMESİ Mehmet Karan Elektrik ve Elektronik Mühendisliği Bölümü Master Tez Yöneticisi: Prof. Dr. M. Erol Sezer Ocak, 1990 Bu tezde, doğrusal, zamana göre değişmeyen, sürekli ve ayırtık zamanlı sistemlerin kararlılığının gürbüzlüğü araştırılmıştır. Yalnızca durum uzayı düşünülmüştür. Sürekli zamanlı sistemlerin gürbüz kararlılığına ilişkin varolan sonuçlar, Liapunov yaklaşımı ve özdeğerlerin sürekliliği açısından gözden geçirilmiştir. Ayrıca, tek parametreli ya da çok parametreli sistem belirsizlikleri altında ayırtık zamanlı sistemler için de benzer sonuçlar elde edilmiştir. Ayırtık zamanlı sistemlerin gürbüzlük alanlarını tanımlayan eşitsizlikler içinde belirsizlik parametrelerinin doğrusal ve ikildoğrusal gözükmelerinden kaynaklanan doğal bîr zorluk da, problemi daha yüksek boyutlu sürekli zamanlı bir sistemin gürbüzlüğüne dönüştürülerek aşılmıştır. Son olarak, ayırtık zamanlı birbirine bağli sistemlerin kararlılık gürbüzlüğü çalışılmış ve değişik yöntemler karşılaştırılmıştır. Anahtar sözcükler: Gürbüz kararlılık, Ayırtık zamanlı sistemler, Toplam sal belirsizlikler, Liapunov kararlılığı, Bağlı sistemler. ABSTRACT STABILITY ROBUSTNESS ANALYSIS OF LINEAR SYSTEMS Mehmet Karan M. S. in Electrical and Electronics Engineering Supervisor; Prof. Dr. M. Erol Sezer February, 1990 In this thesis, robustness of stability of linear, time-invariant, continuous- and discrete-time systems is investigated. Only state-space models and additive perturbations are considered. Existing results concerning stability robustness of continuous- time systems, based on Liapunov approach and continuity of eigenvalues, are reviewed; and similar results for discrete- time systems under single- and multi-parameter additive perturbations are derived. An inherent difficulty which originates from mixed linear and bilinear appearance of perturbation parameters in inequalities defining robustness regions of discrete-time systems is resolved by transforming the problem to robustness of a higher order continuous-time system. Finally, stability robustness of discrete-time interconnected systems is studied, and various approaches are compared. Keywords: Robust Stability, Discrete-time systems, Additive perturba tions, Liapunov stability, Interconnected systems. IV 69
- Published
- 1990
46. Information-theoretic stability and evolution criteria in irreversible thermodynamics
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Pfaffelhuber, E.
- Published
- 1977
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47. Detumbling of Two Inter-Connected Rigid Bodies Using Liapunov Methods for Feedback Control.
- Abstract
A satellite is modeled as two rigid bodies interconnected by a joint with one degree of freedom. In a tumbling situation, the satellite is brought to a condition of stabilized spin about its maximum moment of inertia axis. This is accomplished by a torque applied to the internal joint by an electronic motor. The control law which supplies the input to the torque motor is derived using both nonlinear and linear feedback. Liapunov stability theory is used to develop both methods. Various models are devised with varying rigid body physical properties in order to draw comparisons, and results are presented for each case. While the nonlinear control law was unsuccessful for the case presented, due to a violation in theory, the linear feedback approach produced favorable outcome for a variety of linear control law cases. Keywords: Theses; Attitude control; Spacecraft recovery; Equations of motion; Computer program verification. (Author)
- Published
- 1986
48. On a Functional Equation Arising in the Stability Theory of Difference-Differential Equations
- Abstract
The functional differential equation Q'(t) = AQ(t) + B((Q(tau - t)) to the T-th power), - infinity t infinity, where A,B are n x n constant matrices, tau or = 0, Q(t) is a differentiable n x n matrix and (Q(t)) to the T-th power is its transpose, is studied. Existence, uniqueness and an algebraic representation of its solutions is given. This equation, of considerable interest in its own right, naturally arises in the construction of Liapunov functionals of difference differential equations of the type dx(t)/dt = Cx(t) + Dx(t-tau), where C,D are constant n x n matrices. The role played by the matrix Q(t) is analogous to the one played by a positive definite matrix in the construction of Liapunov functions for ordinary differential equations. In this paper, we show that, in spite of the functional nature of this equation, the linear vector space of its solutions is n squared; moreover, ,we give a complete algebraic characterization of its solutions and indicate computationally simple methods for obtaining these solutions, which we illustrate through an example. Finally, we briefly indicate how to obtain solutions for the nonhomogeneous problem, through the usual variation of constants method.
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- 1976
49. A Liapunov Functional for Linear Volterra Integrodifferential Equations.
- Abstract
Liapunov functionals of quadratic form have been used extensively for the study of the stability properties of linear ordinary, functional and partial differential equations. In this paper, a quadratic functional V is constructed for a linear Volterra intergrodifferential equation. This functional, and its derivative, is more general than previously constructed ones and still retains desirable computational qualities; moreover, it represents a natural generalization of the Liapunov function for ordinary differential equations. The method of construction used suggests functionals which are useful for more general equations.
- Published
- 1982
50. BENDIX TECHNICAL JOURNAL, VOL. 1, NO. 3, AUTUMN 1968: AUTOMATIC CONTROLS
- Abstract
Contents: Learning control systems; Computing process identifiers with pattern-recognition techniques; A fail-operational stability augmentation system; A format method for generating Liapunov functions; A fluidic closed-loop position control system; Control theory and the nuclear arms race; The intermediate-amplitude selective gate; Nonlinear effects on servomechanism dynamics; and The effect of compliant sensor mounting on missile control.
- Published
- 1968
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