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Topic: Lyapunov stability

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	Aleksandr Lyapunov - Wikipedia, the free encyclopedia
		Lyapunov had already begun to study this stability in his previous two-years attempts at solving the task.
		Lyapunov lectured at the university on themes from theoretical mechanics, integrals of differential equations and the theory of probability.
		His main preoccupations were the stability of equilibria and the motion of mechanical systems, the model theory for the stability of uniform turbulent liquid, and particles under the influence of gravity.
	en.wikipedia.org /wiki/Aleksandr_Lyapunov (1661 words)

	Lyapunov stability - Wikipedia, the free encyclopedia
		The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behaviour of different but "nearby" solutions to differential equations.
		The general study of the stability of solutions of differential equations is known as stability theory.
		Lyapunov's realisation was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function can be found to satisfy the above constraints.
	en.wikipedia.org /wiki/Lyapunov_stability (841 words)

	Aleksandr Lyapunov
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857 - November 3, 1918) was a Russian mathematician, mechanician and physicist.
		Lyapunov had lectured already from 1880 at the faculty for mechanics and this had taken him a lot of time.
		His main preoccupations were the stability of equilibrium and the motion of mechanical system, the model theory for the stability of uniform turbulent liquid, and the particles under the influence of gravity.
	www.ebroadcast.com.au /lookup/encyclopedia/ly/Lyapunov.html (1609 words)

	Lyapunov stability - Encyclopedia, History, Geography and Biography
		Lyapunov stability is applicable to only unforced (no control input) dynamical systems.
		A system is said to be stable about the equilibrium point "in the sense of Lyapunov" if for every ε, there is a δ such that:
		Lyapunov stability, Lyapunov stability theorems, Lyapunov second theorem on stability, Stability for state space models, Example, Barbalat's lemma and stability of time-varying systems, See also and Dynamical systems.
	www.arikah.com /encyclopedia/Lyapunov_stability (754 words)

	Lyapunov biography
		Lyapunov was not the only one being coached by Sechenov who was teaching his own daughter Natalia Rafailovna Sechenov at the same time.
		He showed that a sufficient condition for stability is that the second and higher variations of the potential energy are positive.
		Lyapunov established that with variation in the angular velocity of revolution Maclaurin ellipsoids pass into Jacobi ellipsoids.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Lyapunov.html (1229 words)

	Stability - Wikipedia, the free encyclopedia
		Directional stability, or the tendency for a body moving with respect to a medium to point in the direction of motion.
		Stability theory, the study of the stability of solutions to differential equations and dynamical systems
		Stable polynomial, a polynomial whose roots all lie in the open left half-plane or whose roots all lie in the open unit disk
	en.wikipedia.org /wiki/Stability (283 words)

	System Stability (Site not responding. Last check: 2007-10-08)
		We know that a system is Lyapunov stable if its eigenvalues are the left-half plane, and we know that a system is BIBO stable if its poles are in the left-half plane.
		This means that it is possible for a system to be both BIBO stable and Lyapunov unstable.
		Therefore, the system is Lyapunov unstable (one of its eigenvalues is in the right-half plane.
	academic.csuohio.edu /simond/linearsystems/stability (273 words)

	Lyapunov Stability (Site not responding. Last check: 2007-10-08)
		A Lyapunov stable system is a system for which the states will remain bounded for all time, for any finite initial condition.
		A continuous-time linear time-invariant system is Lyapunov stable (internally stable) if and only if all the eigenvalues of A have real parts less than or equal to 0, and those with real parts equal to 0 are nonrepeated.
		Lyapunov stability requires that the state remain bounded for all time, for all initial conditions - not just for some specific initial condition.
	academic.csuohio.edu /simond/linearsystems/stability/lyapunov (351 words)

	UK Nonlinear News 14 (November 1998): Book Review
		stability with respect to part of the variables, is distinct from extensions of the classical stability (with respect to all the variables) to cases when only part of the variables exhibit stability whilst the remaining variables are bounded.
		Stability with respect to part of the variables is shown to be equivalent to Lyapunov stability with respect to all the variables of the auxiliary system.
		Stability with respect to all of the variables of the linearisation of the extended system then implies stability with respect to part of the variables for the original system.
	www.amsta.leeds.ac.uk /Applied/news.dir/issue14.dir/art/review2.html (972 words)

	Lyapunov
		BK11CH12 CHAPTER XII The Rostovs De estabilidad lyapunov remained in Moscow till the Chaos lyapunov matlab first of September, that is, till the eve of the enemy's entry into the city.
		De estabilidad lyapunov Nicholas was somewhere Lyapunov stability system uncertain with the army and had not sent a word since his last letter, in Exponent lyapunov which he had given a detailed account of his meeting with De estabilidad lyapunov Princess Mary.
		De estabilidad lyapunov He got Petya transferred from Obolenski's regiment to Bezukhov's, which was in Lyapunov stability system uncertain training near Moscow.
	alshain.thraddash.com /article/lyapunov.html (645 words)

	abstract1.html (Site not responding. Last check: 2007-10-08)
		Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability.
		Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given.
		Given an equilibrium and a compact inner bound on the domain of attraction, we use Lyapunov techniques to compute an explicit lower bound on the observer gain such that the specified equilibrium is asymptotically stable for the closed-loop system, with a domain of attraction that contains the specified inner bound.
	www.aero.iitb.ac.in /~bhat/jpubabs.html (1322 words)

	4.3 Lyapunov Exponent
		This number, called the Lyapunov exponent "λ" [lambda], is useful for distinguishing among the various types of orbits.
		A Lyapunov exponent of zero indicates that the system is in some sort of steady state mode.
		The logistic equation is superstable at this point, which makes the Lyapunov exponent equal to negative infinity (the limit of the log function as the variable approaches zero).
	hypertextbook.com /chaos/43.shtml (1075 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Lyapunov, while studying the asymptotic stability of solutions of differential systems, proved a theorem which yields a necessary and sufficient condition for stability of a matrix.
		Lyapunov's theorem was a breakthrough in the research of stability.
		The corresponding generalizations of the concepts of Lyapunov scaling factors are applied to study the Lyapunov semistability of block triangular matrices.
	www.siam.org /confpart/showmin.cfm?SESSIONCODE=1067 (319 words)

	Welcome to University of Monaco (Site not responding. Last check: 2007-10-08)
		Asymptotic and Lyapunov Stability of constrained and Poisson equilibria
		First, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account the asymptotically stable behavior that may occur in certain constrained systems, such as Poisson and Leibniz dynamical systems.
		Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al.
	www.monaco.edu /MonthlyNews/Articles/VictorPlanas/victor.php (171 words)

	Lyapunov (print-only)
		The importance of this thesis is emphasised in several articles such as [8], [12], [16] which were all written to celebrate the centenary of the publication of this fundamental contribution.
		In 1901 Lyapunov was elected to the Russian Academy of Sciences in St Petersburg and in the following year became an academician in applied mathematics of the Academy.
		Topics considered in [8] include: stability, particularly the stability of critical points; the construction and the application of the Lyapunov function; stability of functional- differential equations; the second Lyapunov method; and the method of the Lyapunov vector function in stability theory and nonlinear analysis.
	www-groups.dcs.st-and.ac.uk /history/Printonly/Lyapunov.html (1092 words)

	Paper: Lyapunov Stability Analysis of the Quantization Error for DCS :: (Site not responding. Last check: 2007-10-08)
		The Lyapunov function works in parallel during DCS learning, and is able to provide a measure of the effective placement of neural units during the NN s approximation.
		The objective of mathematical theory of nonlinear stability analysis is often misunderstood as the process of nding a solution for the differential equation(s) that govern the dynamics, either analytically or numerically (simulation studies).
		It s often seen that if such systems are stable under one de nition of stability they may tend to become unstable under other de nitions [5].
	computing.breinestorm.net /dynamic+cell+cmu+vol+structures (576 words)

	How Should Ecological Stability be Defined?
		Despite these advantages, I argue ecological stability should not be defined as Lyapunov stability, for two reasons.
		Second, analysis of different criteria for Lyapunov stability demonstrates that the conditions an ecological system must satisfy to be Lyapunov stable are biologically unrealistic.
		A close correspondence between the mathematical theory of Lyapunov stability and well-confirmed theoretical principles ensures this in physics, but the lack of the latter within ecology precludes Lyapunov stability from playing a similarly fruitful role.
	www.ishpssb.org /ocs/viewpaper.php?id=150&print=1 (221 words)

	Stability and Bifurcation
		In the time-continuous case, this stability area is the half-plane left of the imaginary axis, whereas in the time-discrete case it is the unit circle around the origin.
		It helps to reduce the dimensionality of the phase space to the dimensionality of the so-called center manifold which in the bifurcation point is tangentially to the eigenspace of the marginal modes of the linear stability analysis (i.e., the eigenfunctions which neither decay nor expand exponentially).
		The bifurcation diagrams of a Hopf and a period doubling bifurcation are similar to the diagram of a pitchfork bifurcation.
	monet.physik.unibas.ch /~elmer/pendulum/bif.htm (1853 words)

	IEEE TAC: Scanning the Issue
		Sufficient conditions for uniform stability, uniform asymptotic stability, exponential stability, and instability of an invariant set of hybrid dynamical systems are established.
		The main concentration is on building the foundations of a Lyapunov-like stability theory that is applicable to hybrid systems: "multiple Lyapunov functions" are introduced as a tool for analyzing their Lyapunov stability, and iterated function systems are proposed for Lagrange stability.
		The use of piecewise quadratic Lyapunov functions is a powerful extension of quadratic stability that covers polytopic Lyapunov functions as a special case.
	www.nd.edu /~pantsakl/scanningtac.htm (1506 words)

	CRC Press Online
		Domains of Lyapunov stability properties and practical stability properties are concepts that are taking the stability theory of nonlinear dynamical systems in new directions.
		Stability Domains is an up-to-date account of stability theory with a unique emphasis on stability domains.
		It sets forth and proves stability criteria in their complete form and presents various approaches based on those criteria to estimating stability domains.
	www.crcpress.com /shopping_cart/products/product_detail.asp?sku=TF1667&parent_id=1181&pc= (411 words)

	[No title]
		On the Input-to-State Stability Property Eduardo D. Sontag\Lambda Department of Mathematics Rutgers University, New Brunswick, NJ 08903y Abstract The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations.
		Second, boundedness (finite gain) is far too strong a requirement for general nonlinear operators, and it must be replaced by "nonlinear gain estimates," in which the norms of output signals are bounded by a nonlinear function of the norms of inputs; the definition of iss incorporates such gains in a natural way.
		Stability is clear: for small x and z, trajectories coincide with those that would result if uniformity would hold globally on x (cf.
	www.math.rutgers.edu /~sontag/iss-ejc.html (7051 words)

	BrainDex the knowledge source - Free Online Encyclopedia - Lyapunov stability (Site not responding. Last check: 2007-10-08)
		Let us consider that the origin is the equilibrium point (EP) of the system.
		A system is said to be stable "in the sense of Lyapunov" (i.s.L.) if for every ε, there is a δ such that:
		A state space model $\dot{\textbf{x}} = A\textbf{x} is asymptotically$ stable iff
	www.braindex.com /encyclopedia/index.php/Lyapunov_stability (285 words)

	Nonlinear, Time-Varying DWNs
		wave variable, a Lyapunov function can be defined simply as the sum of wave energies associated with the state variables.
		As long as the infinite-precision filter is asymptotically stable, the quantized filter must be also.
		In practice, Lyapunov stability in a DWN is normally guaranteed by ensuring that wave-variables are never amplified by nonlinear operations or time-varying gains.
	ccrma-www.stanford.edu /~jos/wgj/Nonlinear_Time_Varying_DWNs.html (860 words)

	FDN Stability
		Stability of the FDN is assured when the norm of the state vector
		That is, a sufficient condition for FDN stability is
		An alternative stability proof may be based on showing that an FDN is a special case of a passive digital waveguide network (derived in §G.13).
	ccrma-www.stanford.edu /~jos/smithbook/FDN_Stability.html (263 words)

	Report
		On the criterion of asymptotical stability for index-1 tractable DAEs.
		The Lyapunov stability of the trivial solution is discussed.
		As a criterion of the asymptotical stability we propose a numerical parameter $\mbox{\sl\ae}(A,B)$ characterizing the property of the index-1 matrix pencil $\{A, B\}$ to have all finite eigenvalues within the negative complex half-plane.
	www.math.tu-berlin.de /preprints/abstracts/T.Stykel_MonNov12114138.rdf.html (78 words)

	cpubabs (Site not responding. Last check: 2007-10-08)
		We consider the group of phase space symmetries of a stable linear Hamiltonian system, and characterize the subgroup of symmetries whose elements preserve the time averages of quadratic functions along the trajectories of the system.
		More specifically, we show that a system is convergent if all of its orbits have finite arc length, while an equilibrium is Lyapunov stable if the arc length (considered as a function of the initial condition) is continuous at the equilibrium, and semistable if the arc length is continuous in a neighborhood of the equilibrium.
		stable if and only if it is asymptotically stable and has a negative degree of homogeneity.
	www.aero.iitb.ac.in /~bhat/cpubabs.html (2002 words)

	MATHnetBASE: Mathematics Online
		Stability Domains is an up-to-date account of stability theory with particular emphasis on stability domains.
		It also introduces classical Lyapunov and practical stability theory for time-invariant nonlinear systems in general and for complex (interconnected, large scale) nonlinear dynamical systems in particular.
		This is a complete treatment of the theory of stability domains useful for postgraduates and researchers working in this area of applied mathematics and engineering.
	www.mathnetbase.com /ejournals/books/book_summary/summary.asp?id=1511 (126 words)

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