scholarly journals On the Stability of Some Continuous Systems Subjected to Random Excitation

1970 ◽  
Vol 37 (3) ◽  
pp. 623-628 ◽  
Author(s):  
R. H. Plaut ◽  
E. F. Infante
Keyword(s):  
Sufficient Conditions ◽  
Random Force ◽  
Random Excitation ◽  
Continuous Systems ◽  
Almost Sure Stability ◽  
Varying Coefficients ◽  
Linear Partial Differential Equations ◽  
Time Varying Coefficients ◽  
The Stability

A method for the determination of sufficient conditions for the almost-sure stability of some continuous systems of physical interest is presented. The motions of the systems under consideration are assumed to be described by linear partial-differential equations with time-varying coefficients of a random nature. The method presented, which is of a rather general form, is restricted for the sake of simplicity and ease of computations and is applied to problems of elastic columns and plates, a cantilever beam subjected to a random follower force, and a string excited by a pressure-type random force. The emphasis both in the computations and in the nature of the method is on simplicity of computations and in the determination of stability conditions with a minimum of assumptions.

Nonlinear Marine Structures With Random Excitation

1988 ◽  
Vol 110 (3) ◽  
pp. 246-253 ◽  
Author(s):  
E. R. Jefferys
Keyword(s):  
Narrow Band ◽  
Equations Of Motion ◽  
Random Excitation ◽  
Wave Excitation ◽  
Offshore Structure ◽  
Chaotic Motions ◽  
Varying Coefficients ◽  
Excitation Time ◽  
Vessel Response ◽  
Time Varying Coefficients

Various important types of offshore structure contain significant nonlinearities or time-varying coefficients in their equations of motion. Well-known examples include tension leg platforms, free-hanging risers, single-buoy moorings, ships moored against fenders and vessels constrained by stiffening moorings. When subject to sinusoidal wave excitation, time domain mathematical models of these structures can display large subharmonic or chaotic motions. This paper shows that such behavior is often an artifact of the regularity of the excitation and is usually unlikely to present a significant problem in a random sea. Narrow-band vessel response can, however, generate near-harmonic motions to create conditions in which these instabilities may become important.

Almost sure stability of maxima

1973 ◽  
Vol 10 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

scholarly journals CONVERGENCE OF SOLUTIONS OF TIME-VARYING LINEAR SYSTEMS WITH INTEGRABLE FORCING TERM

2008 ◽  
Vol 78 (3) ◽  
pp. 445-462 ◽  
Author(s):  
JITSURO SUGIE
Keyword(s):  
Sufficient Conditions ◽  
Zero Solution ◽  
Time Varying ◽  
Forcing Term ◽  
Varying Coefficients ◽  
Perturbed System ◽  
All Solutions ◽  
Time Varying Coefficients ◽  
Image Position ◽  
Homogeneous Linear

AbstractThe following system is considered in this paper: The primary goal is to establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) and a forcing term p(t) for all solutions to converge to the origin (0,0) as $t \to \infty $. Here, the zero solution of the corresponding homogeneous linear system is assumed to be neither uniformly stable nor uniformly attractive. Sufficient conditions are given for asymptotic stability of the zero solution of the nonlinear perturbed system under the assumption that q(t,0,0)=0.

scholarly journals Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tian Zhang ◽  
Chuanhou Gao
Keyword(s):  
Differential Equations ◽  
Decay Rate ◽  
Broad Class ◽  
General Decay ◽  
Polynomial Decay ◽  
Almost Sure Stability ◽  
General Decay Rate ◽  
Varying Coefficients ◽  
Highly Nonlinear ◽  
Time Varying Coefficients

<p style='text-indent:20px;'>This paper focuses on the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L<inline-formula><tex-math id="M3">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, <inline-formula><tex-math id="M4">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.</p>

Stability and Robust Stabilization of 2-D Continuous Systems in Roesser Model Based on GKYP Lemma

2017 ◽  
Vol 8 (3) ◽  
pp. 990 ◽  
Author(s):  
Ismail Errachid ◽  
Abdelaziz Hmamed
Keyword(s):  
Robust Stabilization ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Continuous Systems ◽  
Roesser Model ◽  
Model Based ◽  
Inequality Technique ◽  
The Stability ◽  
Conditions Of Stability ◽  
Linear Matrix Inequality Technique

This paper is concerned with the stability and Robust stabilization problem for 2-D continuous systems in Roesser model, based on Generalized Kalman$-$Yakubovich$-$Popov lemma in combination with frequency-partitioning approach. Sufficient conditions of stability of the systems are formulated via linear matrix inequality technique. Finally, numerical examples are given to illustrate the effectiveness of the proposed method.

scholarly journals Almost Sure Stability of Stochastic Neural Networks with Time Delays in the Leakage Terms

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Mingzhu Song ◽  
Quanxin Zhu ◽  
Hongwei Zhou
Keyword(s):  
Neural Networks ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Stochastic Neural Networks ◽  
Stochastic Delay Differential Equations ◽  
Almost Sure Stability ◽  
Infinitesimal Operator ◽  
Lasalle Invariant Principle ◽  
The Stability ◽  
Delay Differential

The stability issue is investigated for a class of stochastic neural networks with time delays in the leakage terms. Different from the previous literature, we are concerned with the almost sure stability. By using the LaSalle invariant principle of stochastic delay differential equations, Itô’s formula, and stochastic analysis theory, some novel sufficient conditions are derived to guarantee the almost sure stability of the equilibrium point. In particular, the weak infinitesimal operator of Lyapunov functions in this paper is not required to be negative, which is necessary in the study of the traditional moment stability. Finally, two numerical examples and their simulations are provided to show the effectiveness of the theoretical results and demonstrate that time delays in the leakage terms do contribute to the stability of stochastic neural networks.

Global exponential stability of cellular neural networks with multi-proportional delays

2015 ◽  
Vol 08 (06) ◽  
pp. 1550071 ◽  
Author(s):  
Liqun Zhou ◽  
Yanyan Zhang
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Varying Coefficients ◽  
Proportional Delays ◽  
Time Varying Coefficients ◽  
Delay Differential ◽  
Delay Dependent

In this paper, a class of cellular neural networks (CNNs) with multi-proportional delays is studied. The nonlinear transformation yi(t) = xi( e t) transforms a class of CNNs with multi-proportional delays into a class of CNNs with multi-constant delays and time-varying coefficients. By applying Brouwer fixed point theorem and constructing the delay differential inequality, several delay-independent and delay-dependent sufficient conditions are derived for ensuring the existence, uniqueness and global exponential stability of equilibrium of the system and the exponentially convergent rate is estimated. And several examples and their simulations are given to illustrate the effectiveness of obtained results.

The Stability of Ejected Beams

1997 ◽  
Author(s):  
Anthony Renshaw
Keyword(s):  
Free Vibration ◽  
Lyapunov Functional ◽  
Stability Boundary ◽  
Positive Definiteness ◽  
Dimensionless Time ◽  
Time Varying ◽  
System Equation ◽  
Varying Coefficients ◽  
Time Varying Coefficients ◽  
The Stability

Abstract This paper examines the free vibration and stability of an Euler-Bernoulli beam ejected from or, equivalently, drawn into an orifice at an arbitrary angle relative to gravity. A stability boundary for this system is presented in terms of two dimensionless, time varying parameters, one describing the beam bending stiffness and the other indicating the axial tension induced by gravity. This stability boundary is the limit of positive definiteness of a Lyapunov functional for the system. The Lyapunov functional is the Jacobi integral of the system, which qualifies as a Lyapunov functional for many gyroscopic systems. The ejected beam system is gyroscopic when the time varying coefficients in the system equation are held constant. It is also shown that initially, the free vibration problem for the ejected beam has the same vibration modes shapes as an ordinary cantilever beam but that the magnitude and period of vibration grow as the square root of time.

scholarly journals Model of time-varying linear systems and Kolmogorov equations

2015 ◽  
Vol 25 (2) ◽  
pp. 201-214
Author(s):  
Assen V. Krumov
Keyword(s):  
Analytical Solution ◽  
Linear Systems ◽  
Sufficient Conditions ◽  
Original System ◽  
Approximate Model ◽  
Time Varying ◽  
Kolmogorov Equations ◽  
Varying Coefficients ◽  
Method Of Solution ◽  
Time Varying Coefficients

Abstract In the paper an approximate model of time-varying linear systems using a sequence of time-invariant systems is suggested. The conditions for validity of the approximation are proven with a theorem. Examples comparing the numerical solution of the original system and the analytical solution of the model are given. For the system under the consideration a new criterion giving sufficient conditions for robust Lagrange stability is suggested. The criterion is proven with a theorem. Examples are given showing stable and non stable solutions of a time-varying system and the results are compared with the numerical Runge-Kutta solution of the system. In the paper an important application of the described method of solution of linear systems with time-varying coefficients, namely analytical solution of the Kolmogorov equations is shown.

Almost sure stability of maxima

1973 ◽  
Vol 10 (02) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n≧ 1} unbounded above setMn= max {X1,X2, …,Xn}. When do normalizing constantsbnexist such thatMn/bn→1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for alland in this casebn∼F–1(1 – 1/n) Necessary and sufficient conditions for lim supn→∞,Mn/bn= l &gt;1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn= 1 a.s. except whenl= 1. When theXnare r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

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