Sufficient conditions for the stability of a class of second order systems

2015 ◽  
Vol 84 ◽  
pp. 1-6 ◽  
Author(s):  
Marco M. Nicotra ◽  
Roberto Naldi ◽  
Emanuele Garone
Keyword(s):  
Sufficient Conditions ◽  
Second Order ◽  
The Stability ◽  
Second Order Systems

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

On the Stability of Second Order Time Varying Linear Systems

2005 ◽  
Vol 128 (3) ◽  
pp. 408-410 ◽  
Author(s):  
M. Tadi
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Stiffness Matrix ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Second Order ◽  
Time Varying ◽  
The Stability ◽  
Second Order Systems

This note considers the stability of linear time varying second order systems. It studies the case where the stiffness matrix is a function of time. It provides sufficient conditions for stability and asymptotic stability of the system provided that certain conditions on the stiffness matrix are satisfied.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

scholarly journals On the problem of stability of a motion of essentially nonlinear systems

2021 ◽  
pp. 3-12
Author(s):  
A.A. Martynyuk ◽  
V.O. Chernienko
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Systems Of Equations ◽  
Affine Systems ◽  
The Stability ◽  
Second Order Systems

This article discusses essentially nonlinear systems. Following the approach of applying the pseudolinear inequalitiesdeveloped in a number of works, new estimates for the variation of Lyapunov functions along solutionsof the considered systems of equations are obtained. Based on these estimates, we obtain sufficient conditionsfor the equiboundedness of solutions of second-order systems and sufficient conditions for the stability of anessentially nonlinear system under large initial perturbations. Conditions for the stability of affine systems arealso obtained.

An Asymptotical Stability in Variational Sense for Second Order Systems

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Sobolev Space ◽  
Differential System ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Second Order ◽  
Functional Parameter ◽  
Asymptotical Stability ◽  
Parameter Control ◽  
Initial Condition ◽  
Second Order Systems

AbstractIn this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

A Linearization-Based Approach to Vibrational Control of Second-Order Systems

2013 ◽  
Author(s):  
I. P. M. Wickramasinghe ◽  
Jordan M. Berg
Keyword(s):  
Inverted Pendulum ◽  
Second Order ◽  
Periodic Forcing ◽  
Averaging Methods ◽  
Vibrational Control ◽  
Electrostatic Mems ◽  
Mathieu’S Equation ◽  
Comb Drives ◽  
The Stability ◽  
Second Order Systems

We present an alternative to averaging methods for vibrational control design of second-order systems. This method is based on direct application of the stability map of the linearization of the system at the desired operating point. The paper focuses on harmonic forcing, for which the linearization is Mathieu’s equation, but somewhat more general periodic forcing functions may be handled. When it is applicable, this method achieves significantly greater functionality than previously reported approaches. This is demonstrated on two sample systems. One is the vertically driven inverted pendulum, and the other is an input-coupled bifurcation control problem arising from electrostatic MEMS comb drives.

A Procedure for Investigating the Liapunov Stability of Nonautonomous Linear Second-Order Systems

1973 ◽  
Vol 40 (4) ◽  
pp. 1103-1106 ◽  
Author(s):  
S. E. Jones ◽  
T. R. Robe
Keyword(s):  
Differential Equation ◽  
Ad Hoc ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Liapunov Function ◽  
Null Solution ◽  
Liapunov Stability ◽  
The Stability ◽  
Second Order Systems ◽  
Liapunov's Direct Method

Utilizing Liapunov’s direct method a procedure is presented which generates sufficient conditions for the stability of the null solution of the nonautonomous differential equation x¨ + b(t)x˙ + a(t)x = 0. This procedure systematically leads to the construction of a Liapunov function for a given differential equation and thus eliminates the normally ad hoc nature of the direct method. Four examples illustrating the procedure are discussed.

scholarly journals The Exponential Behaviour of Nonlinear Stochastic Functional Equations of Second Order in Time

2003 ◽  
Vol 03 (02) ◽  
pp. 169-186 ◽  
Author(s):  
Tomás Caraballo ◽  
María J. Garrido-Atienza ◽  
José Real
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Partial Differential ◽  
Mean Square Stability ◽  
Exponential Behaviour ◽  
Second Order In Time ◽  
The Stability ◽  
Exponential Mean Square Stability

Sufficient conditions for exponential mean square stability of solutions to delayed stochastic partial differential equations of second order in time are established. As a consequence of these results, some on the pathwise exponential stability of the system are proved. The stability results derived are applied also to partial differential equations without hereditary characteristics. The results are illustrated with several examples.

The Stability of a Second Order Linear Periodic System

1969 ◽  
Vol 91 (2) ◽  
pp. 207-210 ◽  
Author(s):  
E. J. Davison
Keyword(s):  
Linear System ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Order Linear ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Linear Periodic System

Necessary and sufficient conditions are obtained for the stability of the following second order linear system: x˙=θ(t)x,θ(t)=θt+∑i=1lTi and θ(t) =A1,0<t<T1=A2,T1<t<T1+T2⋮=Al,∑i=1l−1Ti<t<∑i=1lTi in terms of the eigenvalues and elements of the matrices Ai, i = 1, 2…l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.

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