A theorem on the strong asymptotic stability and determination of stabilizing controls

2001 ◽  
Vol 333 (8) ◽  
pp. 807-812 ◽  
Author(s):  
Grigory Sklyar ◽  
Alexander Rezounenko
Keyword(s):  
Asymptotic Stability

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

Determination of Global Regions of Asymptotic Stability for Difference Dynamical Systems

1977 ◽  
Vol 44 (1) ◽  
pp. 147-153 ◽  
Author(s):  
C. S. Hsu ◽  
H. C. Yee ◽  
W. H. Cheng
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Complex Shape ◽  
Stable Equilibrium ◽  
Stable Equilibrium State ◽  
Global Properties ◽  
Backward Mapping ◽  
Basic Ideas ◽  
Second Order Systems

In this paper certain global properties of dynamical systems governed by nonlinear difference equations are studied. When an asymptotically stable equilibrium state or periodic solution exists, it is desirable to be able to determine a global region of asymptotic stability in the state space. In this paper an effective method is presented for the determination of such a region. It will be seen that once certain features of the backward mapping have been properly delineated, the development of the method becomes a rather simple one. The method is mainly presented for second-order systems but the basic ideas are also applicable to higher-order systems. Through the development of the theory and examples, one also sees that, in general, the region of asymptotic stability for a nonlinear difference system is of extremely complex shape.

scholarly journals New approach to asymptotic stability: time-varying nonlinear systems

1997 ◽  
Vol 20 (2) ◽  
pp. 347-366 ◽  
Author(s):  
L. T. Grujić
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Time Varying ◽  
Uniform Asymptotic Stability ◽  
Stability Time ◽  
Necessary And Sufficient

The results of the paper concern a broad family of time-varying nonlinear systems with differentiable motions. The solutions are established in a form of the necessary and sufficient conditions for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability domain. They permit arbitrary selection of a functionp(⋅)from a defined functional family to determine a Lyapunov functionv(⋅),[v(⋅)], by solvingv′(⋅)=−p(⋅){or equivalently,v′(⋅)=−p(⋅)[1−v(⋅)]}, respectively. Illstrative examples are worked out.

scholarly journals DYNAMICS CONTROL AND CONSTRAINT STABILIZATION

2017 ◽  
Vol 19 (9.1) ◽  
pp. 67-75
Author(s):  
N.V. Abramov ◽  
R.G. Mukharlyamov
Keyword(s):  
Asymptotic Stability ◽  
Exponential Stability ◽  
Systems Dynamics ◽  
System Of Differential Equations ◽  
Dynamics Modeling ◽  
Physical Systems ◽  
Constraint Equations ◽  
Control Actions ◽  
Physical Elements

Results of researchers on dynamics modeling of the systems containing different physical elements are proposed. The construction method of the physical systems dynamics equations, providing constraints stabilization, is discussed. The problem of corresponding constraints reactions or determination of control actions is reduced to the construction of the system of differential equations, assuming that the partial integrals are given. The conditions of asymptotic stability and exponential stability an integral manifold's corresponding constraint equations are defined.

Determination of the region of asymptotic stability for a CSTR

1988 ◽  
Vol 12 (2-3) ◽  
pp. 237-241 ◽  
Author(s):  
L. Pellegrini ◽  
G. Biardi ◽  
M.G. Grottoli
Keyword(s):  
Asymptotic Stability

The Stability of a Moving Elastic Strip Subjected to Random Parametric Excitation

1979 ◽  
Vol 46 (2) ◽  
pp. 404-410 ◽  
Author(s):  
F. Kozin ◽  
R. M. Milstead
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Equations Of Motion ◽  
Stochastic Equations ◽  
Thin Strip ◽  
Elastic Strip ◽  
Moving Strip ◽  
The Stability ◽  
Sufficiency Conditions

The dynamic stability of a thin strip, traveling axially, at a constant speed between two roller supports is investigated for the case of zero mean random in-plane loading. Galerkin’s method is used to reduce the equations of motion to a set of fourth-order stochastic equations. An extention of the method first proposed by Wu and Kozin is developed which allows determination of the sufficiency conditions to guarantee Almost Sure Asymptotic Stability of stochastic systems of order greater than two. Using this method, results in terms of the variance of the random loadings on the moving strip are derived. It is found that the critical noise level to guarantee stability of the strip decreases with increasing mode, approaching asymptotically a level determined solely by the strip stiffness.

A Method for the Determination of the Domain of Stability of Second-Order Nonlinear Autonomous Systems

1964 ◽  
Vol 31 (2) ◽  
pp. 315-320 ◽  
Author(s):  
E. F. Infante ◽  
L. G. Clark
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Autonomous Systems ◽  
Second Order

A method for the determination of the domain of asymptotic stability of second-order nonlinear systems is presented. The essence of the method is the construction of Liapunov-like functions. Several simple but important examples illustrate the application of the method.

scholarly journals Symbolic-Numerical Modeling of the Influence of Damping Moments on Satellite Dynamics

2018 ◽  
Vol 173 ◽  
pp. 05008
Author(s):  
Sergey A. Gutnik ◽  
Vasily A. Sarychev
Keyword(s):  
Numerical Modeling ◽  
Asymptotic Stability ◽  
Gröbner Basis ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Real Roots ◽  
Spatial Oscillations ◽  
Damping Torque ◽  
Orbital Coordinate System

The dynamics of a satellite on a circular orbit under the influence of gravitational and active damping torques, which are proportional to the projections of the angular velocity of the satellite, is investigated. Computer algebra Gröbner basis methods for the determination of all equilibrium orientations of the satellite in the orbital coordinate system with given damping torque and given principal central moments of inertia were used. The conditions of the equilibria existence depending on three damping parameters were obtained from the analysis of the real roots of the algebraic equations spanned by the constructed Gröbner basis. Conditions of asymptotic stability of the satellite equilibria and the transition decay processes of the spatial oscillations of the satellite at different damping parameters have also been obtained.

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