Liapunov's Direct Method Research Articles

Stochastic stability of a rotating shaft

The stochastic stability problem of an elastic, balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section is included in the present formulation. Each force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, damping coefficient, damping ratio, angular velocity, mode number and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and as well as an harmonic process with random phase.

Influence of rotatory inertia on stochastic stability of a viscoelastic rotating shaft

The stochastic stability problem of a viscoelastic Voigt-Kelvin balanced rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section and external viscous damping are included into account. The force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, external damping coefficient, retardation time, angular velocity, and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and variance ?2 as well as for harmonic process with amplitude H.

Dynamic stability of the viscoelastic rotating shaft subjected to random excitation

The dynamic stability problem of a viscoelastic Voigt–Kelvin rotating shaft subjected to action of axial forces at the ends is studied. The shaft is of circular cross-section, it rotates at a constant rate about its longitudinal axis of symmetry. The effect of rotatory inertia of the shaft cross-section is included in the present formulation. Each force consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability conditions are obtained as the function of stochastic process variance, retardation time, angular velocity, and geometric and physical parameters of the shaft. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ 2 as well as for harmonic process with amplitude H.

Dynamic stability of a thin-walled beam subjected to axial loads and end moments

The dynamic stability problem of thin-walled beams subjected to combined action of axial loads and end moments is studied. Each force and moment consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability and uniform stochastic stability conditions are obtained as the function of stochastic process variance, damping coefficient, and geometric and physical parameters of the beam. The almost sure stability regions for I-cross section and narrow rectangular cross section are shown in the plane of variances of stochastic parts axial force and end moment. Uniform stochastic stability regions are shown in intensity of stochastic loadings and constant parts of axial loads and end moments.

P-Moment stability of functional differential equations with random impulses

By means of Liapunov's direct method coupled with Razumikhin technique some sufficient conditions for (uniform, uniform and globally asymptotic, uniformly globally asymptotic, and globally exponential) p-moment stability of the zero solution of functional differential equations with random impulses are presented, where the parameter λ( t) in (d V( t, x( t)))/d t is not required to be negative definite. Thus the obtained results are convenient to apply.

Multiple-arm space free-flying robots for manipulating objects with force tracking restrictions

Multiple Impedance Control (MIC) is an algorithm that enforces designated impedance at various levels, i.e. on the manipulated object, all cooperating manipulators, and the moving platform of a robotic system. In this paper, a force tracking strategy inspired by a human control system is added to the MIC algorithm and the general formulation is revised to fulfill a desired force tracking strategy for object manipulation tasks. The stability analysis of the MIC algorithm based on the Liapunov Direct Method, besides error analysis, shows that a good tracking of cooperative manipulators and the manipulated object is guaranteed. Next, using MAPLE and MATLAB tools, a system of three manipulators mounted on a space free-flying robot is simulated. The task is moving an object based on given trajectories which come across an obstacle, to examine the performance of the developed control law. The results show that, even in the presence of both external disturbances and an impact due to collision with the obstacle, the response of the MIC algorithm is smooth. Moreover, based on the embedded force tracking strategy, the contact force is confined to follow a desired trajectory. Also, it is shown that decreasing the values of the controller mass matrix elements results in reducing both the object position and force tracking error.

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.

Exponential stability of functional differential systems with impulsive effect on random moments

The model of nonlinear functional differential systems with impulsive effect on random moments is brought forward by authors in this paper. Then, by means of Liapunov's direct method coupled with Razumikhin technique and comparison principle, sufficient conditions for p-moment exponential stability of the trivial solution to the systems are present, where the parameter λ( t) in ( dV( t, x( t)))/( dt) are not required to be negative. Finally, an example is discussed to show the applications of the obtained results.

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.

Influence of rotatory inertia and transverse shear on stochastic instability of the cross-ply laminated beam

The stochastic instability problem associated with an axially loaded cross-ply laminated beam is formulated. The effects of shear deformation and rotatory inertia are included in the present formulations. The beam is subjected to time-dependent deterministic and stochastic forces. By using the direct Liapunov method, bounds for the almost sure instability of beams as a function of viscous damping coefficient, variance of the stochastic force, ratio of principal lamina stiffnesses, shear correction factor, number of layers, mode numbers and geometrical ratio, are obtained. Numerical calculations are performed for the Gaussian process with a zero mean and variance σ 2 as well as for harmonic process with an amplitude A.

Influence of randomly varying damping coefficient on the dynamic stability of continuous systems

A method for the determination of sufficient conditions for the almost sure asymptotic stability of some continuous systems, when the damping coefficient is random time-dependent function is studied. In this case, the probabilistic property of the derivative process of the damping coefficient is taken into account. The problem is solved by means Liapunov direct method, and tested on a simply supported elastic beam, compressed by time-dependent stochastic axial force. Regions of almost sure asymptotic stability as functions of the constant part of viscous damping coefficient and magnitude of an axial force are given.

Influence of Transverse Shear on Stochastic Instability of the Elastic Beam

In the case when Kirchhoff–Love hypotheses do not give satisfactory results, we have to take the rotatory inertia and transverse shear into account. This paper studies the elastic beam subjected to stochastic axial load, when transverse shear is taken into account. By using the direct Liapunov method, the bounds of the almost sure instability of beams as the function of the damping coefficient, variance of the stochastic force, geometric parameters, mode number, section shape factor and intensity of the deterministic component of axial loading, are obtained. Calculations are performed for the Gaussian process with zero mean and variance σ2, as well as for the harmonic process with amplitude A.

Stabilizing the absolutely or convectively unstable homogeneous solutions of reaction-convection-diffusion systems.

We study the problem of stabilization of a homogeneous solution in a two-variable reaction-convection-diffusion one-dimensional system with oscillatory kinetics, in which moving or stationary patterns emerge in the absence of control. We propose to use a formal spatially weighted feedback control to suppress patterns in an absolutely or convectively unstable system and pinning control for a convectively unstable system. The latter approach is very effective and may require only one actuator to adjust feed conditions. In the former approach, the positive diagonal elements of the appropriate dynamics matrix are shifted to the left-hand part of the complex plane to ensure linear (asymptotic) stability of the system according to Gershgorin criterion. Moreover, we construct a controller that (with many actuators) will approach the global stability of the solution, according to Liapunov's direct method. We apply two alternative approaches to reveal the unstable modes: an approximate one that is based on linear stability analysis of an unbounded system, and an exact one that uses a traditional eigenstructure analysis of bounded systems. The number of required actuators increases dramatically with system size and with the distance from the bifurcation point. The methodology is developed for a system with learning cubic kinetics and is tested on a more realistic cross-flow reactor model.

Stability of stationary solutions of the Schrödinger–Langevin equation

The stability properties of a class of dissipative quantum mechanical systems are investigated. The nonlinear stability and asymptotic stability of stationary states (with zero and nonzero dissipation, respectively) is investigated by Liapunov's direct method. The results are demonstrated by numerical calculations on the example of the damped harmonic oscillator.

Robust control of robot manipulator by model-based disturbance attenuation

In this letter, a model-based disturbance attenuator (MBDA) for robot manipulators is proposed and the stability of the MBDA in robot positioning problems is proved via Liapunov's direct method. This method does not require an accurate model of a robot manipulator and takes care of disturbances or modeling errors so that the plant output remains relatively unaffected by them. The output error due to the gravity or constant disturbance can be effectively eliminated by this method.

The Stability of Chaos Synchronization of the Japanese Attractors and its Application

The synchronization of chaos of the two identical chaotic motions of the Japanese attractor has been studied. First, the stability of the chaos synchronization of the systems is investigated by Liapunov's direct method. Some sufficient conditions of global asymptotic synchronization are attained from rigorous mathematical theory. It has also been demonstrated numerically that applying two different kinds of one-way coupling technology can synchronize the two identical chaotic systems. The sign of the sub-Liapunov exponent has been used as an indicator for the occurrence of chaos synchronization. Chaos synchronization can be assured as shown well by phase trajectory. In addition, the chaotic signals are used to mask the message function in the secure communication system.

Total stability of perturbed systems of differential equations

The notion of Lipschitz stability for systems of ordinary differential equations (ODE) was introduced. In this paper, we will extend the total stability notion to a new type of stability called total Lipschitz stability. Some criteria and results are given. Our technique depends on Liapunov's direct method.

Dynamic stability of two rigid rotors connected by a flexible coupling with angular misalignment

The effect of misalignment on the stability of two rotors connected by a flexible mechanical coupling subjected to angular misalignment is examined. The study performed is to understand the effect of angular misalignment on the stability of rotating machinery. The dimensionless stability criteria of the non-linear system of differential equations of two misaligned rigid rotors are derived using Liapunov's direct method. A rigid disk is attached at the middle of each rotor, where the rotor–disk assembly is mounted on two hydrodynamic bearings with four stiffness and four damping coefficients. Sets of dimensionless conditions for sufficient whirl stability of the two misaligned rotors are derived. The stability conditions are presented in graphical form for deeper understanding of the effect of the flexible mechanical coupling stiffness and angular misalignment on rotating machinery stability. The results show that an increase in angular misalignment or mechanical coupling stiffness terms leads to an increase of the model stability region.

Criteria for the stability of steady flows induced by diabatic heating

By use of the Liapunov direct method, criteria for the stability of a general steady flow induced by heating are derived for a barotropic model with heating and dissipation. Among these criteria, necessary conditions for the instabilities are identified with the rises of supercritically high speed instability, barotropic instability and some other instabilities.

Nonlinear Waves in Networks of Neurons with Delayed Feedback: Pattern Formation and Continuation

An on-center off-surround network of three identical neurons with delayed feedback is considered, and the effect of synaptic delay of signal transmission on the pattern formation and global continuation of nonlinear waves is described. The spontaneous bifurcation of multiple branches of periodic solutions is discussed, and their spatio-temporal patterns and mode interactions are studied by using the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, Liapunov's direct method, LaSalle's invariance principle, a priori estimates, and differential inequalities.