On the class of proper linear differential systems with unbounded coefficients

Proper linear differential systems (whose coefficients are not necessarily bounded on the half-line) are defined as systems for which there exists a generalized Lyapunov transformation reducing them to a diagonal system with constant coefficients (Basov). We prove that Lyapunov’s original definition of a proper system and the Perron and Vinograd criteria hold for the class of proper systems as well as for the class of proper systems with uniformly bounded coefficients. We show that the Lyapunov properness criterion for a triangular system fails for systems with unbounded coefficients; namely, we construct an improper system with the following properties: the Lyapunov exponents of all nonzero solutions of that system are finite and exact, and for an arbitrary reduction of this system by a generalized Lyapunov transformation to triangular form, its diagonal coefficients have finite exact mean values, whose set with regard of multiplicities is independent of the choice of the transformation. In addition, we show that the main property of proper systems with uniformly bounded coefficients (preservation of conditional exponential stability as well as the dimension of the exponentially stable manifold and the exponent of the asymptotic behavior of solutions under perturbations of higher-order smallness) holds for proper systems with unbounded coefficients as well.     

The exponential dichotomy of solutions to systems of linear difference equations with almost periodic coefficients

By the direct Lyapunov method we prove a sufficient condition for the exponential dichotomy with a weakened (in comparison with the case of arbitrary coefficients) requirement to the difference derivative of the Lyapunov function along the system trajectory. We give an illustrative example.     

Lyapunov reducibility and stabilization of nonstationary systems with an observer

For a linear nonstationary control system with an observer, we assume that the coefficients are locally Lebesgue integrable and integrally bounded on ℝ and construct a linear feedback such that the closed-loop plant-controller system is Lyapunov reducible to the special triangular form corresponding to an independent shift of the diagonal coefficients in the original system and in the system of asymptotic estimation of the state by an arbitrary pregiven quantity. For a periodic system, we prove that the constructed controls and Lyapunov transformation are periodic. We obtain corollaries on the uniform stabilization and global controllability of the central and singular exponents of the system.     

On the theory of stability of matrix differential equations

We establish the conditions of asymptotic stability of a linear system of matrix differential equations with quasiperiodic coefficients on the basis of constructive application of the principle of comparison with a Lyapunov matrix-valued function.     

On the global Lyapunov reducibility of two-dimensional linear systems with locally integrable coefficients

We show that if a two-dimensional linear nonstationary control system with locally integrable and integrally bounded coefficients is uniformly completely controllable, then the corresponding linear differential system closed with a measurable bounded control linear in the state variables has the property of global Lyapunov reducibility.     

基于符号数值混合计算的混成系统Lyapunov函数构造

基于平方和松弛和有理向量恢复,提出了一种符号数值混合计算方法来构造多项式Lyapunov函数以判定非线性混成系统的稳定性,首先,为Lyapunov函数预定一个给定次数的多项式模板,则Lyapunov函数构造问题可转化为相应的带参数的多项式优化问题,然后运用平方和松弛方法求得一个近似的数值多项式Lyapunov函数,再应用高斯-牛顿精化和有理向量恢复将数值多项式转化为验证的有理多项式Lyapunov函数.     

Lyapunov exponents of nilpotent Itô systems with random coefficients

The paper considers the top Lyapunov exponent of a two-dimensional linear stochastic differential equation. The matrix coefficients are assumed to be functions of an independent recurrent Markov process, and the system is a small perturbation of a nilpotent system. The main result gives the asymptotic behavior of the top Lyapunov exponent as the perturbation parameter tends to zero. This generalizes a result of Pinsky and Wihstutz for the constant coefficient case.     

Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients

In this article, we study the problem of estimating the pathwise Lyapunov exponent for linear stochastic systems with multiplicative noise and constant coefficients. We present a Lyapunov type matrix inequality that is closely related to this problem, and show under what conditions we can solve the matrix inequality. From this we can deduce an upper bound for the Lyapunov exponent. In the converse direction, it is shown that a necessary condition for the stochastic system to be pathwise asymptotically stable can be formulated in terms of controllability properties of the matrices involved.     

On the control of Lyapunov exponents of two-dimensional linear systems with locally integrable coefficients

We show that if a two-dimensional linear nonstationary control system with locally integrable and integrally bounded coefficients is uniformly completely controllable, then the complete spectrum of Lyapunov exponents of the corresponding closed system whose feedback is linear in the phase variables is globally controllable.     

Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator

Systems constituted by moving components that make intermittent contacts with each other can be modelled by a system of ordinary differential equations containing piecewise linear terms. We consider a soft impact bilinear oscillator for which we obtain bifurcation diagrams, Lyapunov coefficients, return maps and phase portraits of the response. Besides Lyapunov coefficients diagrams, bifurcation diagrams are represented in terms of both non-dimensional time instants of contact (when the mass impacts the obstacle) and of portions of contact duration (the percentage-time interval when the material point is inside the obstacle) vs. non-dimensional external force frequency (or amplitude). The second kind of diagrams is needed because the contact duration (or the complementary flight time duration) are quantities that can easily be measured in an experiment aiming at confirming the validity of the present model. Lyapunov coefficients are evaluated converting the piecewise linear system of ordinary differential equations into a map, the so-called impact map, where time and velocity corresponding to a given impact are evaluated as functions of time and velocity corresponding to the previous impact. Thus, the usual methods related to this last map are used. The trajectories are represented in terms of return maps (all points in the time-velocity plane involved in the impact events) and phase portraits (the trajectory-itself in the displacement-velocity plane). In the bifurcation diagrams, transition between different responses is evidenced and a perfect correlation between chaotic (periodic) attractors and positive (negative) values of the maximum Lyapunov coefficient is found.     

Statistical characteristics of attainability set of controllable systems with random coefficients

For controllable systems with random coefficients we study a property of statistical invariance, satisfied with given probability. We obtain sufficient conditions for invariance of a set with respect to controllable system expressed in terms of Lyapunov functions and shift dynamic system. We study the statistical characteristics of attainability set of a controllable system which is parameterized by metric dynamic system.     

A note on the Lyapunov stability of periodic discrete-time systems

We characterize the stability of discrete-time Lyapunov equations with periodic coefficients. The characterization can be seen as the analog of the classical stability theorem of Lyapunov equations with constant coefficients. It involves quantities readily computable with good accuracy.     

Proof of the strengthened version of simultaneous attainability of central exponents in the class of Hamiltonian systems

The central exponents of a linear Hamiltonian system are moved apart through uniformly small Hamiltonian perturbations of its coefficients, and then they are simultaneously attained by the Lyapunov exponents through infinitesimally small perturbations of the obtained Hamiltonian system.     

ABSOLUTE STABILITY OF THE GENERAL LURIE TYPE ROBUST SYSTEMS

1 IntroductionUP to now, the robust stability of linear interval dynamic systems has beenstudied in many papers [l,ZI. But, the results of Lurie type robust controlsystem with interval coefficient are few [3,4]. The results of [4] were improvedin [3]. In this papers the results of [3,4] will be generalized from Lurie typenonlinear control systems to general Lurie type nonlinear control system, andthe Lyapunov functions and parameters of S-procedure of [3] are adjusted.Hence, our results ar…     

On the properties of Lyapunov exponents of regular linear systems

We establish the sharp Baire class of the Lyapunov exponents on the space of Lyapunov regular linear systems with continuous bounded coefficients equipped with the topology of uniform or compact convergence of the coefficients on the half-line.     

Positive almost periodic solution of discrete nonlinear survival red blood cells model with feedback control

In this paper, we consider a discrete survival red blood cells system with feedback control. Assuming that the coefficients in the system are almost periodic sequences, by using Lyapunov functional approach, we obtain the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.     

Relations between regularity coefficients

For a bounded sequence of matrices defining a nonautonomous dynamics with discrete time, we obtain all possible relations between the regularity coefficients introduced by Lyapunov, Perron and Grobman. This includes considering general inequalities between the coefficients and showing that these inequalities are the best possible, in the sense that for any three nonnegative numbers satisfying them, and for no others, there exists a bounded sequence of matrices having the numbers respectively as Lyapunov, Perron and Grobman coefficients. Moreover, we establish inequalities between the three coefficients and some other regularity coefficients.