Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals

We suggest new approaches to the study of the asymptotic stability of equilibria for equations of the neutral type. Nonmonotone indefinite Lyapunov functionals are used. We investigate the localization of solutions with respect to the level surfaces of a Lyapunov functional and a functional estimating the derivative of the Lyapunov functional along the solutions. By using solution localization tests, we obtain new conditions for the asymptotic stability of equilibria for equations of the neutral type with bounded right-hand side. We present asymptotic stability tests that do not impose any a priori stability condition on the difference operator. A generalization of the Barbashin–Krasovskii theorem for nonmonotone indefinite Lyapunov functionals is proved for autonomous equations.     

Lyapunov functionals of constant signs in the stability problem for a functional-differential equation of neutral type

We present a generalization of the Lyapunov functional method and use it to study the stability of nonautonomous functional-differential equations of neutral type with finite delay. This generalization is based on constructing the limit equations and the limit functionals.     

Nonmonotone functionals in the Lyapunov direct method for equations of the neutral type

We present new tests for the stability and asymptotic stability of trivial solutions of equations with deviating argument of the neutral type. Unlike well-known results, here we use nonmonotone indefinite Lyapunov functionals. Our class of functionals contains both Lyapunov-Krasovskii functionals and Lyapunov-Razumikhin functions as natural special cases. This class of functionals is broad enough that, in a number of stability tests, we have been able to omit the a priori requirement of stability of the corresponding difference operator. In addition, we present tests for the asymptotic stability of solutions of equations of the neutral type with unbounded right-hand side and new estimates for the magnitude of perturbations that do not violate the asymptotic stability if it holds for the unperturbed equation. The obtained estimates single out domains of the phase space in which perturbations should be small and domains in which essentially no constraints are imposed on the perturbation magnitude.     

Lyapunov functionals,weak sequential stability,and uniqueness analysis for energy-transport systems

A class of strongly coupled parabolic systems, modeling the energy transport of electrons in semiconductors, is analyzed. The variables are the electron density and the thermal energy. First, some Lyapunov functionals are derived, which yields the weak sequential stability for smooth solutions in the sense of Feireisl, using weak compactness results. Second, by the H ⁻¹ method, the uniqueness of bounded weak solutions is proved.     

Delay-independent stabilization of uncertain linear systems of neutral type

In this paper, we show that a sufficient condition for the delay-independent stabilizability of linear delay systems, which had been obtained by Amemiya et al., is also valid for linear neutral systems with measurable state variables by a new differential-difference inequality.The authors express their appreciation to Professor G. Leitmann for his useful comments.     

Equivalence between the Lyapunov–Krasovskii functionals approach for discrete delay systems and that of the stability conditions for switched systems

The stability of discrete-time systems with time varying delay in the state can be analyzed by using a discrete-time extension of the classical Lyapunov–Krasovskii approach. In the networked control systems domain a similar delay stability problem is treated using a switched system transformation approach. The paper aims to establish a relation between the switched system transformation approach and the classical Lyapunov–Krasovskii method. It is shown that using the switched systems transformation is equivalent to using a general delay dependent Lyapunov–Krasovskii functionals. This functional represents the most general form that can be obtained using sums of quadratic terms. Necessary and sufficient LMI conditions for the existence of such functionals are presented.     

Asymptotic stability of differential systems of neutral type

We offer sufficient conditions for the asymptotic stability of the equilibrium point of linear neutral differential systems. An application of our results to a family of artificial neural networks of neutral type is also illustrated.     

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.     

Lyapunov inequalities and stability for linear Hamiltonian systems

In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be proved.     

On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovski functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either , the norm of the instantaneous value of the solutions or , the -norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.

     

Lyapunov stability for impulsive control affine systems

This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets.     

The method of Lyapunov functionals in stability analysis of functional-differential equations

We develop the method of Lyapunov functionals in the stability analysis of linear nonautonomous functional-differential equations of neutral type. The approach is based on the construction of limit equations and limit Lyapunov functionals. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 323–331, September, 2000.     

Method of Lyapunov functionals construction in stability of delay evolution equations

The investigation of stability for hereditary systems is often related to the construction of Lyapunov functionals. The general method of Lyapunov functionals construction which was proposed by V. Kolmanovskii and L. Shaikhet and successfully used already for functional differential equations, for difference equations with discrete time, for difference equations with continuous time, is used here to investigate the stability of delay evolution equations, in particular, partial differential equations.