Converse Lyapunov–Krasovskii theorems for systems described by neutral functional differential equations in Hale's form

In this article, we show that the existence of a Lyapunov–Krasovskii functional is necessary and sufficient condition for the uniform global asymptotic stability and the global exponential stability (GES) of time-invariant systems described by neutral functional differential equations in Hale's form. It is assumed that the difference operator is linear and strongly stable, and that the map on the right-hand side of the equation is Lipschitz on bounded sets. A link between GES and input-to-state stability is also provided.     

On Liapunov-Krasovskii functionals under Carathéodory conditions

In [Driver, R. D. (1962). Existence and stability of solutions of a delay-differential system. Archive for Rational Mechanics and Analysis 10, 401-426] a proper definition, not involving the solution, of the derivative of the Liapunov-Krasovskii functional for retarded functional differential equations with continuous right side is given and it is showed that this definition coincides with the non-constructive one given in Krasovskii [1956. On the application of the second method of A. M. Lyapunov to equations with time delays (in Russian). Prikladnaya Matematika i Mekhanika 20, 315-327] involving the solution, for functionals V which are locally Lipschitz (and not only continuous, as it is considered in most literature). In this paper, the result by Driver is extended to a general class of retarded functional differential equations coupled with continuous time difference equations with more general right sides, verifying the Carathéodory conditions. Such result is applied to build a new Liapunov-Krasovskii theorem for studying the input-to-state stability of time-invariant neutral functional differential equations with linear difference operator. An example taken from the literature, concerning transmission lines, is reported, showing the effectiveness of the methodology.     

Input-to-State Stability of Time-Delay Systems: A Link With Exponential Stability

The main contribution of this technical note is to establish a link between the exponential stability of an unforced system and the input-to-state stability (ISS) via the Liapunov–Krasovskii methodology. It is proved that a system which is (globally, locally) exponentially stable in the unforced case is (globally, locally) input-to-state stable when it is forced by a measurable and locally essentially bounded input, provided that the functional describing the dynamics in the unforced case is (globally, on bounded sets) Lipschitz and the functional describing the dynamics in the forced case satisfies a Lipschitz-like hypothesis with respect to the input. Moreover, a new feedback control law is provided for delay-free linearizable and stabilizable time-delay systems, whose dynamics is described by locally Lipschitz functionals, by which the closed-loop system is ISS with respect to disturbances adding to the control law, a typical problem due to actuator errors.      

延时系统输入状态稳定性的Lyapunov逆理论

研究了延时系统输入状态稳定性的局部Lipschitz连续的Lyapunov逆理论. 针对含有任意可测局部本质有界扰动的延时系统, 一个局部Lipschitz连续的Lyapunov泛函被证实是存在的, 如果该系统是鲁棒渐进稳定的. 根据该结论, 延时系统输入状态稳定性的Lyapunov特征被进一步得到.     

Direct and converse Lyapunov theorems for functional difference systems

New Lyapunov criteria for asymptotic stability and input-to-state stability of infinite dimensional systems described by functional difference equations are provided. Conditions in terms of both Lyapunov–Razumikhin functions defined on Euclidean spaces and of Lyapunov–Krasovskii functionals defined on infinite dimensional spaces are found. For the case of Lyapunov–Krasovskii functionals, necessary and sufficient conditions are provided for the asymptotic stability, in both the local and the global case, and for the input-to-state stability. This is the first time in the literature that converse Lyapunov theorems are provided for the class of nonlinear systems here studied.     

Serre’s reduction of linear partial differential systems with holonomic adjoints

Given a linear functional system (e.g., an ordinary/partial differential system, a differential time-delay system, a difference system), Serre’s reduction aims at finding an equivalent linear functional system which contains fewer equations and fewer unknowns. The purpose of this paper is to study Serre’s reduction of underdetermined linear systems of partial differential equations with either polynomial, formal power series or locally convergent power series coefficients, and with holonomic adjoints in the sense of algebraic analysis. We prove that these linear partial differential systems can be defined by means of only one linear partial differential equation. In the case of polynomial coefficients, we give an algorithm to compute the corresponding equation.     

一类前馈系统的鲁棒镇定

Two kinds of saturated controllers are designed for a class of feedforward systems and the closed-loop resulted is locally input-to-state stable and input-to-state stable, respectively. By the word "locally", it is meant that there are restrictions on the amplitude of inputs. At first, under the guidance of suitable energy functions, two kinds of saturated controllers are designed as locally input-to-state stabilizers for a class of perturbed linear systems, from which explicit gain estimations can be obtained for the subsequent design. Then under the conditions that two subsystems of the feedforward system are respectively of locally input-to-state stability and input-to-state stability, the small gain theory is used to determine saturated degrees for corresponding robust stabilizers. The stability proofs are given by using a new characterization of input-to-state stability that is based on the concept of ultimate boundedness. As an application, saturated controllers are designed for the partial dynamics of a certain inverted pendulum.     

Connections between Razumikhin-type theorems and the ISS nonlinearsmall gain theorem

A Razumikhin-type theorem that guarantees input-to-state stability for functional differential equations with disturbances is established using the nonlinear small-gain theorem. The result is used to show that input-to-state stabilizability for nonlinear finite-dimensional control systems is robust, in an appropriate sense, to small time delays at the input. Also, relaxed Razumikhin-type conditions guaranteeing global asymptotic stability for differential difference equations are given     

Stability analysis of extended block boundary value methods for linear neutral delay integro-differential equations

This paper is concerned with the stability of extended block boundary value methods (B₂VMs) for the linear neutral delay integro-differential-algebraic equations (NDIDAEs) and the linear neutral delay integro-differential equations (NDIDEs). It is proved that every A-stable B₂VM can preserve the asymptotic stability of the exact solution of NDIDAEs under some certain conditions. A necessary and sufficient condition of the B₂VMs to be asymptotically stable for NDIDEs is also obtained. A few numerical experiments confirm the expected results.     

Stability and control of nonlinear systems described by retarded functional equations: a review of recent results

This paper reports on recent results in a series of the work of the authors on the stability and nonlinear control for general dynamical systems described by retarded functional differential and difference equations. Both internal and external stability properties are studied. The corresponding Lyapunov and Razuminkhin characterizations for input-to-state and input-to-output stabilities are proposed. Necessary and sufficient Lyapunov-like conditions are derived for robust nonlinear stabilization. In particular, an explicit controller design procedure is developed for a new class of nonlinear time-delay systems. Lastly, sufficient assumptions, including a small-gain condition, are presented for guaranteeing the input-to-output stability of coupled systems comprised of retarded functional differential and difference equations.     

On exponential stability of linear non-autonomous functional differential equations of neutral type

General linear non-autonomous functional differential equations of neutral type are considered. A novel approach to exponential stability of neutral functional differential equations is presented. Consequently, explicit criteria are derived for exponential stability of linear non-autonomous functional differential equations of neutral type. A brief discussion to the obtained results and illustrative examples are given.     

Local ISS of large-scale interconnections and estimates for stability regions

We consider interconnections of locally input-to-state stable (LISS) systems. The class of LISS systems is quite large, in particular it contains input-to-state stable (ISS) and integral input-to-state stable (iISS) systems.Local small-gain conditions both for LISS trajectory and Lyapunov formulations guaranteeing LISS of the composite system are provided in this paper. Notably, estimates for the resulting stability region of the composite system are also given. This in particular provides an advantage over the linearization approach, as will be discussed.     

Stability of solutions of dynamic systems with randomly structured aftereffect

The relation is established between the asymptotic stochastic stability of a linear functional differential equation and exponential stability of the trivial solution to this equation. The direct and inverse Lyapunov theorems on the stability of linear differential equations are proved. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 31–38, March–April 2006.     

Local input-to-state stability: Characterizations and counterexamples

We show that a nonlinear locally uniformly asymptotically stable infinite-dimensional system is automatically locally input-to-state stable (LISS) provided the nonlinearity possesses some sort of uniform continuity with respect to external inputs. Also we prove that LISS is equivalent to existence of a LISS Lyapunov function. We show by means of a counterexample that if this uniformity is not present, then the equivalence of local asymptotic stability and local ISS does not hold anymore. Using a modification of this counterexample we show that in infinite dimensions a uniformly globally asymptotically stable at zero, globally stable and locally ISS system possessing an asymptotic gain property does not have to be ISS (in contrast to finite dimensional case).     

New criteria on exponential stability of neutral stochastic differential delay equations

Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see e.g. [V.B. Kolmanovskii, V.R. Nosov, Stability and Periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, 1981; X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems Control Lett. 26 (1995) 245–251; X. Mao, Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28(2) (1997) 389–401; X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997]). More recently, Mao [Asymptotic properties of neutral stochastic differential delay equations, Stochastics and Stochastics Rep. 68 (2000) 273–295] provided with some useful criteria on the exponential stability for NSDDEs. However, the criteria there require not only the coefficients of the NSDDEs to obey the linear growth condition but also the time delay to be a constant. One of our aims in this paper is to remove these two restrictive conditions. Moreover, the key condition on the diffusion operator associated with the underlying NSDDE will take a much more general form. Our new stability criteria not only cover many highly non-linear NSDDEs with variable time delays but they can also be verified much more easily than the known criteria.     

Explicit Wei–Norman formulae for matrix Lie groups via Putzer's method

The Wei–Norman formula locally relates the Magnus solution of a system of linear time-varying ODEs with the solution expressed in terms of products of exponentials by means of a set of nonlinear differential equations in the parameters of the two types of solutions. A closed form expression of such formula is proposed based on the use of Putzer's method.     

Input to state set stability for pulse width modulated control systems with disturbances

New results on set stability and input-to-state stability in pulse-width modulated (PWM) control systems with disturbances are presented. The results are based on a recent generalization of two time scale stability theory to differential equations with disturbances. In particular, averaging theory for systems with disturbances is used to establish the results. The nonsmooth nature of PWM systems is accommodated by working with upper semicontinuous set-valued maps, locally Lipschitz inflations of these maps, and locally Lipschitz parameterizations of locally Lipschitz set-valued maps.     

Discretized LKF method for stability of coupled differential‐difference equations with multiple discrete and distributed delays

Time‐delay systems described by coupled differential‐functional equations include as special cases many types of time‐delay systems and coupled differential‐difference systems with time delays. This article discusses the discretized Lyapunov–Krasovskii functional (LKF) method for the stability problem of coupled differential‐difference equations with multiple discrete and distributed delays. Through independently dividing every delay region that the plane regions consists in two delays to discretize LKF, the exponential stability conditions for coupled systems with multiple discrete and distributed delays are established based on a linear matrix inequality (LMI). The numerical examples show that the analysis limit of delay bound in which the systems are stable may be approached by our result. Copyright © 2011 John Wiley & Sons, Ltd.