On the asymptotic stability of coupled delay differential and continuous time difference equations

A two-step Liapunov-Krasovskii methodology for checking the asymptotic stability of nonlinear coupled delay differential and continuous time difference equations is proposed here. The feasibility of such methodology is shown by means of Liapunov-Krasovskii functionals with nonconstant kernels in the integrals, for instance discretized Liapunov-Krasovskii ones. An illustrative example taken from the literature, showing the effectiveness of the proposed method, is reported.     

A Lyapunov–Krasovskii methodology for ISS and iISS of time-delay systems

This paper presents a Lyapunov–Krasovskii methodology for studying the input-to-state stability and the integral input-to-state stability of nonlinear time-delay systems. An integral input-state estimate which takes into account non-zero initial conditions is also proposed.     

Stability analysis for coupled systems of fractional differential equations on networks

This paper considers the stability problem for some coupled systems of fractional differential equations on networks (CSFDENs). We provide a systematic method for constructing a Lyapunov function for CSFDENs by using graph theory and the Lyapunov method. Consequently, some sufficient conditions for stability, uniform stability and uniform asymptotic stability of CSFDENs are obtained. Finally, an example and some numerical simulations are presented to verify the effectiveness of the theoretical results.     

Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control

This paper is concerned with the problem of exponential mean-square stabilization of hybrid neutral stochastic differential delay equations with Markovian switching by delay feedback control. A delay feedback controller is designed in the drift part so that the controlled system is mean-square exponentially stable. We discussed two types of structure controls; that is, state feedback and output injection. The stabilization criteria are derived in terms of linear matrix inequalities.     

Discretized Lyapunov-Krasovskii functional for coupled differential-difference equations with multiple delay channels

Many practical systems have a large number of state variables, but only a few components involve time delays. These components are often scalar or low dimensional, and typically only one delay is present in each such component. A special form of coupled differential-difference equations with one delay per channel proposed recently is well suited to formulate such systems. This article extends the discretized Lyapunov-Krasovskii functional method to this class of systems. In addition to generality, this formulation also drastically reduces the computational cost for a typical system, and therefore is appropriate to use even for time-delay systems of retarded type. The discretized formulation is also simpler than the previous formulation for systems with multiple delays.     

Small-gain theorem for ISS systems and applications

We introduce a concept of input-to-output practical stability (IOpS) which is a natural generalization of input-to-state stability proposed by Sontag. It allows us to establish two important results. The first one states that the general interconnection of two IOpS systems is again an IOpS system if an appropriate composition of the gain functions is smaller than the identity function. The second one shows an example of gain function assignment by feedback. As an illustration of the interest of these results, we address the problem of global asymptotic stabilization via partial-state feedback for linear systems with nonlinear, stable dynamic perturbations and for systems which have a particular disturbed recurrent structure.The final version of this work was finished when Z.-P. Jiang held a visiting position at I.N.R.I.A., Sophia Antipolis.     

Oscillation criteria for second order quasilinear neutral delay differential equations on time scales

In this paper, we consider the second-order quasilinear neutral dynamic equation

(r(t)|Z^Δ(t)|^α−1Z^Δ(t))^Δ+q(t)|x(δ(t))|^β−1x(δ(t))=0,     

Stability of 2h-step spline method for delay differential equations

In this paper we consider a linear test equation to study the stability analysis of 2h-step spline method for the solution of delay differential equations. We prove that, this method is P-stable for cubic spline.     

Delay-dependent criteria for robust stability of linear neutral systems with time-varying delay and nonlinear perturbations

This article investigates the robust stability of linear neutral systems with time-varying delay and nonlinear perturbations. Using a new Lyapunov–Krasovskii functional and employing some free weighting matrices, less conservative delay-dependent robust stability conditions for such systems in terms of linear matrix inequalities are derived. Numerical examples are given to indicate significant improvements over some existing results.     

Multiplicative stability criteria based on difference solutions of ordinary differential equations

For computer analysis of Lyapunov stability, multiplicative criteria are proposed that are based on difference approximations to solutions of the Cauchy problem. These criteria can be applied to ordinary differential equations in normal form and include the necessary and sufficient stability conditions. For a system of linear equations with constant coefficients, information on the characteristic polynomial of the coefficient matrix and its roots is not used. The stability analysis is combined with difference solution and simulation of error accumulation. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 127–142, January–February 2006.     

On characteristic roots and stability charts of delay differential equations

This work is devoted to the analytic study of the characteristic roots oftextitscalar autonomous delay differential equations with either real or complex coefficients. The focus is placed on the robust analysis of the position of the roots in the complex plane with respect to the variation of the coefficients, with the final aim of obtaining suitable representations for the relevant stability boundaries and charts. While the real case is almost standard (and known), the investigation of the complex case is not as immediate. Hence, a preliminary shift of the coefficients is proposed, which reduces the number of free parameters. This allows to extend the techniques used for the real case, also allowing for useful graphical visualization of the relevant stability charts. The present research is motivated on the basis of studying the stability of systems with delay. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

Exponential stability of nonlinear differential delay equations

The aim of this paper is to investigate the exponential stability of a nonlinear differential delay equation ) Introduce the corresponding differential equation (without delay) ) and assume it is exponentially stable. It will be shown in this paper that the differential delay equation will remain exponentially stable provided the time lag τ is small enough.     

Set membership state and parameter estimation for systems described by nonlinear differential equations

This paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous-time systems, in a bounded-error context. A state estimator based on a prediction-correction approach is given, where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation (IVP for ODE) using interval analysis and high-order Taylor models, while the correction step uses a set inversion technique. The state estimator is extended to solve the parameter estimation problem. An illustrative example is presented for each part.     

Point delay methods applied to the investigation of stability for a class of distributed delay systems

The main target of this paper is to present criteria (adapted from the point delay systems) for the different kinds of stability, boundedness and existence of a stochastic attractor for a class of distributed delay differential stochastic equations. The new results are based on a technique of reduction of distributed delay to a lumped delay and Mao criteria for point delay equations.     

Stability and convergence of the two parameter cubic spline collocation method for delay differential equations

In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters α,β which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results.     

An alternative use of fuzzy transform with application to a class of delay differential equations

Fuzzy transforms (or F-transforms for short) are an approximation technique recently introduced. The main application is referred to image and data compression. There are really few works devoted to the use of F-transform for solving ordinary differential equations. In the present paper, an F-transform-based Picard-like numerical scheme is proposed in order to solve a class of delay differential equations. For linear cases, the proposed approach leads to a non-recursive approximate solution by means of operational matrices and vectors of known quantities. Numerical results show the good performance of the proposed method against known solutions.     

On robust stability of LTI fractional-order delay systems of retarded and neutral type

This paper deals with the analysis of robust BIBO-stability of LTI fractional order delay systems in the presence of real parametric uncertainties. Two large classes of these systems, namely retarded and neutral types, are considered. Two theorems are given to check the robust BIBO-stability of these two families of fractional order systems. One of these theorems provides necessary and sufficient conditions for the case of retarded type and another one presents only sufficient conditions for the case of neutral type. Furthermore, upper and lower bounds (cutoff frequencies) are provided for drawing the value sets. To illustrate the results, two numerical examples are presented.