V.B. Kolmanovskii

Stability of some linear systems with delays

Asymptotic stability of a class of linear equations with arbitrary discrete and distributed delays is investigated. Both delay-independent and delay-dependent stability conditions are formulated in terms of existence of positive definite solutions to Riccati matrix equations. The approach of deriving various Riccati equations using the direct Lyapunov method is proposed.

Neutral Stochastic Differential Delay Equations with Markovian Switching

Abstract Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see Kolmanovskii, V.B. and Nosov, V.R., Stability and Periodic Modes of Control Systems with Aftereffect; Nauka: Moscow, 1981 and Mao X., Stochastic Differential Equations and Their Applications; Horwood Pub.: Chichester, 1997). Given that many systems are often subject to component failures or repairs, changing subsystem interconnections and abrupt environmental disturbances etc., the structure and parameters of underlying NSDDEs may change abruptly. One way to model such abrupt changes is to use the continuous‐time Markov chains. As a result, the underlying NSDDEs become NSDDEs with Markovian switching which are hybrid systems. So far little is known about the NSDDEs with Markovian switching and the aim of this paper is to close this gap. In this paper we will not only establish a fundamental theory for such systems but also discuss some important properties of the solutions e.g. boundedness and stability.

Stability of difference Volterra equations: Direct Liapunov method and numerical procedure

Abstract A procedure to construct Liapunov functionals for discrete Volterra equations is proposed. Using this procedure stability conditions are derived for general Volterra difference equations. Some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for Volterra integro-differential equations are presented.

Mean square stability of difference equations with a stochastic delay

The paper describes mean-square stability conditions for nonlinear delay difference equations with a stochastic delay. The first part develops a formula for the infinitesimal operator. Using this formula asymptotic mean square stability conditions are derived. A final example is provided.

Stability of discrete volterra equations of hammerstein type

Stability conditions for Volterra equations with discrete time are obtained using direct Liapunov method, without usual assumption of the summability of the series of the coeffcients. Using such conditions, the stability of some numerical methods for second kind Volterra integral equation is analyzed.

Some remarks on the stability of linear systems with delayed state

This paper focuses on mixed delay-independent/delay-dependent asymptotic stability problems of a class of linear systems described by delay-differential equations involving multiple constant delays. Sufficient conditions for characterizing unbounded stability regions in the delays parameter space are given. The proposed approach makes use of some appropriate Liapunov-Krasovskii functionals and the obtained results are expressed in terms of matrix inequalities. Note that these results allow to recover (or to improve) as limit cases previous sufficient delay-independent or/and delay-dependent conditions from the control literature.

Constructive filtering algorithms for delayed systems with uncertain statistics

A stochastic optimal guaranteed estimation problem for dynamic delayed systems with uncertain statistics is considered. The solution of this problem reduces to a complex nonsmooth extremal problem. To obtain an approximate solution, the nonsmooth problem is replaced by a smooth one. Constructive filtering algorithms are obtained from an approximate solution of the smooth problem under the assumption that the delay is small in comparison with the observation time. Estimates for the nonoptimality levels of the proposed filtering algorithms are derived.

Estimation of the solutions of Volterra difference equations

Volterra equations, whose solution is defined by the whole previous history, are widely used in the modelling of the processes in continuous mechanics and biomechanics, problems of control and estimation and also some schemes of numerical solutions of integral and integral–differential equations [1–8]. In this connection, there is an essential interest in such properties of the solutions as stability, limiting periodicity, boundedness and various estimates of the solutions defined by the acting perturbations. In [9], such estimates were obtained for the solutions of nonlinear scalar integral Volterra equations of convolution type

Stability of Systems With Pure, Discrete Multi-Delays

Abstract This paper introduces three different delay-dependent Riccati equations for the study of linear systems with several constant delays, which can be non-commensurate. The considered models can be purely retarded (this means, without any instantaneous feedback). In a first part, the basic idea is presented in the particular case of a single delay, and in a second one the general multi-delay case is solved. Our approach is based on the direct Liapunov-Krassovskii method, and on some model transfonnations. The three obtained Riccati equations involve the values of the delays in an explicit way, and the existence of a positive solution to any of these three equations is sufficient to ensure the global asymptotic stability. Moreover, these sufficient conditions are shown to be necessary when delays tend to zero: this allows thinking that the results may not be too conservative. As an illustrative example, we finally study the particular case of a second order system.

Guaranteed filtering in delayed dynamic systems

A stochastic optimal guaranteed estimation problem for dynamic delayed systems with uncertain statistics is considered. The solution of this problem reduces to a complex nonsmooth variational problem. To obtain an approximate solution the nonsmooth problem is replaced by a smooth one. Constructive filtering algorithms are found from an approximate solu- tion of the smooth problem under the assumption that the delay is small in comparison with the observation time. Estimates for the nonoptimality levels of the proposed filtering algorithms are derived. In the paper, the nonoptimality levels for new con- structive algorithms are presented. These algorithms together with the nonoptimality levels give a useful tool for the engineering design of filters. So, the idea outlined in (KOL 01) is realized completely in the present work. Moreover, a significant numerical example of filtering problem for an oscillation system with delay is considered in detail.