Anatoli F. Ivanov

Oscillations in Singularly Perturbed Delay Equations

This paper presents some recent results on the scalar singularly perturbed differentila delay equation

Global behavior in nonlinear systems with delayed feedback

[v\dot x(t) + x(t) = f(x(t - 1)).

Observation and Observers for Systems from Delay Convoluted Observation

(1)

Global stabilization in nonlinear discrete systems with time-delay

Some issues in the stability of differential delay systems in the linear and the nonlinear case are investigated. In particular, sufficient robustness conditions are derived under which a system remains stable, independent of the length of the delay(s). Applications in the design of delayed feedback systems are given. Two approaches are presented, one based on Lyapunov theory, the other on a transformation to Jordan form. In the former, sufficient conditions are obtained in the form of certain Riccati-type equations.

Periodic solutions of a discretized differential delay equation

The problem of global stability in scalar delay differential equations of the form x/spl dot/(t)=f(x(t-/spl tau/))-g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size /spl tau/ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as /spl tau//spl rarr//spl infin/) difference equation g(x/sub n+1/)=f(x/sub n/): the latter always implying the global stability in the differential equation for all values of the delay /spl tau//spl ges/0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay /spl tau/.

Global Dynamics and Periodic Solutions in a Singular Differential Delay Equation

Article history: Received 28 October 2013 Available online 9 May 2014 Submitted by R. Popovych

Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima

Abstract This paper analyzes finite dimensional linear time-invariant systems with observation of a delay, where that delay satisfies a particular implicit relation with the state variables, rendering the entire problem nonlinear. The objective is to retrieve the state variables from the measured delay. The first contribution involves the direct inversion of the delay, the second is the design of a finite dimensional observer, and the third presents properties of the delay - state relation. Realistic examples treat vehicles with ultrasonic position sensors.

On global stability in a nonlinear discrete model

A class of scalar nonlinear difference equations with delay is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are given. Applications in economics and other fields lead to consideration of associated optimal control problems. An optimal control problem of maximizing a consumption functional is stated. The existence of optimal solutions is established and their stability (the turnpike property) is proved.