Nonlinear Variation Of Constants Formula Research Articles

Non-asymptotic error bounds for constant stepsize stochastic approximation for tracking mobile agents

This work revisits the constant stepsize stochastic approximation algorithm for tracking a slowly moving target and obtains a bound for the tracking error that is valid for the entire time axis, using the Alekseev nonlinear variation of constants formula. It is the first non-asymptotic bound for the entire time axis in the sense that it is not based on the vanishing stepsize limit and associated limit theorems unlike prior works, and captures clearly the dependence on problem parameters and the dimension.

Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay. As an application, we discuss asymptotic properties of perturbed nonlinear differential equations with state-dependent delays.

Euclidean and Function Space Null-Controllability of Nonlinear Delay Systems with Restrained Controls

The basic tool we use in our investigation is the nonlinear variation of constants formula for delay systems whilst we determine the controllability of nonlinear functional differential systems. This paper also touches on capture in nonlinear functional differential games of pursuit.

Attractors for strongly damped wave equations with critical nonlinearities

In this paper we obtain global well-posedness results for the strongly damped wave equation utt +( −∆) θ ut =∆ u + f (u), for θ ∈ � 1 , 1 � ,i nH 1 0 (Ω) × L 2 (Ω) when Ω is a bounded smooth domain and the map f grows like |u| n+2 n−2 .I ff = 0, then this equation generates an analytic semigroup with generator −A(θ). Special attention is devoted to the case when θ =1 since in this case the generator −A(1) does not have compact resolvent, contrary to the case θ ∈ � 1 , 1 � . Under the dissipativeness condition lim sup |s|→∞ f (s) s ≤ 0 we prove the existence of compact global attractors for this problem. In the critical growth case we use Alekseev’s nonlinear variation of constants formula to obtain that the semigroup is asymptotically smooth.

Invariant Manifolds for Retarded Semilinear Wave Equations

By means of a nonlinear variation of constants formula it is shown that, under suitable assumptions, there exists a global invariant manifold for semilinear hyperbolic evolution equations with a retarded perturbation, provided that the time-delay is small or the magnitude and the Lipschitz constant of the perturbation are small. By construction, this invariant manifold is weaved by trajectories of an associated nonretarded evolution equation; it is also locally exponentially attracting. As applications of the theory, we treat retarded perturbutions of the Sine-Gordon equation and the dissipative Klein-Gordon equation.

G-Representations and differential equations

Earlier results concerning g- and h-representations for nonlinear input-output maps are extended so that vector-valued inputs and outputs can be treated. It is established that the range of applicability of the extended results is very wide, and it is illustrated how they can be used, within the contexts of systems governed by differential equations. In particular, it is shown that the nonlinear variation of constants formula is a special case of the main representation theorem. >

Global attractivity in the delay logistic equation with variable parameters

AbstractUsing a non-linear variation of constants formula, sufficient conditions are derived for the global asymptotic stability of the trivial solution of

A unified approach to stability in integrodifferential equations via Liapunov functions

In this paper we investigate the problems of stability, uniform stability, uniform Lipschitz stability, asymptotic stability, and uniform asymptotic stability in three classes of integrodifferential systems. These are linear nonautonomous systems, nonlinear systems, and nonlinearly perturbed nonlinear systems. A simple method for constructing Liapunov functions will be used to unify the treatment of all of the above-mentioned stability notions relative to the various classes of integrodifferential systems. In our investigation of linear systems we benefited from the work in [4-8, 123. For nonlinear systems and their perturbations, we have used some of the results obtained recently in [IS, 101. In particular, we have used an extension of Alekseev’s [l] nonlinear variation of constants formula which was developed in [lo]. The approach we use here was first formulated in [3] and utilized to study uniform Lipschitz stability [2] of nonlinear differential systems. The method has been extended in [12] to cover all other stability notions in nonlinear differential systems. We now give a description of the contents of the paper. Section 1 includes all the basic definitions and preliminaries. In Section 2, we investigate the various stability notions of nonautonomous integrodifferential systems. This extends the work in [6-81, which mainly dealt with autonomous systems. We recall here from [4] that, in linear systems, the notions of uniform stability and uniform Lipschitz stability are equivalent. It was shown in [4], however, that, for nonlinear systems, uniform

Perturbations of nonlinear systems of ordinary differential equations

In this paper the asymptotic properties of solutions of systems of ordinary differential equations with strong nonlinearity are discussed. The main tools used here are Alekseev's nonlinear variation of constants formula and two nonlinear integral inequalities of the Bellman type established by the present author. Theorems 1 and 2 proved here generalize the main results obtained by I.M. Vlasov ( Izv. Vysš. Učebn. Zaved.) Matematika 69 (2) (1968) , 27–34). As applications of the results a series of corollaries for perturbed linear differential systems are also stated which are extensions of some known theorems. Although there are many other articles concerned with the same problem as that in the present paper (for example, see [9–16]), the results obtained here seem to be new.

On Asymptotic Behaviour of Certain Integro-Differential Equations

This paper deals with the asymptotic behaviour of some integro-differential equations which can be seen as perturbations of some nonlinear systems of ordinary differential equations. The main tools used here are two generalizations of the Pachpatte’s integral inequality and an Alekseev type nonlinear variation of constants formula.

On asymptotic behaviour of certain integro-differential equations

This paper deals with the asymptotic behaviour of some integro-differential equations which can be seen as perturbations of some nonlinear systems of ordinary differential equations. The main tools used here are two generalizations of the Pachpatte’s integral inequality and an Alekseev type nonlinear variation of constants formula.

On Nonlinear Perturbations of Nonlinear Dynamical Systems, and Applications to Control

The nonlinear variation of constants formula is generalised to infinite dimensional systems and applied to the stability problem and to obtain bounds on the states of a nonlinear control system when the controls are bounded. The theory is illustrated by a simple model from nuclear reactor dynamics

A note on non-linear observers

A general non-linear exponential observer is presented for non-linearly perturbed nonlinear systems. This is done by using the non-linear variation of constants formula.

Perturbations of non-linear difference equations

Abstract Consider the non-linear difference equations where x, y, j, and g are; q-vectors, fx(n, x) is continuous in x, and the origin is asymptotically self-invariant relative to (1). It is shown that if the origin is uniformly stable in variation (uniformly asymptotically stable in variation), and if the perturbation g satisfies then the origin is eventually uniformly stable (eventually uniform-asymptotically stable) for (2). These stability results are obtained via a non-linear variation of constants formula.

A nonlinear variation of constants formula for volterra equations

Abstract : A variation of constants formula is developed for nonlinear Volterra integral and integro-differential equations. This formula is suited to the study of various perturbation problems for Volterra equations. (Author)