当前位置：文档库 › Vector Lyapunov functionspractical stability nonlinear impulsive functional differential equations

Vector Lyapunov functionspractical stability nonlinear impulsive functional differential equations

J.Math.Anal.Appl.325(2007)612–623

https://www.wendangku.net/doc/865399564.html,/locate/jmaa

Vector Lyapunov functions for practical stability

of nonlinear impulsive functional differential equations

Ivanka M.Stamova

Department of Mathematics,Bourgas Free University,8000Bourgas,Bulgaria

Received 29September 2005

Available online 9March 2006

Submitted by William F.Ames

Abstract

This paper studies the practical stability of the solutions of nonlinear impulsive functional differential equations.The obtained results are based on the method of vector Lyapunov functions and on differential inequalities for piecewise continuous functions.Examples are given to illustrate our results.

Keywords:Practical stability;Impulsive functional differential equations;Vector Lyapunov functions

1.Introduction

Impulsive differential equations arise naturally from a wide variety of applications such as air-craft control,inspection process in operations research,drug administration,and threshold theory in biology.There has been a signi?cant development in the theory of impulsive differential equa-tions in the past 10years (see monographs [3,4,13,20]).Now there also exists a well-developed qualitative theory of functional differential equations [2,9–12].However,not so much has been developed in the direction of impulsive functional differential equations.In the few publications dedicated to this subject,earlier works were done by Anokhin [1]and Gopalsamy and Zhang [8].Recently,some qualitative properties (oscillation,asymptotic behavior and stability)are investi-gated by several authors (see [5–7,18,21,23,24]).

The ef?cient applications of impulsive functional differential equations to mathematical sim-ulation requires the ?nding of criteria for stability of their solutions.

E-mail address:stamova@bfu.bg.

doi:10.1016/j.jmaa.2006.02.019

I.M.Stamova /J.Math.Anal.Appl.325(2007)612–623613

In the study of Lyapunov stability,an interesting set of problems deal with bringing sets close to a certain state,rather than the state x =0.The desired state of a system may be mathemati-cally unstable and yet the system may oscillate suf?ciently near this state that its performance is acceptable.Many problems fall into this category including the travel of a space vehicle between two points,an aircraft or a missile which may oscillate around a mathematically unstable course yet its performance may be acceptable,the problem in a chemical process of keeping the temper-ature within certain bounds,etc.Such considerations led to the notion of practical stability which is neither weaker nor stronger than Lyapunov stability.The main results in this prospect are due to Martynyuk [14,16,17].

It is well known that employing several Lyapunov functions in the investigation of the quali-tative behavior of the solutions of differential equations is more useful than employing a single one,since each function can satisfy less rigid requirements.Hence,the corresponding theory,known as the method of vector Lyapunov functions,offers a very ?exible mechanism [15].

In this paper,we use piecewise continuous vector Lyapunov functions to study practical stabil-ity of the solutions of nonlinear impulsive functional differential equations.The main results are obtained by means of the comparison principle coupled with the Razumikhin technique [14,19].Examples are given to illustrate our results.

2.Statement of the problem.Preliminary notes and de?nitions

Let R n be the n -dimensional Euclidean space with norm |x |=( n i =1x 2i )

1/2,Ωbe a bounded domain in R n containing the origin and R +=[0,∞).Let t 0∈R ,τ>0.Consider the system of impulsive functional differential equations ˙x(t)=f (t,x(t),x t ),t >t 0,t =t k , x(t k )=x(t k +0)?x(t k )=I k (x(t k )),t k >t 0,k =1,2,...,

(1)where f :(t 0,∞)×Ω×D →R n ;D ={φ:[?τ,0]→Ω,φ(t)is continuous everywhere ex-

cept at ?nite number of points ?t

at which φ(?t ?0)and φ(?t +0)exist and φ(?t ?0)=φ(?t )};I k :Ω→Ω,k =1,2,...;t 0t 0,x t ∈D is de?ned by x t =x(t +s),?τ s 0.Let ?0∈D .Denote by x(t ;t 0,?0)the solution of system (1)satisfying the initial conditions: x(t ;t 0,?0)=?0(t ?t 0),t 0?τ t t 0,x(t 0+0;t 0,?0)=?0(0).

(2)The solution x(t)=x(t ;t 0,?0)of the initial value problem (1),(2)is characterized by the fol-lowing:

(a)For t 0?τ t t 0the solution x(t)satis?ed the initial conditions (2).

(b)For t 0

˙x(t)=f t,x(t),x t ,t >t 0,

x t 0=?0(s),?τ s 0.

At the moment t =t 1the mapping point (t,x(t ;t 0,?0))of the extended phase space jumps momentarily from the position (t 1,x(t 1;t 0,?0))to the position (t 1,x(t 1;t 0,?0)+I 1(x(t 1;t 0,?0))).

614I.M.Stamova /J.Math.Anal.Appl.325(2007)612–623

(c)For t 1t 1,

y t 1=?1,?1∈D,

where ?1(t ?t 1)=????0(t ?t 1),t ∈[t 0?τ,t 0]∩[t 1?τ,t 1],

x(t ;t 0,?0),t ∈(t 0,t 1)∩[t 1?τ,t 1],x(t ;t 0,?0)+I 1(t ;t 0,?0),t =t 1.

At the moment t =t 2the mapping point (t,x(t))jumps momentarily,etc.

The solution x(t ;t 0,?0)of problem (1),(2)is a piecewise continuous function for t >t 0with points of discontinuity of the ?rst kind t =t k ,k =1,2,...,at which it is continuous from the left.

Introduce the following notations:

I =[t 0?τ,∞);I 0=[t 0,∞);G k = (t,x)∈I 0×Ω:t k ?1

k =1,2,...;G =∞

k =1G k ; φ =sup s ∈[?τ,0]

φ(s) is the norm of the function φ∈D.Together with system (1)we shall consider the system ˙u =g(t,u),t >t 0,t =t k ,

u(t k )=B k (u(t k )),t k >t 0,k =1,2,...,(3)

where g :(t 0,∞)×R m +→R m +,B k :R m +→R m +,k =1,2,....Denote by u +(t ;t 0,u 0)the maximal solution of system (3)satisfying the initial condition u +(t 0+0;t 0,u 0)=u 0∈R m +