STABILITY OF NONLINEAR SYSTEMS UNDER CONSTANTLY ACTING PERTURBATIONS 273

In this paper, we investigate total stability, attractivity and uniform stability in 
terms of two measures of nonlinear differential systems under constant perturbations. Some sufficient 
conditions are obtained using Lyapunov's direct method. An example is also worked out.


INTRODUCTION
When we model a physical system by means of a differential equation, it is not generally possible to take into account all the causes which determine the evolution.In other words, we have to admit that there are small perturbations permanently acting which cannot be accurately estimated consequently the validity of the description of the evolution, as given by a corresponding solution of the differential equation, requires that this solution be "stable" not only with respect to the small perturbations of the initial conditions, but also with respect to the perturbations, small in a suitable sense, of the right hand side of the equation.This kind of stability is called total stability, which we shall define in the next section.There are several different concepts of stability studied in the literature, such as eventual stability, partial stability, conditional stability, etc.To unify these varieties of stability notions and to offer a general basis for investigation, it is convenient to introduce stability in terms of two different measures.Following Movchan [4], Salvadori [5] has successfully developed the theory of stability in terms of two measures.In the recent years much work has been done using two measures.See [2,3] and references therein.
In this paper we investigate the total stability, attractivity and uniform stability of perturbed systems in terms of two measures.In view of the generality of the present approach, our results improve and include some of the earlier findings and may be suitable for many applications.
Let us begin by defining the following class of functioni for future use.
F Definition 2.1 Let ho, h E 1', then we say tlat ho is uniformly finer than h if there exists a 6>0 and a function (1) /f such that ho(t,x) < 6 implies h(t,x) <_ #P(ho(t,x)).
Definition 2.2 Let VeC[R+ ttN, R+] and ho, h F then V(t,x) is said to be (i) h-positive definite if there exist a p > 0 and a function b E K such that h(t,x) < p implies b(t(t,z)) < V(t,x).
(ii) ho decrescent if there exist a 6 > 0 and a function a E K such that ho(t,x) < 6 implies V(t,x) < a(ho(t,x)).
Let ho, h ( F. We shall now define the stability concepts for the system (2.1) in terms of two measures (h0, h).Let S(h,p)= [(t,x) R+ R', h(t,x) < p].

MAIN RESULTS
In this section we shall investigate the stability and attractivity properties of the differential system.
THEOREM 3.1 Assume that (i) h0, h E F and h0 is uniformly finer than h.
which implies V(to + T,y(to + T)) < a(ho(to + TI y(to + T))) C(6,) (T T) < 0 This contradiction shows the existence of t" and it follows from (ho, h, T) total stability of (2.1) that the system (2.2) is (ho, h) attractive, which completes the proof of the theorem.The next result is on (ho, h, T2) stability.THEOREM 3.3 Assume that (i) ho, h (E 1-' and ho is uniformly finer than h. (ii V.,(t,x) <_ -C(V(t,z)), (t,x)(S(h,p), C K.
Then the system (2.1) is (ho, h,T) totally stable.
In the previous threms, in order to prove total stability properties of (2.2) we sumed the uniform asymptotic stability properties of (2.1) (the unperturbed system).In the following threm we prove (h0, h) stability of (2.2) under weaker sumptions on (2.1) by avoiding using norm on the perturbed term.THEOREM 3.4 Assume that (i) ho, h F and ho is uniformly finer than h.