STABILITY AND ASYMPTOTIC STABILITY IN IMPULSIVE SEMIDYNAMICAL SYSTEMS

In this paper we generalize two results of Lasalle’s, the invariance theorem and asymptotic stability theorem of discrete and continuous semidynamical systems, to impulsive semidynamical systems.

Furthermore, the following two properties are easy to check: (i) (x,O) x for any x and (ii) ( (x, t), x) (x, + s), for x( and and s[O,T(x)), such that t+se [O,T(x)).Thus from here on we assume that ,M,I and N are as considered above in a dynamical system (X,), and call (,), with as defined above, an impulsive semidynamical system associated with (X, r)(see [6]).
Let 0 {x G f:(I)(x) < oc}.Then there exists a function g:Fto--,N defined by g(x) (r, (I)(x)) for any x G f0" Assumption I: We assume throughout the paper that (I) is a continuous function on f.It follows then that for any x eft o, g(x)= I(r(x, (I)(x)))is a continuous function on ft 0.
In this paper, as in [3] and [4], we continue to study dynamical notions defined in (,) by relating them to similar notions in (N,g).
The following notations are used throughout.For any A C f2 and d(x,A) < }, where d(x, A) -inf{d(x, y): y a}, cl denotes the closure operator in f; a a f N, and for any xea, Co(x)-{r(x,t)-(x,t):0_<t<O(x)}.We extend the known definitions of these sets in a natural way to (f,).
(2.1.1)The limit set of x in (f, is defined by L (x) {y e a" (x, tn)-y for some tnT(x and t n < T(x)}.
(2.1.2)The prolongation limit set of x in (,) is defined by J (x) {y e a: (xn, tn)y for xnx and tn---*T(x), t n < (2.1.3)The prolongation set of x in (a,) is defined by D(x) {y _ a:' (xn, tn)-y for xn-,x and t n z The definition of these sets in a discrete semidynamical sys..tem are well known.In (N, g) we shall denote them by L(x), J(x) and D(x) respectively for x E N. Furthermore, we denote the orbit of a point x Eft in (,) by C(x)-{(x^,t)" t e [0, Tx))} and its closure in by K(x).These sets in (N,g) are denoted by C(x) and g(x) for x N.
Generally one uses a + sign with the above symbols for these sets for flows in positive direction, but as we are dealing with flow only in positive direction we have dispensed with it for convenience.Definition 2.2: A subset A of is said to be positively invariant if for any x A, C (x) C A, and it is said to be invariant if, it is positively invariant and, furthermore, given x A and E [0, T(x)) there exists a y A such that (y, t) x.
A subset A of N is said to be positively invariant if g(A)C A and is said to be invariant if A-g(A).
It is clear that is not a continuous function.However, the following lemma holds.
Lemma 2.3: Suppose (xn} is a sequence in , convergent to a point y E .Then for any E [0, T(y)), there exists a sequence of real numbers, (n}, n ----0' such that, tA-en < T(xn) and Proof: First suppose that (I,(y)< that (xn) < Then clearly, we may assume, since is continuous, Case 1: If 0 < t < (I)(y), let e < O(y)-t.Then from the continuity of we may conclude that (I)(y)-< O(xn) for all n, so that < O(xn) and (xn, t)-r(xn, t), and the continuity of r implies the result for n-0.
Lemrna 2.4: Suppose A C f is positively invariant in (f, ), then clA is positively invariant in Proof: Suppose x clA and t [0, T(x)).Then there exists a sequence {xu} in A with xnx.tIence by Lemma 2.3, (xn, t+en)-(x,t for some sequence n--0 such that + n [0, T(xn)).Since A is invariant, (xn, + en) A, hence (x, t) clA.This completes the proof.
Lemma 2.5: quppose .pG , and ft'(p) is compact, then T-qp(p: is infinite, where p_po + n-o Proof: Suppose T is finite. Then (I'(pn +)40 Assuming that p +-q , n k r(p4, O(p+k))zr(q,O --q.But, r(Pnk, O(pk))e M, which is a closed subset of X. Hence q E M implying M N (I).But this is a contradiction since f is open in X and hasno point in common with its boundary M in X.
Assumption II: From now on we assume that if x N, then T(x)cx; so that if for some x , (x, (I'(x)) N, then T(x) is also infinite.Note that if (x, (I)(x)) N then T(x) cx anyway.Thus from here on T(x) will be assumed to be infinity for all x , and [0, T(x)) will be replaced by R +.
Lemma 2.6: For any x , (x) is closed and positively invariant in (, ).
Proof: To show that it is closed is trivial.To prove that L (x) is positively invariant, let y e .(x)Then there exists a sequence {t} in R + t --cx, such that (x, tn)-xn--y.Now, given R +, there exists a sequence {en} as in Lemma 2.3, such that, pn+-q as nk---o where P4 gnk(p).Then (P4't + e) r(q,t)-y by Lemma 2.3.n k tk E I'(P/+ + O(x), then (x, k + t + n)y, and y e (x).This completes the proof. i--0
Corollary 2.15: Let x e a and (x,(x))-p e .If (p) has compact closure in , the L (x) is invariant.
Lastly, we note a trivial result for future reference.
Lemma 2.16: If A C and D(A) A in (a,), then A is positively invariant in (a, ).
Proof: By definition for any x A, C (x)C K(x)C D(x)C A. Hence the result.

Stability
Definition 3.1: A subset A of Q is called a cylindrical set if for any x A, Co(x C A, where Co(z)t).o< < Note: For any x , Co(x is a cylindrical set. The following is trivial.Lemma 3.2: If A C Q is a cylindrical set, then so is clA.Lemma 3.3: Let V be an open set in , then Co(V)-t0{Co(x); x Y}-W is a cylindrical open set in .
Proof: W is clearly a cylindrical set.Let z W. Then there is a y V such that z--(y,t), 0_t<(I)(y).Let 0<<(I)(y)-t.By continuity of (I) there exists an open set U containing.y and U C V, such that, if x E U, then (I)(x)-(I)(y) < e. Hence t < (I)(y)-e < (I)(x), and t(x) (x, t) Co(x C W. Since wt is a homeomorphism, y rt(U C W implies that W is open.Definition 3.4: A subset A of is said to be stable in (, if given any cylindrical open set U containing A there exists an open set V, such that A C V C U, and for any x G V, C (x) C U. Theorem 3.5: x e A, (P(x) < oc.U(A,).
Let be locally compact.Let A be a cylindrical subset of and for each Then for any > 0 there exists an open set U D A, such that, Co(U C Proof: Let x G A. Then, since " is continuous and is locally compact, there exists a compact neighborhood U(x) of x and a real number 5(x)> 0, such that 5(x)is less than both and (I)(x), and for any y E such that d(x,y)< 5(x), and any t, t' in [0,(I)(x)+ ], satisfying It-t' < 5(x), we have d((x,t),(y,t'))< .And, since (I)is continuous, there exists a such that for any y G for which d(x, y) < u(x), (I)(x)-(I)(y) < 5(x)/2 and U(x, u(x)) C U(x).
Hence, d(r(x, t', r(y, t)) < e.This proves the claim.Hence if U U {U(x, (x))'x A}, then U is open, contains A and Co(U C_ U(A,e).
Example 1" The following example shows that if a set A C does not satisfy the condition that (I)(x) < oc for each x G A, then the above theorem is note true.Consider the dynamical system (R2, ) shown in the Figure 1 below" Figure 1 Let X be the open set in R 2 obtained by deleting the set {(x,y):xy-+ k,y > 0} for some k > 1.The two curves, .xy-4-kdefine trajectories in the original dynamical system.Hence (X,r) is a dynamical system, with metric space x.Let M-{(x,y): 0 < y < 1, x-4-k}, and N-{(x,y):-i <x< 1, and y-k}.Let -X-M, and define I: M---,N by I(x,y)-(y2, k)if x k, and I(x,y)-(-y2, k)if x -k.
Notice that deleting the orbits as we did from the original dynamical system to get X, makes (I)   continuous on .
Now suppose A {(0, y):0 < y < k}.Then A is a cylindrical set. is locally compact.But for each x E A, (I)(x)cx.Note also that for this A the above theorem fails, because any cylindrical open set containing A has finite diameter of 2k.Lemma 3.6: Let A..C be positively invariant in ., and for each x A, let 4p(x) < oc.
Then p (x, (P(x)) A. Consequently, A f3 N A f3 g.Proof: Since (I)(x)< oc, p N. Since A is positively invariant p A also.Repeating the above argument we get pl + N and p EA. Thus, inductively, pn + AVIN for n-1,2, Consequently, p N and therefore p A A '1N.
If, furthermore, x N, then clearly x E N, and the last result follows.
Lemma 3.7: Let A Co and D(A)-A.If N has compact closure in , then there exists an open set V containing A such that V N N-V N.
Proof: Suppose no such 2pen set exists.Then there exists a sequence {xn} in N-N, xn---xft.Since x nN-N, there exists a t nR+, such that, (xn, tn)-qn N, and   (I)(qr-C.Since N has compact closure in , we may assume that qn---+q .However, q D(A)-A, and, therefore, (I)(q)< c)c.But this contradicts the continuity of (I) at q, and completes the proof.
Lemma 3.8: Suppose A C is posilivel..y invariant and for each x G A, 4p(x) < cxz.Suppose V is an open set containing A such that V-VN-V N. Then there exists a cylindrical open set U D A such that for any y G U, ep(y) < cx and " (y,(y)) G Y.
Proof: Let x G A. Then (I)(x)< oc imp.lies 7r(x,(P(x))x 1 M. Since A is positively invariant, by Lemma 3.7 (x, gp(x))-Xl + A. Now I being continuous, there exists an open set U(Xl) in M, containing Xl, such that I(U(Xl) C V. Again, by.the continuity of r and (I), there exists an open set U(x)in 12 containing x, such that, for any y U(x), r(y,(y)) U(Xl) and (I)(y)< oc.Furthermore, we may assume that U(x)lies in V. Hence U-C0[t2 {U(/): x E A}] is a cylindrical open set (Lemma 3.3), and satisfies the requirements of the lemma.Theorem 3.9: Let f be locally compact.Let A C f be a closed cylindrical set and for each x A, let (P(x) < oc.IrA is stable in (, ), then 1.
there exists an open set U in f containing A-A VI N, such that, U V1N-U VI N.
Proof: 1. Follows from the usual argument using Theorem 3.5.
2. From continuity of (I), each x A is contained in an open set V x such that for each y Vx, (I)(y) < oc.Set V t2 {Co(Vx):X A}.Then V is a cylindrical open set (Lemma3.3) con- taining A. Hence, by stability~o f A there exists an open set U, A C U C V such that C(x) C V for each x E U. Set W t2 {C(x): x U}.Then clearly W is positively invariant and since for each xV, (I)(x)<oc, by Lemma3.6,WNN-WVIN.Hence, UV1N-UV1N.This completes the proof.
2. Limit Sets, Prolongation Limit Sets and Prolongation Sets