S. Leela
UNCERTAIN DYNAMIC SYSTEMS ON TIME SCALES
Abstract. Utilizing the framework of the theory of dynamic sys- tems on time scales for measure chains, stability of moving invariant sets is discussed. These results include both continuous and discrete dynamic systems.
reziume. naSromSi ganxil ul ia iseTi dinamiuri sistemebis moZravi invariantul i simravl eebis mdgradobis sakiTxi, romel Ta drois Skal a SeiZl eba iKos rogorc uCKveti, aseve diskretul i.
1. Introduction
Nonlinear dierential equations with uncertain parameters may cause change of equilibrium states. To investigate such situations, Siljak, Ikeda and Ohata [8] have introduced the notion of parametric stability and dis- cussed its study which is interesting in itself.
A fundamental feedback control problem is that of obtaining some desired behavior from the given system which has uncertain information. Leitmann and associates [1, 2, 9] have dealt with such a problem in a series of papers.
They have investigated continuous and discrete uncertain systems by means of Lyapunov functions.
Recently, a theory known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, namely, time as an arbitrary closed set of reals, and permit us to handle both systems simultaneously [6]. This theory allows one to get some insight into and bet- ter understanding of the subtle dierences between discrete and continuous systems.
To study uncertain systems, a dierent idea is employed recently [5], which exhibits moving invariant sets as the parameter changes. By reduc- ing the problem to a simpler comparison problem, the stability of moving invariant sets is discussed employing comparison method. The derivative of the Lyapunov function involved is estimated from opposite directions rela- tive to suitable sets in phase space that depend on the moving parameter.
1991 Mathematics Subject Classication. 34D28, 34A99.
Key words and phrases. Stability oriteria, dynamic systems on time scales, uncertian systems.
In this paper, utilizing the framework of the theory of dynamic systems on time scale, we will investigate uncertain dynamic systems on time scale relative to stability of moving invariant sets. As an application of our results, we will consider the control of uncertain dynamic system on time scales and obtain the desired stability behavior of moving invariant sets.
2. Preliminaries
LetTbe a time scale (any subset ofRwith order and topological struc- ture dened in a canonical way) with t0 0 as a minimal element and no maximal element. Since a time scaleT may or may not be connected, we need the following concept of jump operators.
Denition 2.1. The mappings;:T!Tdened by
(t) =inf[s2T:s>t] and (t) =sup[s2T:s<t] are called the jump operators.
Denition 2.2. A nonmaximal elementt 2T is called right-dense (rd) if
(t) = t, right-scattered (rs) if (t) > t, left-dense (ld) if (t) = t, left- scattered (ls) if (t) < t. In the case T = R, we have (t) = t, and if
T=hZ, then(t) =t+h.
Denition 2.3. The mapping : T ! R+ dened by (t) = (t) t is called graininess. If T=R, then (t) = 0, and whenT=Z, we have
(t) = 1.
Denition 2.4. The mappingu: T!X, where X is a Banach space is called rd-continuous if it is continuous at each right-dense t 2 T, and at each left-dense t, the left-sided limitu(t ) exists.
LetCrd[T;X] denote the set of rd-continuous mappings fromTtoX. It is clear that a continuous mapping is rd-continuous. However, ifTcontains left-dense and right scattered points, then rd-continuity does not imply continuity. But on a discrete time scale the two notions coincide.
Denition 2.5. A mappingu:T!X is said to be dierentiable att2T, if there exists an2X such that for any>0 there exists a neighborhood
N oftsatisfying
ju((t)) u(s) ((t) s)jj(t) sj for all s2N:
Letu(t) denote the derivative ofu. Note that ifT=R, then=u=
du(t)
dt and ifT=Z, then=u=u(t+ 1) u(t). It is easy to see that if
uis dierentiable att, then it is continuous att, ifuis continuous attand
tis right-scattered, then uis dierentiable and
u
(t) =u((t)) u(t)
(t) :
Denition 2.6. For each t 2T, let N be a neighborhood of t. Then we dene the generalized derivative (or Dini derivative), D+u(t), to mean that, given>0, there exists a right neighborhoodNN oft such that
u((t)) u(s)
(t;s) <D+u(t)+ for s2N; s>t; where (t;s) =(t) s:
In case t is rs and u is continuous at t, we have, as in the case of the derivative,
D +
u
(t) = u((t)) u(t)
(t) :
Denition 2.7. Lethbe a mapping fromTtoX. The mappingg:T!X is called the antiderivative ofhonTif it is dierentiable onTand satises
g
(t) =h(t) fort2T.
Following Denition 2.6, deneD+V(t;x(t)) forV 2Crd[TRn;R+] to mean that, given >0, there exists a right neighborhood N N of t such that
1
(t;s)[V((t);x((t))) V(s;x((t)) (t;s)f(t;x(t)))]<
<D +
V
(t;x(t)) +
for eachs2N,s>t. As before, if tis rs andV(t;x(t)) is continuous att, this reduces to
D +
V
(t;x(t)) = V((t);x((t))) V(t;x(t))
(t) :
We need the following comparison results in terms of Lyapunov-like func- tions. See [6].
Theorem 2.1. LetV 2Crd[TRn;R+],V(t;x) be locally Lipschitzian in
x for eacht2Twhich is rd, and let
D +
V
(t;x)g(t;V(t;x));
where g 2Crd[TR+;R], g(t;u)(t) +uis nondecreasing in u for each
t2T, andr(t) =r(t;t0;u0) is the maximal solution ofu=g(t;u),u(t0) =
u
0
0, existing on T. Then, V(t0;x0) u0 implies that V(t;x(t))
r(t;t0;u0), t2T, tt0.
A result giving the lower estimate is also true.
Theorem 2.2. LetV 2Crd[TRn;R+],V(t;x) be locally Lipschitzian in
x for eacht2Twhich is rd, and let
D +
V
(t;x)g(t;V(t;x));
where g2Crd[TR+;R], g(t;u)(t) +uis nondecreasing in u for each
t2T, and(t) =(t;t0;u0) is the minimal solution ofu=g(t;u),u(t0) =
u
0
0, existing onT. Then, V(t0;x0)u0 implies that V(t;x(t))(t),
t2T, tt0.
We need both comparison results in our discussion below.
3. Main Results Consider the dynamic system on time scales
x
=f(t;x;); x(t0) =x0; t02T; (3.1) wheref 2Crd[TRnRd;Rn], 2Rd is an uncertain parameter andT is a time scale. Consider also the comparison dynamic equation
u
=g(t;u;); u(t0) =u00; (3.2) where g2Crd[TR2+;R] and=()0 is a parameter depending on
.
Let 0 r0 r be depending on . Then we will say that the set B = [x 2Rn :0 jxj] is conditionally invariant with respect to
A= [x2Rn :r0 jxjr] and is uniformly asymptotically stable (UAS) relative to (2.1) if
(I) r0jx0jrimplies0jx(t)j,t2T,tt0; (ii) given>0 andt02T,
(a) there exists a =()>0 such that r0 jx0jr+ implies0 <jx(t)j<+,tt0,t2T;
(b) there exist a0>0 and aT =T()>0 such thatr0 0
jx
0
jr+0implies0 <jx(t)j<+,tt0+T,t2T; wherex(t) =x(t;t0;x0) is any solution of (3.1).
Relative to the comparison equation (3.2), we will say that = [u 2
R
+:R0uR] is invariant and is UAS relative to (3.2) if (I) R0u0RimpliesR0u(t)R,tt0,t2T; (ii) given>0 andt02T,
(a) there exists a =()>0 such that R0 u0 R+ impliesR0 <u(t)<R+, tt0,t2T;
(b) there exists a0>0 and aT =T()>0 such thatR0 0
u
0
R+0 impliesR0 <u(t)<R+,tt0+T,t2T, whereu(t) =u(t;t0;u0) is any solution of (3.2).
Let us dene the usual classKof functions byK= [a2C[R+;R+] :a(u) is strictly increasing inuwitha(0) = 0 anda(u)!1asu!1].
We can now prove the following result on UAS of the conditionally in- variant setB with respect toA, relative to the system (3.1). Let us dene the sets r, r0 by r = [x 2Rn : x 2 A and jxjr], r0 = [x 2 Rn :
x2Aandjxjr0].
Theorem 3.1. Assume that
(A0) for each 2Rd, there existr=r(), r0=r0(), r0r satisfying
r!0 as jj!0 and r0!1asjj!1;
(A1) there exists V 2Crd[TRn;R+] such that V(t;x) is locally Lips- chitzian inxfor each right denset2Tand forai,bi2K,I = 1;2,
b
1(jxj)V(t;x)a1(jxj) if x2r;
b
2(jxj)V(t;x)a2(jxj) if x2r0;
(A2) ifx2r,D+V(t;x)g(t;V(t;x);r), and ifx2r0,D+V(t;x)
g(t;V(t;x);r0), whereg2Crd[TR2+;R],g(t;u;)(t)+uis non- decreasing inufor each(t;u);
(A3) for each r0 r, there exists R0 R such that R=a1(r) = b1() andR0 =b2(r0) =a2(0), where 0 r0 r and R! 0 as
r!0,R0!1asr0!1;
(A4) the set is invariant and is UAS with respect to (3:2).
Then the set B is conditionally invariant with respect to A and is UAS relative to the system(3:1).
Proof. We will rst prove thatB is conditionally invariant with respect to
Aand (3.1). If not, there would exist a solution x(t) =x(t;t0;x0) of (3.1) withr0jx0jrandt0<t2 such that either
(i) jx(t2)j>andr0jx(t)j,t2[t0;t2]\T, (ii) orjx(t2)j<0andjx(t)jr,t2[t0;t2]\T.
Because of (A2), using comparison Theorems 2.2, 2.3, we get either
V(t;x(t))r(t;t0;V(t0;x0)); or
V(t;x(t))(t;t0;V(t0;x0));
for t 2 [t0;t2]\T, where r(t;t0;u0) and (t;t0;u0) are the maximal and minimal solutions of (3.2). Hence using (A3) and (A4), in the case (i) we have
b
1()<b1(jx(t2)j)V(t2;x(t2))r(t2;t0;V(t0;x0))
r(t2;t0;a1(jx0j))r(t2;t0;a1(r))a1(r) =b1(); or, in the case (ii), we get
a
2(0)>a2(jx(t2)j)V(t2;x(t2))(t2;t0;V(t0;x0))
(t2;t0;b2(jx0j))(t2;t0;b2(r0))b2(r0) =a2(0):
Thus, we have a contradiction in both cases and hence B is conditionally invariant with respect toAand (3.1).
Let 0 < < 0 and t0 2 T be given. Since (A4) holds and a1(r) =
b
1() =R,R0=b2(r0) =a2(0), givena2(0 ),b1(+), there exist1,
1,>0 such that
R
0+1=a1(r+)<b1(+) =R+1; and
R
0
=a2(0 )<b2(r0 ) =R0 1; satisfying
R
0
1
<u
0
<R+1 implies R0 1<u(t)<R+1; tt0; t2T;
where u(t) = u(t;t0;u0) is any solution of (3.2). We claim that with this
>0, the setB is US relative toA, that is,
r
0
<jx
0
j<r+ implies 0 <jx(t)j<+; tt0; t2T:
If this is not true, there would exist a solutionx(t) of (3.1) with r0 <
jx
0
j<r+and at2>t0 such that either
(a) jx(t2)j+andjx(t)jr0, [t0;t2]\T, (b) orjx(t2)j0 andjx(t)jr, [t0;t2]\T. Consider (a). As before, we obtain
V(t;x(t))r(t;t0;V(t0;x0)); [t0;t2]\T;
and therefore, we arrive at the contradiction
b
1(+)b1(jx(t2)j)V(t2;x(t2))r(t2;t0;V(t0;x0))
r(t2;t0;a1(jx0j))r(t2;t0;a1(r+))<b1(+): Similarly, in case (b), we rst get
V(t;x(t))(t;t0;V(t0;x0)); [t0;t2]\T;
and then
a
2(0 )>a2(jx(t2)j)V(t2;x(t2))(t2;t0;V(t0;x0))
(t2;t0;b2(jx0j))(t2;t0;b2(r0 ))b2(r0) =a2(0 ); which is again a contradiction. Hence the setB is US relative toA.
To prove UAS of the setB relative toA, let us x =0 and designate
0=(0) so that we have
r
0
0
<jx
0
j<r+0 implies 0<jx(t)j<+0; tt0; t\T:
Let 0<<0 andt02T. Since is UAS, givena2(0 ),b1(+) there exists aT =T()>0, witht0+T 2Tsuch that
b
2(r0 0)<u0<a1(r+0) implies a2(0 )<u(t)<b1(+); tt0+T:
We claim that wheneverr0 0<jx0j<r+0, we have
0
<jx(t)j<+;tt0+T; t2T:
If this is not true, there would exist a solutionx(t) of (3.1) such that (a) jx(t2)j+,t2t0+T,t22T,
(b) jx(t2)j0 ,t2t0+T,t22T, wherer0 0<jx0j<r+0. As before, using (A2) and (A3), we get successively
b
1(+)V(t2;x(t2))r(t2;t0;a1(r+0))<b1(+); and
a
2(0 )V(t2;x(t2))(t2;t0;b2(r0 0))>a2(0 ); which are contradictions. Hence we haveBis UAS with respect to
Arelative to the system (3.1) and the proof is complete.
Remarks. IfT=R, then (3.1), (3.2) reduce to the continuous dierential systems. Since, in this case,(t) = 0, the results of Theorem 3.1 reduce to those in [4]. Note that the conditions (A1) and (A2), which are sucient to prove UAS, are then weaker. If, on the other hand, T = Z, so that
(t) = 1, (3.1) and (3.2) reduce to dierence equations, and consequently, one needs stronger conditions (A1), (A2). Theorem 3.1 oers results in this special case.
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(Received 9.06.1997) Author's address:
SUNY at Geneseo, Department of Mathematics Geneseo, NY 14454, USA
