Florida Institute of Technology. Melbourne, FL 32901, U.S.A.. (Received ... Certain Lyapunov-like functions were used to study the stability criteria of. SDE in [2].
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MATHEMATICAL
AND COMPUTER MODELLING
Mathematical and Computer Modelling 41 (2005) 1371-1378 www.elsevier.com/locate/mcm
Stability Criteria for Set Differential Equations T . G N A N A B H A S K A R AND J . V A S U N D H A R A D E V I * Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, U.S.A.
(Received and accepted January 2004) A b s t r a c t - - N o t i o n s of stability for the solutions of set differential equations, using Lyapunov-like functions are considered. Criteria for the equistability, equiasymptotie stability, uniform and uniform asymptotic stability are presented. @ 2005 Elsevier Ltd. All rights reserved.
K e y w o r d s - - S e t differential equations, Equistability, Uniform stability.
1. I N T R O D U C T I O N The study of set differential equations (SDEs) in a suitable metric space was initiated in [1], and the basic theory, comparison results and the stability considerations for hybrid dynamical systems were discussed therein. Much progress has since been made in studying various fundamental aspects of SDEs. For some recent work on SDEs, see [2-4]. In particular, using Lyapunov-like functions t h a t are lower semicontinuous, the Lyapunov stability for the Euler solutions of SDEs was studied in [5]. Certain Lyapunov-like functions were used to study the stability criteria of SDE in [2]. Here in this paper, we study the Lyapunov stability for the solutions of SDEs, using Lyapunovlike functions t h a t are continuous. We employ an important comparison result in terms of Lyapunov functions, and investigate the qualitative behaviour of the solutions of the IVP for the SDE
DHU = F(t, U),
U(to) = Uo • Kc(]R~),
(1.1)
where F E C[R+ x Kc(R~), K~(R~)]. These Lyapunov-like functions serve as a vehicle to transform the set differential equations into scalar comparison differential equations, and therefore, it is enough to consider the qualitative properties of the simpler comparison equation under suitable conditions for Lyapunov-like functions. The authors express their gratitude to Prof. V. Lakshmikantham, for the stimulating discussions and insightful remarks on the contents of the paper. *On leave from GVP College for PG courses, Visakhapatnam, India. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.01.012
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1372
T. CNANA BHASKAI~AND J. VASUNDHARADEVI
2. P R E L I M I N A R I E S Let K~(N ~) denote the collection of all nonempty, compact and convex subsets of R ~. Define the Hausdorff metric r ]
D[A,B] = max/sup d(x,A),sup d(y,B//, "
(2.1)
yEA
where d[x, A] = inf[d(x, y) : y C A], A, B are bounded sets in ]Rn. We note that Kc(R n) with this metric is a complete metric space. It is known that if the space K~(R n) is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then Kc(R ~) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space. The Hausdorff metric (2.1) satisfies the following properties:
D[A + C, B + C] = D[A, B]
D[A, B] = D[B, A],
and
D[AA, ABI = AD[A, BI, Did, B] < D[d, C] + D[C, B],
(2.2) (2.3) (2.4)
for all A, B, C E Kc(R n) and A E R+. Let A, B E K~(R~). The set C E K , ( R '~) satisfying A = B + C is known as the Hukuhara difference of the sets A and B and is denoted by the symbol A - B. We say that the mapping F : I ~ K~(R n) has a Hukuhara derivative DHF(to) at a point to E I, if lim F(to
+ h) -
h-,O+
F(to)
and
h
lim F(to) h--+o+
-
F(to
-
h)
h
exist in the topology of K~(R ~) and are equal to DHF(to). Here I is any interval in R. It follows from (2.1) that for A E Kc(R~),
D[A, O]
=
IIAII = sup Ilall, aCA
where 0 is the zero element of R ~ which is regarded as a one-point set. If F : I -~ K c ( ~ '~) is a continuous function, then it is integrable and the integral
a(t) = G(to) +
F(s) ds,
t
I,
is Hukuhara differentiable and DHG(t) = F(t). 3. R E S U L T S
ON
SET
DIFFERENTIAL
EQUATIONS
In order to discuss the equistability and uniform stability criteria for SDE (1.1), we state some known results for SDEs, in this section, from [1]. Let K: be the class of functions ~r E C[[0, p),]R+] such that (r(0) = 0 and or(w) is increasing in w.
Assume that (i) Y E C[R+ x Kc(Rn),R+] and IV(t,A) - V(t,B)I < LD[A,B], where n is the local Lipschitz constant, for A, B E Kc(]~n), t E ]~+; (ii) g E C[R~_, R] and for t E R+, d e Kc(R~),
THEOREM 3.1.
D+V(t,A)-limsupl[y(t+h,d+hF(t,A))-V(t,A)] h--~O+
re
0 and to E R+, there exists a positive function 5 = 5(to, e) that is continuous in to for each s such that IlW01I < 5 implies Ilg(t)ll < s, t >_ to, where U(t) = U(t, to, Wo) is the solution of (4.2). The other notions of stability can easily be formulated following the standard ones given in [6]. Before presenting the stability results, in this new setup, we present the following example which is discussed in [2]. Consider the ODE u' = - u , u(0) = uo E JR, (4.3) and the corresponding SDE
DHU
=
-[7,
U(0)
=
[7o E
Kc(R).
(4.4)
Since the values of the solution (4.4) are interval functions, equation (4.4) can be written as
[ul, u~] = ( - 1 ) u = [-~2,-~11,
(4.5)
where U(t) : [ul(t), u2(t)] and U(0) = [Ul0, u20]. Relation (4.5) is equivalent to the system of equations It i = --iS2,
U 1(0) :
U~ :
lt2(O ) = U20 ,
--Ul,
Ul0 ,
Stability Criteria
1375
whose solution for t _> 0 is 1
1
ul(t) = g[~10 + ~2o]~-' + ~[~lo 1
-
u~0]~ ~,
1
u~(t) = ~ [ ~ o + ~1o]~ -~ + g[~2o - ~o]~ ~. Given Uo E K~(R), if there exists Vo, Wo E K~(R) such t h a t Uo = Vo + Wo, then the H u k u h a r a difference Uo - Vo = Wo exists. Choosing 1
Uo ~-- [trio, u20], so t h a t
Vo -- ~ [(UlO - u20), (tt20 - ~tlO)], 1
Wo = ~[(ltlo -~- u20), (u20 -t- ulO)], if UlO ~ -u2o, we get for t _> O, 1
1
u(t, go) = ~[-(~2o - ~1o), (~o - ~lo)]e ~ + ~[(~1o + ~o), (~o + ~o)]e -~, 1
U(t, Vo) = ~ [(ulo - u2o), (u2o - ulo)]e t,
and
1
u(t, w0) = ~[(~10 + ~20), (~0 + ~20)]~ -~. If on the other hand, ul0 = -u20, which implies we choose the special Uo = [-d,d], where d = u2o, then U0 = V0 and TWo = [0, 0]. This choice eliminates the t e r m with e - t and we have only the undesirable part of the solution. The other situation is to choose Uo = [c, e] for some c, which eliminates the t e r m with e t and retains only the desirable part of the solution compared with the ODE. Even when U0 is chosen as U0 = I - d , d], we can find Vo = [c - d, c + ~ for some c so t h a t we have V0 = Uo + W0, where W0 = [c, c]. We note t h a t for any general initial value U0, the solution of SDE (4.2) contains both desired and undesired parts compared to the solution of the O D E (4.1). In order to isolate the desired part of the solution U(t, Uo) of (4.2) which matches the solution of the ODE (4.1), we need to use the initial values satisfying the desired H u k u h a r a difference of the given two initial values. If, on the other hand, we have the SDE as
DHU = A(t)U,
U(0) = Uo,
(4.6)
which is generated by u' = ~(t)u,
u(0) = u0,
(4.7)
where A(t) > 0 is a real-valued function from R+ to R+ such t h a t A E LI(IR+), then we see that, with similar computation,
u(t, Uo) = Uo exp
[/0 ]
X(s) as ,
t_>0,
for any Uo E Kc(Rn). Hence, we get the stability of the trivial solution of (4.6). In this case, we note t h a t the solutions of b o t h equations (4.6) and (4.7) match, providing the same stability results. There is no necessity, therefore, to choose the initial values as before, since the undesirable part of the solution does not exist among solutions of (4.6). Consequently, it does not m a t t e r whether we use the H u k u h a r a difference or not, we get the same conclusion. In order to be consistent and take care of all the situations, we formulate the results in terms of Hukuhara differences of initial values. We begin the following result on equistability.
1376
T. GNANA
BHASKAR
AND
J. VASUNDHARA
DEVI
THEOREM 4.1. Assume that the following hold: (i) V E C[R+ x S(p),lI~+], }V(t, gl) - V(t, U2)] 0 and for (t,U) E N+ x S(p), where S(p) : [U E Kc(lRn) • Ilgll < p],
D+V(t, U) - lim sup l [v(t + h, U + hF(t, U)) - V(t, U)] _< 0; A-*O+
/L
(4.8)
(ii) b([]Uii ) < V(t, U) to such that IIg(h)ll
= E
and
(4.9)
If not, there would exist a solution U(t) =
Ilg(t)ll _< s < p,
to _< t _< tl,
(4.10)
whenever IIW011 < ~. By Corollary 3.1, we then have
v(t, u(t))
