Lakshmikantham, V

Leela, S

Sivasundaram, S

Elsevier - Publisher Connector 

JOURNAL OF MATHEMATICAL ANALYSIS AiiV APPLICATIONS 154, 391402 (1991) 
Lyapunov Functions on Product Spaces and Stability 
Theory of Delay Differential Equations 
V. LAKSHMIKANTHAM 
Florida institute qf Technology, 
Melhourne, Floridu 32901 
S. LEE~A 
SUNY College at Geneseo, 
Geneseo. Neu, York 14454 
AND 
S. SIVASUNDARAM 
Embry-Riddle Aeronautical University, 
Daytona Beach. Florida 32014 
Submitted hy /he Editors 
Received August 8. 1989 
1. INTRODUCTION 
If we examine the Lyapunov functionals constructed for all the-examples 
that have been discussed so far in the literature, we find that the 
investigators, inadvertently, employ a combination of a Lyapunov function 
and a functional in such a way that the corresponding derivative can be 
estimated suitably without demanding minimal classes of functions or the 
knowledge of solutions as in the case of Lyapunov functions or Lyapunov 
functionals, respectively [3-61. The method of Lyapunov functionals, 
however, demands the knowledge of solutions of delay differential equa- 
tions considered and consequently the discussion of examples is not really 
in the spirit of the method. This observation leads us to develop the 
method of Lyapunov functions on product spaces for studying stability 
properties of equations with delay, where, except conceptually, the 
knowledge of solutions is not demanded. We shall also develop stability 
theory in terms of two measures which unifies several known stability con- 
cepts. Our presentation demonstrates the advantage of utilizing Lyapunov 
functions on product spaces. 
391 
0022-247X/91 $3.00 
CopyrIght ( 1991 by Academic Press. Inc 
All rights of reproducrmn ,n any form reserved 
392 LAKSHMIKANTHAM, LEELA, AND SIVASUNDARAM 
2. PRELIMINARIES 
Let %? = C[ [ -r, 01, R”] and for any 4 E %?, let us use the norm 
1410=max-,.,., I&)l. If x~C[[t,-T, co),R”], t,ER+, we define 
x, E %? by x,(s) = x(t + s), -z 6 s < 0. We consider the initial value problem 
-x’(t) =f(t, *XI), -x,, = do E g, (2.1) 
where f~ C[R+ x%‘, R”]. It is known that if f maps bounded sets into 
bounded sets, then for each (to, do), there exists a solution x(t) = 
x(t,, q&)(t) defined on an interval [to, t, + c1), c( > 0. We wish to 
employ Lyapunov functions on the product space R”pif? and develop 
corresponding theory for studying stability criteria for the system (2.1). If 
VE C[R, ~R”fif?, R,], then we define 
D+ V(t, d(O), 4) 
=hliy+ SUPi CV(t+h,4(0)+ hf(49),x,+,(t,4))- V(t,qw), 4)1, 
(2.2) 
where it is understood that x( t, d) is any solution of (2.1) with the initial 
function 4 at time t. To unify several different concepts of stability studied 
in the literature such as partial stability, conditional stability, eventual 
stability, it is convenient to introduce stability concepts in terms of two 
measures [ 1, 8-111. We shall therefore discuss stability properties of (2.1) 
with respect to two measures. We need the following definitions and the 
classes of functions: 
X = {a E C[ [0, p), R + ] a(O) = 0 and a(u) is strictly increasing 
in u}, 
ofi?= {QEC[R+, R,]: a(t) is decreasing with lim a(t) =O}, 
,-?c 
~?X={~EC[R+X[O,~),R+]: a(t,U)EXforeach tER+}, 
r= (lz~C[[t~-~, co)xR”, R,]: infh(t,x)=O). 
* 
DEFINITION 2.1. Let h, ho E I- and define for 4 E G$, 
hot4 4) = -yy<, hO(f + & 4(s)), . . 
k(t,#)= max h(t+s,&s)). 
-- r<sC” 
(2.3) 
LYAPUNoVFUNCTloNS 393 
Then h, is said to be finer than h’ if there exists a p > 0 and a $ E X” such 
that 
hot4 d)<P implies @c d) d Il/(ho(f, 9)). 
DEFINITION 2.2. Let VE C[R + ~R”fi%‘, R + 1. Then V is said to be 
(i) h-positive definite if there exists a p > 0 and a b E X such that 
W(f, -x)1 d 4~ -x, $1, q5 E V whenever h(l, .x) < p; 
11 
that (“) 
weakly A,-decrescent if there exists a p > 0 and a $ E CX such 
vc d(O)> 4) d tit4 ho(f, 4)) if hot4 0) <P; 
(iii) h,-decresent if there exists a p > 0 and a $ E X such that 
WC do), 4) G bwo(c 4)) if h,(f, 4) <P; 
(iv) asymptotically A,-decrescent if there exists a p > 0 and a 
$ E X9 = { o(t, u): o( ., U) E .X and a(t, .) E Y} such that 
vc 4(O), 4) G Il/(ho(G ($1, f) if hot4 d) <P; 
(v) (ho, A*)-decresent if there exists a p >O and two functions 
a,, a, E X such that 
vt, x> d) 6 Qo(hOU, xl) + a,(h*(t, 4)) 
whenever h”(t, X) < p and h*(t, 4) < p, where /I* E C[R+ x%7, R + 1. 
DEFINITION 2.3. Let I E C[R+ , R,]. Then i is said to be integrally 
positive if j,A(s) ds= co, whenever I= tJ,?=, [IX;, /?,I, c1,</3,<~,+, and 
p, - ai > 6 > 0. 
DEFINITION 2.4. The system (2.1) is said to be (ho, h)-stable if given 
E>O and t,eR+, there exists a 6 = s(t,, E) > 0 such that 
hO(fO> do) < 6 implies h( t, x(t)) < E, t 3 to. 
Based on Definition 2.4 and the usual stability notions, it is easy to 
formulate other kinds of stability concepts in terms of two measures (ho, h). 
We shall give below a few choices of (ho, h) to demonstrate the generality 
of Definition 2.4. Definition 2.4 reduces to 
(1) the well-known stability of the trivial solution of (2.1) if 
h(t, -x) = l-xl and h,(f, 4) = Mlo; 
394 LAKSHMIKANTHAM, LEELA, AND SIVASUNDARAM 
(2) stability of the prescribed solution y(t) = y(to, tjo)(t) of (2.1) if 
h(f, xl = Ix - y(t)1 and ho(t, d) = Id - solo; 
(3) partial stability of the trivial solution of (2.1) if h(t, X) = Ixlk, 
1 <k<n and h,(t, #)= IdlO; 
(4) stability of conditionally invariant set B with respect o A where 
A c B c R” if h( t, x) = d(x, B) and h,( t, 4) = max ~ ~ < s ~ O d(&s), A ), d being , . 
the distance function; 
(5) eventual stability of (2.1) if h(t, x) = 1x1 and h,(t, 4) = IdlO + a(t), 
CT E L. 
Remark. Note that in case (5) the set of initial times to is to be 
prescribed as in [6, Defn. 2.41. 
3. MAIN RESULTS 
Let us begin by proving a result on non-uniform stability in terms of two 
measures under weaker assumptions, which includes other interesting spe- 
cial cases and also shows that in situations where the Lyapunov function 
employed does not satisfy all the desired conditions, it is more fruitful to 
perturb it by a family of Lyapunov functions than to discard it [7]. 
Let for any IIEI’, S(h, p) = {(t, x): h(r, x) < p} and for any h* E 
CCR, xg!, R, I, S(h*, P) = {Cc 4): h*(f, 4) <P). 
THEOREM 3.1. Assume that 
(A,) h, ho E r and ho isfiner than z, where ho, h” are defined by (2.3); 
(A,) F’, E C[S(h, p) xS(& p), R,], V,(t, x, 4) is locally Lipschitzian 
in x and weakly ho-decrescent; 
(AZ) for every 0 < q < p, there exists a VZq E C[n, R + 1, Vz,l(t, x, 4) is 
locally Lipschitzian in .Y and for (t, 4(O), 4) E 9, 
b(h(t, d(O))) d V2,jf’ 4(O)), 4) < a(h,(t, 4)), 
where a, b E I‘ and Q = S(h, p) n S”(h’, r]) xS(h”, p) n S’(h,, q), S’(h’, q) 
and S’(h,, q) being the complements qf S(h”, q) and S(h,, q), respectively; 
(A,) for (t7 4(O), #)~a. 
LYAPUNOV FUNCTIONS 395 
where g, E C[ R + XR + , R + 1, gz( t, 0) s 0 and the trivial solution of 
w’ = g2(tr w), w( to) = w() 3 0 (3.1) 
is uniformly stable; 
(Ad) D+V,(t, cWM) d g,(t, v,(t, d(O), 4)) on S(h, P) xS(k PI, where 
g,fCCR+xR+, R, 1, g,(t, 0) E 0 and the trivial solution of 
u’ = g, (1, u), u(t()) = ug 3 0 (3.2) 
is stable. 
Then, the delay differential system (2.1) is (h,, h)-stable. 
Proof: Since V, is weakly h,-decrescent, here exists a 0 < p1 f p and a 
tiO E CX such that 
V,(t, d(O), 4) d Il/o(t, ho(t, 4)) if hot& 4)<p,. (3.3) 
Also, ho is finer than h’ implies that there exists a 0 < p0 6 p, and a II/ E X 
such that 
h”(t, 4) d $(ho(t, 4)) if hot& 4) < po, (3.4) 
where p0 is such that t&q,) d p. Hence, by (2.3) we have 
46 4(O)) f &t, 4) Q Il/(ho(t, 4)) if h,(t, 4) < po. (3.5) 
Let 0 < E < p and to E R + be given. Since the trivial solution of (3.1) is 
uniformly stable, given b(c) > 0 and to E R + , there exists a 6, = &,(E) such 
that 
w(t, to, ~0) <b(E), t2 to if ua,<6,, (3.6) 
where w(t, to, wo) is any solution of (3.1). Since a and $ belong to class X, 
we can find a S, = S,(e) > 0 such that 
46,) <do/2 and $(6,) c.5. (3.7) 
The stability of the trivial solution of (3.2) yields that given Jo/2 > 0 and 
t,sR., there exists a 6* = d*(t,, E) > 0 such that 
U,<d* implies 46 to, %) -=c 4J2, fk to, (3.8) 
where u(t, to, uo) is any solution of (3.2). Choose Y,(to, @o(O), 40) = uo. 
Since $o~ CY and (3.3) holds, there exists a 6,=6,(t,, s)>O such that 
&E (0, min(b,, P,)) and 
hotto, 40) < 6, implies V,(to, 4o(OL 40) 6 $o(fo, hotto, 40)) -~a*. 
(3.9) 
409.154’2-7 
396 LAKSHMIKANTHAM, LEELA. AND SIVASUNDARAM 
We set 6 = min(b,, 6,) and suppose that h,(t,, do) < 6. We note that we 
obtain 
NfO> do(O)) G wo> do) G ICl(ho(to, 40)) < Icl(S,) < 6 
hO(to, do(O)) 6 ho(fo, do) < 6, I 
(3.10) 
in view of (3.4) and (3.7). 
We now claim that h,(t,, do) < 6 implies that h(t, x(t)) <E, t > f,, where 
x(t) = x(t,, do)(t) is any solution of (2.1). If this is not true, because of 
(3.10), there exists a solution x(t) of (2.1) with h,(t,, do) < 6 and t, > t, > to 
such that 
hO(t,, x(t,))=6,, h(b> X(fZ)) = &, 
x(t) E S(h, E) f-l S”(hO, 6,) for tE [r,, t,]. 1 
(3.11) 
This also implies that h”(t,, x,~) = E, h,(t, , x,,) = 6, and x, E S(&, E) n 
S”(h,, 6,). Setting q = 6,) we see by (A*) that there exists a V,, and for 
TV [tl, f2], we have 
where m(t) = I’,( t, x(t), xt) + V2( t, x(t), x,). Hence, by comparison theorem 
[6], we obtain 
m(j) G r,(f, t, 2 m(t,)), tE Cf,, f21, (3.12) 
where r,(t, t,, m( ti)) is the maximal solution of (3.1). We can obtain 
similarly the estimate 
v,(c 4th x,1 G r,(t, 207 V,(to, do(O) do)), tE [to, t,l, (3.13) 
where r,(t, to, uo) is the maximal solution of (3.2). Hence, by (3.8) (3.9) 
and (3.13), we have 
Also, by (AZ) and (3.7) we obtain 
Hence, it follows that m(t,)<~?, and therefore (3.6) and (3.12) yield that 
(3.14) 
But m(t,) > Vz,(tz, x(t*), x,*) > b(h(t,, x(tZ))) = b(s), which is a contra- 
diction to (3.14). Hence, the proof is complete. 
LYAPUNOV FUNCTIONS 397 
Remark. Let V, E 0 and g, E 0 in Theorem 3.1. Then, we obtain (h,, h)- 
uniform stability showing the advantage of using a family of Lyapunov 
functions in proving uniform stability. If, on the other hand, Vzq E 0 and 
g, E 0, assuming P’, to be h-positive definite guarantees (h,, h)-stability by 
Theorem 3.1. As it is, Theorem 3.1 is an extension of [7, Theorem 2; and 
8, Theorem 3.11. 
We shall next consider a result on asymptotic stability in the same spirit 
as that of Theorem 3.1 which is an extension of [8, Theorem 3.21. 
THEOREM 3.2. Assume that (A,)-(A,) of Theorem 3.1 hold. Suppose 
further that 
(A:) there exist V,, V4eC[S(h, p)xS(z, p), R,] such that V, = 
V, + V,, V, is h-positive definite and on S(h, p) xS(x, p) 
D+ f’l(t, d(O), 9) d -i(t) ?/(v,(t, 4(O)> 4)) 
holds, where 1” is integrally positive and 7 E X; 
(A,) for every YE C[ [t,-T, oc), R”] such that h(t, y(t)) < p, 
t E [to - 5, cc), the funcrion ji,, [D’ V4(s, y(s), y,)] + ds is untformly con- 
tinuous on [to, co), bvhere [ .] k means that either the positive or the negative 
part is considered for all s E [t,, z ). 
Then, the delay differential system (2.1) is (h,, h)-asymptotically stable 
and lim,,, V4(t, x(t), x,) exist and i.s,finite for any solution x(t) of (2.1). 
Proof: Since (A$) implies that D+ V,(t, 4(O), 4) d 0 on S(h, p) xS(h”, p), 
by Theorem 3.1, the assumptions (A,,-(A,) yield (h,, h)-stability for the 
system (2.1). Choosing E = p and designating 6, = 6,(t,, p), it is clear that 
Mto, do) < 60 implies h(t, x(t)) < P, t3 t,, (3.15) 
where x(t)=x(tO, q&)(t) is any solution of (2.1). Set ml(t)= V,(t, x(t), x,), 
mX(t)= V,(t,x(t),x,) and m,(t)= V,(t,x(t),x,) so that m,(t)=m3(t)+ 
m,(t). Assumptions (A,) and (A:) yield that ml(t) is non-increasing and 
bounded from below and thus, lim,, r m,(t) = cr < x. 
We show that lim,, z m,(t) =O. Clearly, lim,,, inf m,(t) =O. If not, 
we arrive at a contradiction because of (A$). Now, suppose that 
lim,, co sup m3( t) > 0. Then there exists a p > 0 such that lim,, ~, sup m3( t) 
> 3~. Since lim, _ r m,(t) = 0 and m,(t) is non-increasing, there exists a 
T> 0 such that 
adm,(t)da+,u, tat,+ T. (3.16) 
398 LAKSHMIKANTHAM, LEELA, AND SIVASUNDARAM 
For the sake of clarity, suppose that assumption (A,) holds with [ .] + . 
Since m,(t) is continuous, we can choose a sequence 
t,+T<t;“<t\“< ... <t;“<tj2’< .., 
such that for i= 1, 2, . . . . 
m,(tj”) = 3p, m(3j(t!2’) = P 
P <m,(t) < 314 t E [t$“. p-j. 
From (3.16) and (3.17), it is easy to see that 
m,(t!“) - m,(t”‘) Gc7-2~~ 
ml(tj2’)-m,(ti?‘)>0-p. I 
Since m,=m,+m,, it follows from (3.18) that 
(21 
O<~~m,(tj”)-rn,(tj”)~S::, [D+m,(s)]+ ds, 
‘i 
which shows by (A,) that there exists a d > 0 such that 
t(2) -I t!‘) > 6, i= 1, 2, . . . . 
By (3.17), (3.19), and (A,), we obtain 
(3.17) 
(3.18) 
(3.19) 
Qm,(t,,+T)--y(p)ji(s)ds= -co, 
where I= u,“= 1 [ti (l), t!“]. This contradiction proves that lim,, J; m,(t) = 0 
and since V3 is h-positive definite, we obtain lim,, 5 h(t, x(t)) = 0. Thus, 
we conclude that the system (2.1) is (h,, h)-asymptotically stable. To prove 
the last assertion of the theorem, note that lim,, m m,(t)= cr and 
lim,,, m3(t) = 0 and consequently, lim, _ 2 m,(t) = CJ. The proof is there- 
fore complete. 
The following corollaries of Theorem 3.2 are interesting. 
COROLLARY 3.1. Let (A,), (A,), (AZ), and (A,) of Theorem 3.2 hold. If 
VI is h-positive definite, then the conclusion of Theorem 3.2 remains valid. 
COROLLARY 3.2. Let (A,), (A,), and (At) of Theorem 3.2 hold. Assume 
further that V, is h-positive definite, V4(t, 4(O), 4) = jyr h*(t + s, b(O), 4) ds 
LYAPUNOVFUNCTIONS 399 
and i; is finer than h*, where h* E C[S(h, p) xS(6, p), R, 1, Then, the conclu- 
sion of Theorem 3.2 remains valid. 
The next corollary uses the facts lim, _ cc V,(t, x(t), x,) = r~ < CC and I/, 
is (ho, I/,)-decresent. 
COROLLARY 3.3. Let the assumptions of Corollary 3.2 hold except that 
V, is not h-positive definite. Assume that V, is (ho, V,)-decrescent and 
lim r-m V,( t, x(t), x,) = 0 implies lim, _ z inf h*( tx( t)) = 0. Then the conclu- 
sion of Theorem 3.2 remains valid. 
Remark. We observe that the special form of V, in Corollaries 3.2 and 
3.3 immediately shows that condition (A,) is satisfied in view of the 
property of h*. This observation makes the direct proofs given in [24] 
redundant. 
The next result shows that when D+ V, is not h-negative definite, it is 
beneficial to find another Lyapunov function relative to which suitable 
conditions yield (ho, h)-asymptotic stability. 
THEOREM 3.3. Assume that (A,) and (A,) of Theorem 3.1 hold. Suppose 
further that 
(i) D+ V,(t, d(O), 4) d -4t) 14 Vdt, 9(O), 6)) on S(h, P) xs(h, P), 
where i is integrally positive, y E X, V, E C[S(h, p) xS(i, p), R + ] and 
V, either satisfies-(A,) or V,(t, 4(O), 4) = jTr h*(t + s, d(O), 4) ds, where 
h* E C[S(h, p) XS(h, p), R,] and h is finer than h*; 
(ii) V, E C[S(h, p) xS(h”, p), R,], V2(t, x, ~5) is locally Lipschitzian in 
x, V, + V, is h-positive definite and on S(h, p) xS(&, p), 
D+ J’,(t, 4(O), 4) 6 -c,(V,(t, 4(O), 4)) + CA J’,(t> 4(O)> 4))s 
where c 1, c2 E 2-x. 
Then the system (2.1) is (ho, h)-asymptotically stable. 
Proof Following the proof of Theorem 3.2, we can prove that 
W + m mj(t) =O, where m3(t) = V3(t, x(t), x,), for any solution x(t) = 
x( to, do)(t) for which we have 
hotto, 40) < 6, implies h(t, x(t)) <P> t> to. 
We shall next prove that lim,, ~ V,( t, x(t), x,) = 0. Letting m2( t) = 
V2(t, x(t), x,), we see that condition (ii) yields 
D+mdt) 6 -c,(m,(t)) + c,(m,(t)), tat,. (3.20) 
400 LAKSHMIKANTHAM, LEELA. AND SIVASUNDARAM 
We claim that lim, _ a, inf m,(t) = 0. If it is not true, then there exists a 
d> 0 and a T, > 0 such that 
m,(t) >/d for t3 t,,+ T,. (3.21) 
Since lim, _ oc’ m3( t) = 0, there exists a T2 > 0 such that 
c,(d) 
4%(t)) G 2 for t3to+Tz. (3.22) 
Choose T= max(T,, T2) so that for t > t, + T, we have by (3.20) and 
(3.22), 
c,(d) 
D+m,(t) d -- 
2 
(3.23) 
The differential inequality (3.23) yields 
m,(t) d m,(t, + T) - +)(t-T-to), t>t,,+T 
which implies a contradiction. Suppose that lim, _ 3c sup m,(t) # 0. Then 
there exists a p > 0 and a sequence 
such that 
~(a;) = P, MPJ = 214 and P 6 m,(t) 6 2c(, t E C%, Pi I, 
(3.24) 
i= 1, 2, . ..I 
Since lim, _ ,x m3( t) = 0, there exists a T, > 0 such that 
Cl(P) 
CA%(t)) G--y t3 t,+ T,. 
From (3.20), (3.24), and (3.25), we obtain for sufficiently large i, 
C,(P) 
D+m,(t) d - 2’ t E La,, PiI5 
(3.25) 
which shows that m,(t) is non-increasing on [cci, /Ii]. Hence m2(Pi)< 
mz(ai) which contradicts (3.24). Thus, lim, _ z mz(t) = 0. Since v/2 + V, 
is h-positive definite, it follows that lim, _ J3 h(t, x(t)) = 0, proving 
(h,, h)-asymptotic stability. The proof is complete. 
LYAPUNOVFUNCTIONS 401 
As in [lo], one can show that Theorem 3.3 includes several interesting 
cases. 
So far, we have considered the estimation of DC V covering the following 
cases: 
(i) D+ Ut, 4(O), #) < .dt, Vt, b(O), 411, 
(ii) D+ Vt, 4(O), 4) f w(t, 4(O), 4). 
In general, we may have the estimation 
(iii) D+ v(t, 4(O), 9) < g(f, v(t, (01, $1, d(O), 41, 
and hence, we shall next discuss this general case. For this purpose, we 
need to define stability concepts relative to the comparison equation 
u’ = KC4 u, x(t), XI), u(t,) = un z 0, (3.26) 
where g E C[ R + ;cR +xR”x%?, R], g( t, 0, x, 4) E 0 and x( 1) = X( t,, &)(t) is 
any solution of (2.1). 
DEFINITION 3.1. The trivial solution of (3.26) is said to be 
h,-conditionally stable if given E > 0 and t, E: R + , there exist 6, = 
fil(to,s)>O and 6,=6,(t,,~)>O such that 
!I,([,, do) < 6, and u0 < 6, implies u(t, t,, do, z+,) < E, t3 I,, 
where u(t, t,, do, ug) is any solution of (3.26). 
We shall now state the following result whose proof may be contructed 
based on the proof of Theorem 3.1 and the proofs of corresponding results 
for ODE in [IS]. 
THEOREM 3.4. Assume that (A,) holds. Suppose further that 
VE C[S(h, p) XS(h, p), R ,. 1, V( t, x, 4) is locally Lipschitzian in x, h-positive 
definite, h,-decrescent and on S(h, p) xS(h”, p), 
holds, where g is defined as in (3.26). Then the stability properties of the 
trivial solution of (3.26) imply the corresponding (h,, h)-stability properties 
of(2.1). 
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402 LAKSHMIKANTHAM, LEELA, AND SIVASUNDARAM 
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Lyapunov functions on product spaces and stability theory of delay differential equations 

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Lyapunov functions on product spaces and stability theory of delay differential equations

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