- Email: [email protected]
Chapter 4: Global Stability of Functional Differential Equations of Neutral Type
Chapter 4
Global Stability of Functional Differential Equations of Neutral 'Qpe
4.1 Definitions [1,2] Suppose f,g are continuous functions taking functional difference operator
D(., .) : [T, 00) x
[T, 00) x
C + En.Define the
c + E" (4.1.1)
for t E
[T, oo),
E C = C([-h, 01, E").Consider the system d -D(t).t dt
= f(t,.#),4
s )
= .(t
+ s),
(4.1.2)
where zt E C. We say z is a solution of (4.1.2) with initial value 4 at u if there exists a E [T,co), a > 0 such that z : [u - h , u a] -, En is continuous z, = 4, D(t)zt is continuously differentiable on (6, u a), and (4.1.2) is satisfied on (6, u a). In this section we shall consider the
+
+
following systems: (4.1 .a);
d dt
+ F ( t ,z t ) , d ;7i[D(f)zt- G(t,zt)] = f(t, z t ) + F ( t , --D(t)zt
= f ( t , .t)
zt),
+
(4.1.3) (4.1.4)
under the following, basic assumptions: D(t)+ = 4(0) - g ( t , 4) is defined in (4.1.1) where g ( t , 4) is linear in 4 and is given by g ( t , 4)
=
J
0
[dep(t,0)14(0).
(4.1.5a)
-h
Here p ( t , 8) is an n x n matrix function o f t E variation in 8 which satisfies the inequality
87
[T, m), 8
E [-h, 01, of bounded
88
Stability and Time-Optimal Control of Hereditary Systems
Whenever (4.1.5) holds for g , we say g is uniformly nonatomic at zero. We also assume that G in (4.1.4) is uniformly nonatomic at zero. F'urthermore, there is a nonnegative constant N such that Ig(t,d)I
5 "ll,
v ( t , d ) E b-,m) x c.
(4.1.6)
Basic in our discussions below are the assumptions that f ,g , D, G, F are smooth enough t o ensure that a solution of (4.1.2) - (4.1.4) exists through each (u,4)E [T,OO) x C,is unique and depends continuously upon (g,t$), and can be continued t o the right as long as the trajectory remains in a bounded set in [r,m) x C. The needed conditions are given in Section 7.1. Our main interest here is the stability theory of (4.1.2), (4.1.3), and (4.1.4). The basic Lyapunov theory on (4.1.2) is first presented. In addition, if in (4.1.2) f has a uniform Lipschitz constant, a converse theorem t o the basic Lyapunov theory of Cruz and Hale [2] is formulated. All these are applied in subsequent sections. Some definitions are needed.
Definition 4.1.1: The zero solution of (4.1.2) is Uniformly Asymptotically Stable in the Large (UASL) if it is uniformly stable and if the solutions of (4.1.2) are uniformly bounded, and if for any a > 0 and any E > 0 and any u E [O,co) there exists a T(6,a) > 0 such that if 11~~5115 a, then 11zt(u,4)II < E for all t 3 u "(€,a).The trivial solution (4.1.2) is said to be Weakly Uniformly Asymptotically Stable in the Large (WUASL) if it is uniformly stable and for every u E [O,co)and every 4 E C we have 11q(u,$)I1 -+ 0 as t + co. The zero solution of (4.1.2) is Exponentially Asymptotically Stable in the Large (EXASL) if there exists c > 0 and for any a > 0 there exists a k ( a ) > 0 such that if 11411 5 a,
+
The zero solution is Uniformly Exponentially Asymptotically Stable in the Large (UEXASL) if there exist c > 0, k > 0 such that
(4.1.7) The trivial solution is Eventually Uniformly Stable (EVUS) if for every 6 > 0 there is a = a(€) such that (4.1.2) is uniformly stable fort 2 u 3 a(€). Other eventual stability concepts can similarly be formulated by insisting that the initial time u satisfy u 2 a. For example, the trivial solution of (4.1.2) is called Eventually Weakly Uniformly Asymptotically Stable in the Large (EVWUASL) if it is (EVUS) and if there exists a0 > 0 such that for every u 2 a. and for every q!~ E C, we have 11zt(a,q5)II 0 as t co.
- -
Global Stability of Functional Digerential Equations of Neutral Type
4.2
89
Lyapunov Stability Theorem
Consider the homogeneous difference equation D(t)zt = 0 t
2D
zU = 4
D ( D )=~0 ,
(4.2.1)
which was discussed in Section 2.4. There it was stated that D is uniformly stable if there are constants P , a > 0 such that for every E [T, co), 4 E C the solution z(u,4) of (4.2.1) satisfies
Ilzt(u, 4111 5 Pe-Q(t-u)l1411,
t
1
Cruz and Hale have given the following two lemmas [2]: Lemma 4.2.1 If D is uniformly stable, then there are positive constants cm,b,c,d such that for any h E C([T,OO),E"), the space of continuous functions from [r,co) -+ En, and u E [ T , co) the solution Z ( D , 4,g) of D(t)zt = M(t), t 2
D,
zU = 4,
(4.2.2)
satisfies
co), Furthermore, a , b , c, d can be chosen so that for any s E [a, I I ~ t ( ~ , 4 , g >5I le-a(t-S)(bl1411
+c
SUP usust
IM(u)I)
+ d s
fort 2 s + h . Lemma 4.2.2 Suppose D is uniformly stable, and M : [ T , O O ) -+ E" is continuous such that M ( t ) + 0 as t ---* 00. Then the solution z ( u , d , M ) of (4.2.2) approaches zero as t -+ 00 uniformly with respect to D in [T,OO) and 4 in closed bounded sets. Let V : [T,OO) x C --+ E be a continuous functional and z(D,+) be a solution of (4.1.2) through (u,4), i.e., a solution of the integral equation
D ( t ) z t = D(u)4 Define V ( t ,4) by
+
f ( s , z d ) d ~ , t >_ u,
2 ,
= 4.
(4.2.3)
90
Stability and Tame-Optimal Control of Hereditary Systems
Theorem 4.2.1 [2] Let D be uniformly stable, f : [T,OO) x C -, En, continuous, and f maps [T,co)x (bounded sets of C)into bounded sets of En.Suppose there are continuous, nonnegative, and nondecreasing functions u(s), u ( s ) , ~ ( s )with u(s), v ( s ) > 0 for s # 0 and u(0) = u ( 0 ) = 0, and there is a continuous function V : [r,m) x C -,E such that (4.2.4a) then the solution x = 0 of (4.1.2) is uniformly stable. I f u ( s ) co as s the solutions of (4.1.2) are uniformly bounded. U W ( S>) 0 for s > 0, then the solution x = 0 of (4.1.2) is uniformly asymptotically stable. The same conclusion holds if the upper bound on V ( t ,4) is given by -w( Ib(0)l.
00,
The following converse theorem to Theorem 4.2.1 was stated and proved in Chukwu [l]. Theorem 4.2.2 In (4.1.2), assume (i) f ( t , O ) = 0 V t , and f has a uniform Lipschitz constant, i.e., there exists a constant L such that for any 4, E C and for all t E [T,00)
I f 0 9 4) - f 4)l 5 414 - 411. (tl
(ii) There is a constant N
> 0 such
that
l W 4 l I Nll4ll v t E [ T , c o ) , 4 E cThe zero solution of (4.1.2) is (UEXASL), so that
IMn,4111 5 ~e-"'"-"'11411
v t 1 g,
for some k > 0, c > 0. Then there are positive constants M , R and a continuous functional V : [r,m) x C -,E such that
lD(Wl I V @ 4) , I Mlldll, (4.2.4b)
We now use Theorem 4.2.1 to study the system
d - [ x ( t ) - A - l z ( t - h)] = f ( t ,X(t),x(t - h ) ) , dt
(4.2.5)
Global Stability of Funciional Diflerentiaf Equations of Neutral Type
91
where A-1 is an n x n symmetric constant matrix and f : [T,OO) x En x En-+ E" and f : [T, OO) x (bounded sets of E") x (bounded sets of En)-+ (bounded sets of En).We also assume that f has continuous partial derivatives. If we let D ( t ) 4 = +(O) - A-1 +(-/a), then (4.2.5) is equivalent to d
-dt w 4 = f (1, W ) +, ( - h ) ) . In (4.2.6) denote by D, (a = 1,2) the n x n symmetric matrices where
(4.2.6) +(d,jj
+
d,ji),
Let
J " --2-1( A D .p
+ DZA),
a = 1,2,
(4.2.7)
where A is a positive definite constant symmetric n x n matrix, and DZ is the transpose of D,. Our stability study of (4.2.6) will be made under various assumptions on J a .
Theorem 4.2.3 In (4.2.5), assume that (i) D+ = 4(0) - A-l+(-/a) is uniformly stable, (ii) f(t,O,O)= 0. (iii) There are some positive definite constant symmetric matrices A and N such that the matrix J = ( ;
;),
(4.2.8a)
where
and J , are defined in (4.2.7), is negative definite. (iv) A-1 is symmetric. Then the solution x G 0 of (4.2.5) is uniformly asymptotically stable. Remark 4.2.3: J in (4.2.8) will be negative definite if
92
Stability and Time-Optimal Control of Hereditary Systems
Global Stability of Functional Differential Equations of Neutral n p e
93
since J is negative definite. It now follows that (4.2.4) of Theorem 4.2.1 is completely verified. Because D is assumed to be uniformly stable, all the requirements of Theorem 4.2.1 axe met: We conclude that the solution z = 0 of (4.2.5) is uniformly asymptotically stable. We now consider some examples.
Example 4.2.1: Consider the scalar equation d
-[z(t) dt
+ c r ( t - h)]+ a z ( t ) = 0 ,
4< > 0 Icl < 1 and a > c / ( l - c2) 2 c < 1. Lemma 3.1 of Cruz and Hale [2]implies that the operator 09 = d(O)+cd(-h) is uniformly stable. We now apply Theorem 4.2.3 with A-1 = a -c, A = 1, J1 = 0 , JZ = -a, N = a/2,2J*+N = -2a+- = -3a/2 = a
where a and c are constants with a
2
p = J1+ J z A - l + NA-1 a ac 2 2 a y = A - 1 N A - 1 - N = -(c2 2
= ac - -c
- 1)
is negative definite. We can also use Remark 4.2.3 with Xz = a , XB = a / 2 , A4 = ;(l - c'), so that 2 X 3 ~ 4= 2 ( a / 2 ) ( a / 2 ) ( 1 - c2) > ac - ac/2,
since a2(1 - c 2 ) / 2 > ac/2 if we assume a ( 1 -
2 )> c.
Example 4.2.2: Consider the equation d
-[z(t) dt
+ c ( t ) z ( t - h)]+ a z ( t ) + b ( t ) z ( t - h ) = 0,
where a > 0 is a constant, c ( t ) , b ( t ) are continuous for t 2 0, and there is a 6 > 0 such that c 2 ( t ) 5 1 - 6. Let D ( t ) + = O(0) c(t)d(-h), so that by a remark in Cruz and Hale [2, p. 3381, D ( t ) is uniformly stable. Observe , - h ) ) = - a z ( t ) - b(t)z(t - h). On taking A = 1, A-1 = that f ( t , z ( t ) z(t -c(t), N = a / 2 , J1 = - b ( t ) , 5 2 = --a = - A z , JZ N = -a/2 =
+
+
94
Stability and Time-Optimal Control of Hereditary Systems
J2A-1 = ac, NA-1 = -ac/2, -X3, We observe that
A4
= N - A-1 NA-1 = a/2 - ac2/2.
ensures uniform asymptotic stability. This inequality will hold if a2/26 Ib(t) ac(t)/21 for all t 2 0.
+
>
Example 4.2.3: Consider the n-dimensional autonomous neutral difference equation
where ~ ( tE) En.A - l , A o , A l are constant n x n matrices h > 0. Let D1 = $(A1 AT), D2 = ~ ( A o A;), Ji = ADi DTA, i = 1,2, A, a positive definite matrix. Assume llA-111 < 1 and A-1 to be symmetric. For some constant positive definite symmetric matrix N , Remark 4.2.3 yields the required condition for uniform asymptotic stability.
+
4.3
+
+
Perturbed Systems
The converse Theorem 4.2.2 will now be employed to prove stability results for the system (4.1.3). Theorem 4.3.1 In System 4.1.3, assume that 01 Is(t,d)I 5 NIl0ll, If(t,d) - r(t,$)lINlld - $11, v 0,$ E t E [T, co); and D is uniformly stable. (ii) Suppose that the zero solution of (4.1.2) is (UEXASL), so that every t , satisfies solution ~ ( 4)
c,
for some constants c > 0 k > 0 . (iii) The function F in (4.1.3) satisfies F = F1 F2 with IFl(t,d)l 5 v(t)ID(t)q5lV t 2 u, 0 E C , where there is a constant > 0 such that for any p > 0
+
<
t+P
iP l
and for some 6 > 0 we have
v(s)ds
< t , 12 u,
(4.3.1)
Global Stability of Functional Differential Equations of Neutral Type
95
Then every solution x(u,4) of (4.1.3) satisfies
uniformly with respect to u E [r,oo),and 4 in closed bounded sets. In particular, the null solution of (4.1.3) is WUASL. Proof: The assumptions we have made in Theorem 4.2.2 guarantee the existence of a Lyapunov function V with the properties of Theorem 4.2.2. Suppose y = y(u,+) and x = z(u,4) are solutions of (4.1.3) and (4.1.2) respectively with initial values 4 at u. If $4.1.3) and $4.1.2) are the derivatives of V along solutions of the systems (4.1.3) and (4.1.2) respectively, then the relations (4.2.4b) yields the inequality $4.1.q(',
4) 5 *4.I.2)(u, 4) + R:iT+
1 ;[Yu+r(',
4) - cc+r(u, 411. (4.3.2)
But then
Because g ( t , + ) is uniformly nonatomic at zero and (4.1.5) holds, there is an ro > 0 such that
for 0 < r
< ro. With this estimate in (4.3.2) we obtain
V t 2 u, 4 E C where Q =
1 - c(hJ
It follows from (4.2.4b) that
96
Stability and Time-Optimal Control of Hereditary Systems
and
and
1 t
(-44
+ Qv(s))ds 5 (-: + QE) ( t - u) 5 -c/8(t
- u).
Using all these in (4.3.4), one obtains V(t,u)
C
5 ----V(t,n) + &v(t)v(t,d); 4
and with r ( t ) = V(t, z t ( u ,d)),
+(t)5
(-: + ~ v ( t ) )r ( t ) .
The solution of this inequality satisfies
Since (4.2.4b) is valid, this last inequality leads to
Because D is assumed uniformly stable, Lemma 4.2.2 can be invoked to ensure that z ( u , $ ) --+ 0 as t + 00, uniformly with respect to B E [r,oo), and 4 in closed bounded sets. To conclude the proof we verify uniform stability as follows: Because D is uniformly stable, there are positive constants al,a2, a3, a4 such that
For any 6 > 0 choose 6 = (az+a3M+a4M)-'c, and observe that if 11q!J11 < 6, then I l Z t ( B , 4)ll < a26 a3M6 a4M6 < c,
+
+
for t 2 u. This proves that z = 0 is uniformly stable, so that the zero solution of (4.1.3) is WUASL.
Global Stability of Functional Differential Equations of Neutral Type
97
Remark 4.3.1: Condition (4.3.1) can be replaced by the assumption 00
VE
v(s)ds
< 00
(4.3.5)
to deduce the same results. When f ( t , d ) A(t,$) is linear in 4 , one c a n obtain a generalization of the famous theorem of Lyapunov on the stability with respect to the first approximation. It is the global stability result that is most useful when dealing with controllability questions of neutral systems with limited power. Consider the linear system d
go(t)zt= A ( t ,z t ) ,
(4.3.6)
and its perturbation d
+
g [ D ( t ) z t- G(t,xt)l = A ( t , z t ) f ( t , z t ) ,
(4.3.7)
where A(t,(6), G(t,(6) are linear in (6. The following result is valid:
Theorem 4.3.2 In Equations (4.3.6) and (4.3.7) assume that (i) the linear system (4.3.6) is uniformly asymptotically stable, so that for some k 2 1, a > 0 the solution x(o,(6) of (4.3.6) satisfies llzt(u,(6)IlL ke-a(*-u)l141L
(ii) The function F = Fl
+ FZ
t L u (6 E c.
satisfies
where
+ +
a 1 - k(X Mo) 2k 1 k(X Mo) and where if t = a / 4 , then for each p > 0 c=-
t +P
1 P
+
v(s)ds t
(iii) The function G = G1 + G2 satisfies
< < V t 2 U.
(4.3.8)
98
Stability and Time-Optimal Control of Hereditary Systems
for t 2 u, q5 E C where ?r(t) is continuous and bounded with a bounded Mo such that
Then every solution z(u,4) of (4.3.7) satisfies
Ilz*(c 4)II Iq q 5 1 1 ~ - a ' 4 ( * - u )
9
t20,
for some N . The proof is contained in [l,p. 8661.
Remark 4.3.2: The requirement of (4.3.8) can be replaced by the hypothesis
Jam
v(s)ds
< 00.
It is possible to weaken the conditions on F , to deduce global eventual weak stability. Because of its importance it is now stated.
Theorem 4.3.3 For the system (4.3.6) and (4.3.7) assume that (i) Equation (4.3.6) is uniformly asymptotically stable. (ii) The function F in (4.3.7) can be written as
where
for all t
2 u 2 To for some sufficientlylarge To and all q5 E C,where
[ is a sufficientlysmall number, and where
+
€=
Q 1 - k(X Mo) Gl+k(X+Mo)'
A Is0 H ( t ) = J,
g(s)ds
+
0 as t
+ 00.
Global Stability of Functional DifferentialEquations of Neutral Type
(5)The function G = GI
99
+ Gz satisfies
fort 2 u, 4 E C,where n ( t ) is continuous and bounded with a bound MO such that
Then every solution of (4.3.7) with initial data (u,4)satisfies
for some To, u 2 is (EVUASL).
TO.As a consequence the trivial solution of (4.3.7)
For the proof consult [l,p. 8691. The above result deals with the linear case f ( t , 6 ) = A ( t , 4). A similar result holds for general nonlinear f . Indeed, we have the following:
Theorem 4.3.4 Assume that (4 Is(t,4)I INll6l1, I f ( t , 6 ) - f(t,$)I 5 Nll4 - $11, v 6,$ E t E [T, w) and D is uniformly stable. (ii) The trivial solution of (4.3.6) is uniformly asymptotically stable. (iii) The function F in (4.3.7) is such that F = Fl F2 F3 where
c,
+ +
for some To,where
with c,[ sufficiently small and for u large. Also
M(t)=
m(s)ds -+ 0 as t
Then there exists some T such that if u
> T,
-+
w.
100
Stability and Time-Optimal Control of Hereditary Systems
Thus the solution x = 0 of (4.3.7) is (EVWUASL). A proofis indicated in [I,p. 8721. Example 4.3.1: Delay Logistic Equation of Neutral Type [5]. Consider the system
where h l , h2, I(, c, r are constants 2 0, with r, ha, K conditions of the type
> 0. We assume initial
If we let N(t)
1q1 + z ( t ) ] ,
+
A(t)= r[l ~ ( t ) ] , B ( t ) = 2ckA(t)1[1 k 2 k 2 ( t- h z ) ] ,
+
(4.3.9) is equivalent to d
dt
LX(') h, t
-
A(s + hl).(s)ds] = -A(t
+ h l ) + B ( t ) k ( t - hz). (4.3.10)
Set
= ( 1 + c ) exp[r( 1 + c ) h J b* = (1 - c)exp[hlr(I - c)]
u*
2cka' b* exp(-u*) ru*
+ 2ckru*hl+
rb* 2(2cku*r)2 exp(-a*)
1
.
b* exp(-a')
Global Stability of Functional Differential Equations of Neutral Type
101
For this we use the Lyapunov functional
V ( t ) = V ( t ,+ ) I , A(s + hl)z(s)ds]
+ 2 ft - h
B(s A(s
2
+
l-h + t
B A(s ( s + h2)t2(s)ds
h 2 ) A 2 (+~h1)2z2(s)ds + hl + h2) +
+
and calculate the upper right derivative
(4.3.10).
D+V of dt
V along solutions of
Proposition 4.3.1 Assume that
E (0, 00); h2 E Pl > 0, P2 > 01 r, hl, k
2ckra*
10, 001,
< 1; 0 < c < 1.
(4.3.11) (4.3.12) (4.3.13)
Then every positive solution of (4.3.9) satisfies
N ( t ) + O as t + m .
Proof: We need only show that if (4.3.11) - (4.3.12) hold, an arbitrary solution of (4.3.10) satisfies t(2) - 0
as t
+ 00.
(4.3.14)
The needed details are contained in [5].
4.4 Perturbations of Nonlinear Neutral Systems [6] Consider a nonlinear system that is more general than (4.1.3), namely
d (4.4.1) -D(t, .*I = f(t,.t> + 9(t,.t). dt We assume the following as basic: Let A C C be an open set, 11 C E an open interval, P = 11 x A. Denote by En' the set of n x n real matrices. Suppose D , f : r + En are continuous functions. We assume also that 0 E A. The main result in this section is stated as follows:
102
Stability and Time-Optimal Control of Hereditary Systems
Theorem 4.4.1 Assume (i) D ,f in (4.4.1) satisfy the following hypothesis: (a) The Fre'chet derivatives of D ,f with respect to I#, denoted by D , , f, respectively exist and are continuous on I? as well as D,,, and (b) for each ( t , 4 ) E I?, 11, E C write
J
~ + ( 4>[41= t, ~ ( 4t , ~ 0 -)
0
-h
dep(t,4, e)[+(e)l
for some functions
A : r -+E"', p : r x [-h,o]
-+
E"'
with A continuous, p ( t , 4, .) of bounded variation on [-A, 01 and such that the map ( t ,4) -+ f hd e p ( t ,4, 0)[11,(0)] is continued for each $j E
c.
(b) Assume for each ( t , d ) E compact sets I< C I?, i.e., det A ( t , 4)
IL
r
D is uniformly atomic at zero on
#0
d e p ( t , ~ , e ) ~ ( d )!(S)II+II ~l<
E [O, h] for all ( t ,4) E I< for some function t : [0, h] + [0, oo) that is continuous and nonincreasing, !(O) = 0. (ii) g is Lipschitzian in 4 on compact subsets of r. (iii) j ( t ,0,O) = 0 for all t E I. (iv) For each (a,4) E r, we have s
a,4111 I e x p ( 4 t )
+ b(a)), t L
6,
where a ( t ) , b(t) are continuous functions from E+ = [0, oo) into E+, i.e., are elements of C(E+,E+). Here T ( t ,a;4) is the solution operator associated with the variational equation
d -D,(t, dt
t t ( U , 4))[ZtI
(v) For each ( a , d )E I',
= f#(t,%(a,4))[.tl,
t E [a, + a).
(4.4.2)
Global Stability of Functional Daflerential Equations of Neutral Type
103
+
where ri E (0,1], c j ( t ) E C(E+, E + ) , j = 1,2,.. . , N 1. (v;) cj(S)eria(8)+*(d), c$)+le*(s) E ~ 1 ( T , u )j, = I,. . . ,N. (vii) a ( t ) +. -00 as t 00. Then every solution of (4.1.1) satisfies IlZt(u1
4)ll
- -t
00.
The proof is contained in Chukwu and Simpson in [6, p. 571. It uses the nonlinear variation of parameter equation
Ju
(4.4.3)
where y t ( u , 4) is the solution of
d
-dtW , Y : ) = f ( t , Y : ) .
(4.4.4)
This formula (4.4.3) holds when the solution of (4.4.1) is unique.
REFEREN cES 1. E. N. Chukwu, “Global Asymptotic Behavior of Functional Differential Equations of the Neutral Type,” Nonlinear Analysis Theory Methods and Applications 5 (1981)853472. 2. M.A. Cruz and J. K. Hale, “Stability of F‘unctional Differential Equations of Neutral Type,” J . Differential Equations 7 (1970)334-355. 3. J. K.Hale, Ordinary Differential Equations, Interscience, New York, 1969.
K. Hale, “Theory of Functional Differential Equations,” Applied Mathematical Sciences 3,Springer-Verlag, New York, 1977. 5. K. Gopalsamy, “Global Attractivity of Neutral Logistic Equations,” in Differential Equations and Applications, edited by A, R. Aftabizadeh, Ohio University Press, 4. J.
Athens, 1989. 6.
E. N. Chukwu and H. C. Simpson, “Perturbations of Nonlinear Systems of Neutral Type,” J . Differential Equations 82 (1989)28-59.
Global Stability of Functional Differential Equations of Neutral 'Qpe
4.1 Definitions [1,2] Suppose f,g are continuous functions taking functional difference operator
D(., .) : [T, 00) x
[T, 00) x
C + En.Define the
c + E" (4.1.1)
for t E
[T, oo),
E C = C([-h, 01, E").Consider the system d -D(t).t dt
= f(t,.#),4
s )
= .(t
+ s),
(4.1.2)
where zt E C. We say z is a solution of (4.1.2) with initial value 4 at u if there exists a E [T,co), a > 0 such that z : [u - h , u a] -, En is continuous z, = 4, D(t)zt is continuously differentiable on (6, u a), and (4.1.2) is satisfied on (6, u a). In this section we shall consider the
+
+
following systems: (4.1 .a);
d dt
+ F ( t ,z t ) , d ;7i[D(f)zt- G(t,zt)] = f(t, z t ) + F ( t , --D(t)zt
= f ( t , .t)
zt),
+
(4.1.3) (4.1.4)
under the following, basic assumptions: D(t)+ = 4(0) - g ( t , 4) is defined in (4.1.1) where g ( t , 4) is linear in 4 and is given by g ( t , 4)
=
J
0
[dep(t,0)14(0).
(4.1.5a)
-h
Here p ( t , 8) is an n x n matrix function o f t E variation in 8 which satisfies the inequality
87
[T, m), 8
E [-h, 01, of bounded
88
Stability and Time-Optimal Control of Hereditary Systems
Whenever (4.1.5) holds for g , we say g is uniformly nonatomic at zero. We also assume that G in (4.1.4) is uniformly nonatomic at zero. F'urthermore, there is a nonnegative constant N such that Ig(t,d)I
5 "ll,
v ( t , d ) E b-,m) x c.
(4.1.6)
Basic in our discussions below are the assumptions that f ,g , D, G, F are smooth enough t o ensure that a solution of (4.1.2) - (4.1.4) exists through each (u,4)E [T,OO) x C,is unique and depends continuously upon (g,t$), and can be continued t o the right as long as the trajectory remains in a bounded set in [r,m) x C. The needed conditions are given in Section 7.1. Our main interest here is the stability theory of (4.1.2), (4.1.3), and (4.1.4). The basic Lyapunov theory on (4.1.2) is first presented. In addition, if in (4.1.2) f has a uniform Lipschitz constant, a converse theorem t o the basic Lyapunov theory of Cruz and Hale [2] is formulated. All these are applied in subsequent sections. Some definitions are needed.
Definition 4.1.1: The zero solution of (4.1.2) is Uniformly Asymptotically Stable in the Large (UASL) if it is uniformly stable and if the solutions of (4.1.2) are uniformly bounded, and if for any a > 0 and any E > 0 and any u E [O,co) there exists a T(6,a) > 0 such that if 11~~5115 a, then 11zt(u,4)II < E for all t 3 u "(€,a).The trivial solution (4.1.2) is said to be Weakly Uniformly Asymptotically Stable in the Large (WUASL) if it is uniformly stable and for every u E [O,co)and every 4 E C we have 11q(u,$)I1 -+ 0 as t + co. The zero solution of (4.1.2) is Exponentially Asymptotically Stable in the Large (EXASL) if there exists c > 0 and for any a > 0 there exists a k ( a ) > 0 such that if 11411 5 a,
+
The zero solution is Uniformly Exponentially Asymptotically Stable in the Large (UEXASL) if there exist c > 0, k > 0 such that
(4.1.7) The trivial solution is Eventually Uniformly Stable (EVUS) if for every 6 > 0 there is a = a(€) such that (4.1.2) is uniformly stable fort 2 u 3 a(€). Other eventual stability concepts can similarly be formulated by insisting that the initial time u satisfy u 2 a. For example, the trivial solution of (4.1.2) is called Eventually Weakly Uniformly Asymptotically Stable in the Large (EVWUASL) if it is (EVUS) and if there exists a0 > 0 such that for every u 2 a. and for every q!~ E C, we have 11zt(a,q5)II 0 as t co.
- -
Global Stability of Functional Digerential Equations of Neutral Type
4.2
89
Lyapunov Stability Theorem
Consider the homogeneous difference equation D(t)zt = 0 t
2D
zU = 4
D ( D )=~0 ,
(4.2.1)
which was discussed in Section 2.4. There it was stated that D is uniformly stable if there are constants P , a > 0 such that for every E [T, co), 4 E C the solution z(u,4) of (4.2.1) satisfies
Ilzt(u, 4111 5 Pe-Q(t-u)l1411,
t
1
Cruz and Hale have given the following two lemmas [2]: Lemma 4.2.1 If D is uniformly stable, then there are positive constants cm,b,c,d such that for any h E C([T,OO),E"), the space of continuous functions from [r,co) -+ En, and u E [ T , co) the solution Z ( D , 4,g) of D(t)zt = M(t), t 2
D,
zU = 4,
(4.2.2)
satisfies
co), Furthermore, a , b , c, d can be chosen so that for any s E [a, I I ~ t ( ~ , 4 , g >5I le-a(t-S)(bl1411
+c
SUP usust
IM(u)I)
+ d s
fort 2 s + h . Lemma 4.2.2 Suppose D is uniformly stable, and M : [ T , O O ) -+ E" is continuous such that M ( t ) + 0 as t ---* 00. Then the solution z ( u , d , M ) of (4.2.2) approaches zero as t -+ 00 uniformly with respect to D in [T,OO) and 4 in closed bounded sets. Let V : [T,OO) x C --+ E be a continuous functional and z(D,+) be a solution of (4.1.2) through (u,4), i.e., a solution of the integral equation
D ( t ) z t = D(u)4 Define V ( t ,4) by
+
f ( s , z d ) d ~ , t >_ u,
2 ,
= 4.
(4.2.3)
90
Stability and Tame-Optimal Control of Hereditary Systems
Theorem 4.2.1 [2] Let D be uniformly stable, f : [T,OO) x C -, En, continuous, and f maps [T,co)x (bounded sets of C)into bounded sets of En.Suppose there are continuous, nonnegative, and nondecreasing functions u(s), u ( s ) , ~ ( s )with u(s), v ( s ) > 0 for s # 0 and u(0) = u ( 0 ) = 0, and there is a continuous function V : [r,m) x C -,E such that (4.2.4a) then the solution x = 0 of (4.1.2) is uniformly stable. I f u ( s ) co as s the solutions of (4.1.2) are uniformly bounded. U W ( S>) 0 for s > 0, then the solution x = 0 of (4.1.2) is uniformly asymptotically stable. The same conclusion holds if the upper bound on V ( t ,4) is given by -w( Ib(0)l.
00,
The following converse theorem to Theorem 4.2.1 was stated and proved in Chukwu [l]. Theorem 4.2.2 In (4.1.2), assume (i) f ( t , O ) = 0 V t , and f has a uniform Lipschitz constant, i.e., there exists a constant L such that for any 4, E C and for all t E [T,00)
I f 0 9 4) - f 4)l 5 414 - 411. (tl
(ii) There is a constant N
> 0 such
that
l W 4 l I Nll4ll v t E [ T , c o ) , 4 E cThe zero solution of (4.1.2) is (UEXASL), so that
IMn,4111 5 ~e-"'"-"'11411
v t 1 g,
for some k > 0, c > 0. Then there are positive constants M , R and a continuous functional V : [r,m) x C -,E such that
lD(Wl I V @ 4) , I Mlldll, (4.2.4b)
We now use Theorem 4.2.1 to study the system
d - [ x ( t ) - A - l z ( t - h)] = f ( t ,X(t),x(t - h ) ) , dt
(4.2.5)
Global Stability of Funciional Diflerentiaf Equations of Neutral Type
91
where A-1 is an n x n symmetric constant matrix and f : [T,OO) x En x En-+ E" and f : [T, OO) x (bounded sets of E") x (bounded sets of En)-+ (bounded sets of En).We also assume that f has continuous partial derivatives. If we let D ( t ) 4 = +(O) - A-1 +(-/a), then (4.2.5) is equivalent to d
-dt w 4 = f (1, W ) +, ( - h ) ) . In (4.2.6) denote by D, (a = 1,2) the n x n symmetric matrices where
(4.2.6) +(d,jj
+
d,ji),
Let
J " --2-1( A D .p
+ DZA),
a = 1,2,
(4.2.7)
where A is a positive definite constant symmetric n x n matrix, and DZ is the transpose of D,. Our stability study of (4.2.6) will be made under various assumptions on J a .
Theorem 4.2.3 In (4.2.5), assume that (i) D+ = 4(0) - A-l+(-/a) is uniformly stable, (ii) f(t,O,O)= 0. (iii) There are some positive definite constant symmetric matrices A and N such that the matrix J = ( ;
;),
(4.2.8a)
where
and J , are defined in (4.2.7), is negative definite. (iv) A-1 is symmetric. Then the solution x G 0 of (4.2.5) is uniformly asymptotically stable. Remark 4.2.3: J in (4.2.8) will be negative definite if
92
Stability and Time-Optimal Control of Hereditary Systems
Global Stability of Functional Differential Equations of Neutral n p e
93
since J is negative definite. It now follows that (4.2.4) of Theorem 4.2.1 is completely verified. Because D is assumed to be uniformly stable, all the requirements of Theorem 4.2.1 axe met: We conclude that the solution z = 0 of (4.2.5) is uniformly asymptotically stable. We now consider some examples.
Example 4.2.1: Consider the scalar equation d
-[z(t) dt
+ c r ( t - h)]+ a z ( t ) = 0 ,
4< > 0 Icl < 1 and a > c / ( l - c2) 2 c < 1. Lemma 3.1 of Cruz and Hale [2]implies that the operator 09 = d(O)+cd(-h) is uniformly stable. We now apply Theorem 4.2.3 with A-1 = a -c, A = 1, J1 = 0 , JZ = -a, N = a/2,2J*+N = -2a+- = -3a/2 = a
where a and c are constants with a
2
p = J1+ J z A - l + NA-1 a ac 2 2 a y = A - 1 N A - 1 - N = -(c2 2
= ac - -c
- 1)
is negative definite. We can also use Remark 4.2.3 with Xz = a , XB = a / 2 , A4 = ;(l - c'), so that 2 X 3 ~ 4= 2 ( a / 2 ) ( a / 2 ) ( 1 - c2) > ac - ac/2,
since a2(1 - c 2 ) / 2 > ac/2 if we assume a ( 1 -
2 )> c.
Example 4.2.2: Consider the equation d
-[z(t) dt
+ c ( t ) z ( t - h)]+ a z ( t ) + b ( t ) z ( t - h ) = 0,
where a > 0 is a constant, c ( t ) , b ( t ) are continuous for t 2 0, and there is a 6 > 0 such that c 2 ( t ) 5 1 - 6. Let D ( t ) + = O(0) c(t)d(-h), so that by a remark in Cruz and Hale [2, p. 3381, D ( t ) is uniformly stable. Observe , - h ) ) = - a z ( t ) - b(t)z(t - h). On taking A = 1, A-1 = that f ( t , z ( t ) z(t -c(t), N = a / 2 , J1 = - b ( t ) , 5 2 = --a = - A z , JZ N = -a/2 =
+
+
94
Stability and Time-Optimal Control of Hereditary Systems
J2A-1 = ac, NA-1 = -ac/2, -X3, We observe that
A4
= N - A-1 NA-1 = a/2 - ac2/2.
ensures uniform asymptotic stability. This inequality will hold if a2/26 Ib(t) ac(t)/21 for all t 2 0.
+
>
Example 4.2.3: Consider the n-dimensional autonomous neutral difference equation
where ~ ( tE) En.A - l , A o , A l are constant n x n matrices h > 0. Let D1 = $(A1 AT), D2 = ~ ( A o A;), Ji = ADi DTA, i = 1,2, A, a positive definite matrix. Assume llA-111 < 1 and A-1 to be symmetric. For some constant positive definite symmetric matrix N , Remark 4.2.3 yields the required condition for uniform asymptotic stability.
+
4.3
+
+
Perturbed Systems
The converse Theorem 4.2.2 will now be employed to prove stability results for the system (4.1.3). Theorem 4.3.1 In System 4.1.3, assume that 01 Is(t,d)I 5 NIl0ll, If(t,d) - r(t,$)lINlld - $11, v 0,$ E t E [T, co); and D is uniformly stable. (ii) Suppose that the zero solution of (4.1.2) is (UEXASL), so that every t , satisfies solution ~ ( 4)
c,
for some constants c > 0 k > 0 . (iii) The function F in (4.1.3) satisfies F = F1 F2 with IFl(t,d)l 5 v(t)ID(t)q5lV t 2 u, 0 E C , where there is a constant > 0 such that for any p > 0
+
<
t+P
iP l
and for some 6 > 0 we have
v(s)ds
< t , 12 u,
(4.3.1)
Global Stability of Functional Differential Equations of Neutral Type
95
Then every solution x(u,4) of (4.1.3) satisfies
uniformly with respect to u E [r,oo),and 4 in closed bounded sets. In particular, the null solution of (4.1.3) is WUASL. Proof: The assumptions we have made in Theorem 4.2.2 guarantee the existence of a Lyapunov function V with the properties of Theorem 4.2.2. Suppose y = y(u,+) and x = z(u,4) are solutions of (4.1.3) and (4.1.2) respectively with initial values 4 at u. If $4.1.3) and $4.1.2) are the derivatives of V along solutions of the systems (4.1.3) and (4.1.2) respectively, then the relations (4.2.4b) yields the inequality $4.1.q(',
4) 5 *4.I.2)(u, 4) + R:iT+
1 ;[Yu+r(',
4) - cc+r(u, 411. (4.3.2)
But then
Because g ( t , + ) is uniformly nonatomic at zero and (4.1.5) holds, there is an ro > 0 such that
for 0 < r
< ro. With this estimate in (4.3.2) we obtain
V t 2 u, 4 E C where Q =
1 - c(hJ
It follows from (4.2.4b) that
96
Stability and Time-Optimal Control of Hereditary Systems
and
and
1 t
(-44
+ Qv(s))ds 5 (-: + QE) ( t - u) 5 -c/8(t
- u).
Using all these in (4.3.4), one obtains V(t,u)
C
5 ----V(t,n) + &v(t)v(t,d); 4
and with r ( t ) = V(t, z t ( u ,d)),
+(t)5
(-: + ~ v ( t ) )r ( t ) .
The solution of this inequality satisfies
Since (4.2.4b) is valid, this last inequality leads to
Because D is assumed uniformly stable, Lemma 4.2.2 can be invoked to ensure that z ( u , $ ) --+ 0 as t + 00, uniformly with respect to B E [r,oo), and 4 in closed bounded sets. To conclude the proof we verify uniform stability as follows: Because D is uniformly stable, there are positive constants al,a2, a3, a4 such that
For any 6 > 0 choose 6 = (az+a3M+a4M)-'c, and observe that if 11q!J11 < 6, then I l Z t ( B , 4)ll < a26 a3M6 a4M6 < c,
+
+
for t 2 u. This proves that z = 0 is uniformly stable, so that the zero solution of (4.1.3) is WUASL.
Global Stability of Functional Differential Equations of Neutral Type
97
Remark 4.3.1: Condition (4.3.1) can be replaced by the assumption 00
VE
v(s)ds
< 00
(4.3.5)
to deduce the same results. When f ( t , d ) A(t,$) is linear in 4 , one c a n obtain a generalization of the famous theorem of Lyapunov on the stability with respect to the first approximation. It is the global stability result that is most useful when dealing with controllability questions of neutral systems with limited power. Consider the linear system d
go(t)zt= A ( t ,z t ) ,
(4.3.6)
and its perturbation d
+
g [ D ( t ) z t- G(t,xt)l = A ( t , z t ) f ( t , z t ) ,
(4.3.7)
where A(t,(6), G(t,(6) are linear in (6. The following result is valid:
Theorem 4.3.2 In Equations (4.3.6) and (4.3.7) assume that (i) the linear system (4.3.6) is uniformly asymptotically stable, so that for some k 2 1, a > 0 the solution x(o,(6) of (4.3.6) satisfies llzt(u,(6)IlL ke-a(*-u)l141L
(ii) The function F = Fl
+ FZ
t L u (6 E c.
satisfies
where
+ +
a 1 - k(X Mo) 2k 1 k(X Mo) and where if t = a / 4 , then for each p > 0 c=-
t +P
1 P
+
v(s)ds t
(iii) The function G = G1 + G2 satisfies
< < V t 2 U.
(4.3.8)
98
Stability and Time-Optimal Control of Hereditary Systems
for t 2 u, q5 E C where ?r(t) is continuous and bounded with a bounded Mo such that
Then every solution z(u,4) of (4.3.7) satisfies
Ilz*(c 4)II Iq q 5 1 1 ~ - a ' 4 ( * - u )
9
t20,
for some N . The proof is contained in [l,p. 8661.
Remark 4.3.2: The requirement of (4.3.8) can be replaced by the hypothesis
Jam
v(s)ds
< 00.
It is possible to weaken the conditions on F , to deduce global eventual weak stability. Because of its importance it is now stated.
Theorem 4.3.3 For the system (4.3.6) and (4.3.7) assume that (i) Equation (4.3.6) is uniformly asymptotically stable. (ii) The function F in (4.3.7) can be written as
where
for all t
2 u 2 To for some sufficientlylarge To and all q5 E C,where
[ is a sufficientlysmall number, and where
+
€=
Q 1 - k(X Mo) Gl+k(X+Mo)'
A Is0 H ( t ) = J,
g(s)ds
+
0 as t
+ 00.
Global Stability of Functional DifferentialEquations of Neutral Type
(5)The function G = GI
99
+ Gz satisfies
fort 2 u, 4 E C,where n ( t ) is continuous and bounded with a bound MO such that
Then every solution of (4.3.7) with initial data (u,4)satisfies
for some To, u 2 is (EVUASL).
TO.As a consequence the trivial solution of (4.3.7)
For the proof consult [l,p. 8691. The above result deals with the linear case f ( t , 6 ) = A ( t , 4). A similar result holds for general nonlinear f . Indeed, we have the following:
Theorem 4.3.4 Assume that (4 Is(t,4)I INll6l1, I f ( t , 6 ) - f(t,$)I 5 Nll4 - $11, v 6,$ E t E [T, w) and D is uniformly stable. (ii) The trivial solution of (4.3.6) is uniformly asymptotically stable. (iii) The function F in (4.3.7) is such that F = Fl F2 F3 where
c,
+ +
for some To,where
with c,[ sufficiently small and for u large. Also
M(t)=
m(s)ds -+ 0 as t
Then there exists some T such that if u
> T,
-+
w.
100
Stability and Time-Optimal Control of Hereditary Systems
Thus the solution x = 0 of (4.3.7) is (EVWUASL). A proofis indicated in [I,p. 8721. Example 4.3.1: Delay Logistic Equation of Neutral Type [5]. Consider the system
where h l , h2, I(, c, r are constants 2 0, with r, ha, K conditions of the type
> 0. We assume initial
If we let N(t)
1q1 + z ( t ) ] ,
+
A(t)= r[l ~ ( t ) ] , B ( t ) = 2ckA(t)1[1 k 2 k 2 ( t- h z ) ] ,
+
(4.3.9) is equivalent to d
dt
LX(') h, t
-
A(s + hl).(s)ds] = -A(t
+ h l ) + B ( t ) k ( t - hz). (4.3.10)
Set
= ( 1 + c ) exp[r( 1 + c ) h J b* = (1 - c)exp[hlr(I - c)]
u*
2cka' b* exp(-u*) ru*
+ 2ckru*hl+
rb* 2(2cku*r)2 exp(-a*)
1
.
b* exp(-a')
Global Stability of Functional Differential Equations of Neutral Type
101
For this we use the Lyapunov functional
V ( t ) = V ( t ,+ ) I , A(s + hl)z(s)ds]
+ 2 ft - h
B(s A(s
2
+
l-h + t
B A(s ( s + h2)t2(s)ds
h 2 ) A 2 (+~h1)2z2(s)ds + hl + h2) +
+
and calculate the upper right derivative
(4.3.10).
D+V of dt
V along solutions of
Proposition 4.3.1 Assume that
E (0, 00); h2 E Pl > 0, P2 > 01 r, hl, k
2ckra*
10, 001,
< 1; 0 < c < 1.
(4.3.11) (4.3.12) (4.3.13)
Then every positive solution of (4.3.9) satisfies
N ( t ) + O as t + m .
Proof: We need only show that if (4.3.11) - (4.3.12) hold, an arbitrary solution of (4.3.10) satisfies t(2) - 0
as t
+ 00.
(4.3.14)
The needed details are contained in [5].
4.4 Perturbations of Nonlinear Neutral Systems [6] Consider a nonlinear system that is more general than (4.1.3), namely
d (4.4.1) -D(t, .*I = f(t,.t> + 9(t,.t). dt We assume the following as basic: Let A C C be an open set, 11 C E an open interval, P = 11 x A. Denote by En' the set of n x n real matrices. Suppose D , f : r + En are continuous functions. We assume also that 0 E A. The main result in this section is stated as follows:
102
Stability and Time-Optimal Control of Hereditary Systems
Theorem 4.4.1 Assume (i) D ,f in (4.4.1) satisfy the following hypothesis: (a) The Fre'chet derivatives of D ,f with respect to I#, denoted by D , , f, respectively exist and are continuous on I? as well as D,,, and (b) for each ( t , 4 ) E I?, 11, E C write
J
~ + ( 4>[41= t, ~ ( 4t , ~ 0 -)
0
-h
dep(t,4, e)[+(e)l
for some functions
A : r -+E"', p : r x [-h,o]
-+
E"'
with A continuous, p ( t , 4, .) of bounded variation on [-A, 01 and such that the map ( t ,4) -+ f hd e p ( t ,4, 0)[11,(0)] is continued for each $j E
c.
(b) Assume for each ( t , d ) E compact sets I< C I?, i.e., det A ( t , 4)
IL
r
D is uniformly atomic at zero on
#0
d e p ( t , ~ , e ) ~ ( d )!(S)II+II ~l<
E [O, h] for all ( t ,4) E I< for some function t : [0, h] + [0, oo) that is continuous and nonincreasing, !(O) = 0. (ii) g is Lipschitzian in 4 on compact subsets of r. (iii) j ( t ,0,O) = 0 for all t E I. (iv) For each (a,4) E r, we have s
a,4111 I e x p ( 4 t )
+ b(a)), t L
6,
where a ( t ) , b(t) are continuous functions from E+ = [0, oo) into E+, i.e., are elements of C(E+,E+). Here T ( t ,a;4) is the solution operator associated with the variational equation
d -D,(t, dt
t t ( U , 4))[ZtI
(v) For each ( a , d )E I',
= f#(t,%(a,4))[.tl,
t E [a, + a).
(4.4.2)
Global Stability of Functional Daflerential Equations of Neutral Type
103
+
where ri E (0,1], c j ( t ) E C(E+, E + ) , j = 1,2,.. . , N 1. (v;) cj(S)eria(8)+*(d), c$)+le*(s) E ~ 1 ( T , u )j, = I,. . . ,N. (vii) a ( t ) +. -00 as t 00. Then every solution of (4.1.1) satisfies IlZt(u1
4)ll
- -t
00.
The proof is contained in Chukwu and Simpson in [6, p. 571. It uses the nonlinear variation of parameter equation
Ju
(4.4.3)
where y t ( u , 4) is the solution of
d
-dtW , Y : ) = f ( t , Y : ) .
(4.4.4)
This formula (4.4.3) holds when the solution of (4.4.1) is unique.
REFEREN cES 1. E. N. Chukwu, “Global Asymptotic Behavior of Functional Differential Equations of the Neutral Type,” Nonlinear Analysis Theory Methods and Applications 5 (1981)853472. 2. M.A. Cruz and J. K. Hale, “Stability of F‘unctional Differential Equations of Neutral Type,” J . Differential Equations 7 (1970)334-355. 3. J. K.Hale, Ordinary Differential Equations, Interscience, New York, 1969.
K. Hale, “Theory of Functional Differential Equations,” Applied Mathematical Sciences 3,Springer-Verlag, New York, 1977. 5. K. Gopalsamy, “Global Attractivity of Neutral Logistic Equations,” in Differential Equations and Applications, edited by A, R. Aftabizadeh, Ohio University Press, 4. J.
Athens, 1989. 6.
E. N. Chukwu and H. C. Simpson, “Perturbations of Nonlinear Systems of Neutral Type,” J . Differential Equations 82 (1989)28-59.