Exponential Stability of
Nonlinear Dierential Delay Equations
X. Mao
Department of Statistics and Modelling Science
University of Strathclyde
Glasgow G1 1XH, Scotland, U.K.
Abstract: The aim of this paper is to investigate the exponential stability of a nonlinear dier-
ential delay equation x_ (t)=f (t;x(t);x(t? )). Introduce the corresponding dierential equation
(without delay) y_ (t)=f (t;y(t);y(t)) and assume it is exponentially stable. It will be shown in
this paper that the dierential delay equation will remain exponentially stable provided the time
lag is small enough.
Key Words: exponential stability, nonlinear dierential delay equations, Gronwall's inequality.
1. Introduction
Many authors (cf. [1{8]) have recently devoted their interests to the asymptotic
stability of linear dierential delay equations since delay systems appear frequently
in science, engineering, physics, biology, economics etc. To explain the mathematical techniques used in the study of the asymptotic stability, let us consider a linear
dierential delay equation
x_ (t) = Ax(t) + Bx(t ? ):
(1:1)
Although there are several methods to study the asymptotic stability of equation (1.1),
we shall only mention two here. One of them is to regard equation (1.1) as the perturbed
system of the corresponding linear dierential equation
y_ (t) = Ay(t);
(1:2)
and then to show equation (1.1) is asymptotically stable provided so is equation (1.2)
and B is small enough. The results obtained in this method on the asymptotic stability
are generally delay-independent, i.e. equation (1.1) is always asymptotically stable no
matter how large the time lag is (cf. [1, 2, 7]). Of course, it is not quite practical to ask
a delay system to have such property, although it is easier to deal with mathematically.
The another method to study the asymptotic stability of equation (1.1) is to re-write
the equation as
x_ (t) = (A + B )x(t) ? B (x(t) ? x(t ? ))
and regard it as the perturbed system of the following equation
y_ (t) = (A + B )y(t):
Partially supported by the Royal Society and the LMS.
1
(1:3)
Obviously, if the time lag is small enough then the perturbation term B (x(t)?x(t? ))
could be so small that the perturbed equation (1.1) would behave in a similar way as
equation (1.3) asymptotically. For example, if equation (1.3) is asymptotically stable
and the time lag is small enough, then equation (1.1) will remain asymptotically
stable. This fact has been proved by several authors recently (cf. [3{6, 8]). Especially,
in 1994 Su, Fong and Tseng [5, 6, 8] obtained several useful criteria for the asymptotic
stability of the linear delay equation.
In the direction of the second method, so far there are few results on the asymptotic
stability of a nonlinear dierential delay equation
x_ (t) = f (t; x(t); x(t ? )):
(1:4)
However, in the same way as above one can re-write this equation as
x_ (t) = f (t; x(t); x(t)) ? [f (t; x(t); x(t)) ? f (t; x(t); x(t ? ))]
and regard it as the perturbed system of the following non-delay equation
y_ (t) = f (t; y(t); y(t)):
(1:5)
It is reasonable to hope that the delay equation (1.4) would behave similarly as equation
(1.5) provided the time lag is small enough. The main aim of this paper it to show
that if equation (1.5) is exponentially stable and the time lag is small enough then
the nonlinear delay equation (1.4) still remains exponentially stable.
2. Main Results
Consider an n-dimensional ordinary dierential delay equation of the form
(
x_ (t) = f (t; x(t); x(t ? )); t > to ;
x(t) = '(t ? to ); to ? t to :
(2:1)
Here f : R+ Rn Rn ! Rn is a continuous function, to 0; > 0 and ' = f'(s) :
? s 0g 2 C ([?; 0]; Rn). Introduce a standing hypothesis:
(H1) There exists a pair of positive constants 1 and 2 such that
jf (t; x; y) ? f (t; x; y)j 1jx ? xj + 2 jy ? yj
for all t 0 and x; y; x; y 2 Rn. Moreover, f (t; 0; 0) 0.
It is well-known that under the standing hypothesis (H1) equation (2.1) has a unique
solution denoted by x(t; to; ') on t to ? . Moreover the equation admits a trivial
solution x(t; to; 0) 0.
Introduce the corresponding ordinary dierential equation (without delay)
(
y_ (t) = f (t; y(t); y(t));
y(to) = '(0):
2
t > to ;
(2:2)
It is again well-known that under the standing hypothesis (H1) equation (2.2) has a
unique solution denoted by y(t; to; '(0)) on t to . Moreover the equation admits
a trivial solution y(t; to; 0) 0. For the purpose of this paper we propose another
standing hypothesis:
(H2) Equation (2.2) is exponentially stable. That is, there exists a pair of positive
constants and such that
jy(t; to; '(0))j j'(0)je?(t?to )
for all to 0 and '(0) 2 Rn .
on t to
The main aim of this paper is to show that under (H1) and (H2) equation (2.1)
remains exponentially stable provided is small enough. This is described as follows.
Theorem 2.1 Let (H1) and (H2) hold. Then equation (2.1) is exponentially stable
provided < (=2) ^ , where = log(=0:9)=(0:9 ) and > 0 is the unique root to
the following equation
0:9e + [( + ) + ]e2(1 +2) 1
2
2
(=0:9)1=9h 2
i
exp 2 (1 + 2)e2(1 +2) = 1:
(2:3)
Proof. We divide the whole proof into two steps.
Step 1. Fix the initial data to and ' = f'(s) : ? s 0g arbitrarily. Write
x(t; to; ') = x(t) and y(t; to; '(0)) = y(t) simply. By hypothesis (H1), it follows from
equations (2.1) and (2.2) immediately that for all t to
jx(t) ? y(t)j (1 + 2 )
Z t
to
jx(s) ? y(s)jds + 2
Z t
to
jx(s) ? x(s ? )jds:
The well-known Gronwall inequality yields that
jx(t) ? y(t)j 2
Hence
e(1 +2 )(t?to )
jx(t)j jy(t)j + 2
e2(1 +2 )
Z t
to
Z t
to
jx(s) ? x(s ? )jds:
jx(s) ? x(s ? )jds
(2:4)
if to t to + 2. On the other hand, if t to + we can derive from equation (2.1)
that Z t
Z t Z s
jx(s) ? x(s ? )jds 1 jx(r)j + 2jx(r ? )j drds:
to + s?
to +
By changing the order of integrations one sees
Z t
Z s
to + s?
1jx(r)jdrds 1
3
Z t
to
jx(r)jdr
and
Z t
Z s
to + s?
2 2 jx(r ? )jdrds 2
Z t
to
jx(r)jdr + 2 2
Consequently,
Z t
to +
jx(s) ? x(s ? )jds (1 + 2 )
Z t
Z t
to
jx(r ? )jdr
sup
to ? sto
jx(s)j :
jx(r)jdr + 2 2
to
sup
to ? sto
jx(s)j : (2:5)
if t to + . We now restrict to ? + t to ? + 2. Substituting (2.5) into (2.4)
and using hypothesis (H2) one sees that
Z t
jx(t)j j'(0)je?(? ) + 2 (1 + 2)e2(1+2 )
+ (2 )2 e2(1+2 )
to
sup
to ? sto
jx(s)j :
jx(s)jds
(2:6)
Note from the denition of that
e?(? ) = (=0:09:e9)1=9 :
Also
Z t
to
jx(s)jds sup
to sto ? +
jx(s)j +
Z t
to ? +
Substituting these into (2.6) one obtains that
jx(t)j 2 (1 + 2
)e2(1 +2)
Z t
jx(s)jds:
jx(s)jds
to ? +
0:9e
2
(
+
)
1
2
+
+ [(1 + 2) + 2 ]e
sup
jx(s)j
(=0:9)1=9 2
to ? sto ? +
By the Gronwall inequality, one sees that
jx(t)j C1
sup
to ? sto ? +
jx(s)j
(2:7)
for to ? + t to ? + 2, where
C1 = (=0:09:e9)1=9 + 2 [(1 + 2) + 2 ]e2(1+2 )
i
h
exp 2 (1 + 2)e2(1+2 ) :
Note that C1 < 1 from (2.3), since < . Write C1 = e?" with " = ? log C1=. It
then follows from (2.7) that
sup
to ? +tto ? +2
jx(t; to; ')j e?"
4
sup
to ? tto ? +
jx(t; to; ')j
(2:8)
holds for any to 0 and ' 2 C ([?; 0]; Rn).
Step 2. Fix to 0 and ' 2 C ([?; 0]; Rn) arbitrarily, and let k = 1; 2; . Denote
x^(to + (k ? 1); to ; ') = fx(to + (k ? 1) + s; to; ') : ? s 0g
which is regarded as a continuous function in C ([?; 0]; Rn). Note that the solution
has the following ow property
x(t; to ; ') = x(t; to + (k ? 1); x^(to + (k ? 1); to; '))
for any t to + (k ? 1). Hence, by (2.8),
sup
jx(t; to; ')j
=
to ? +ktto ? +(k+1)
sup
jx(t; to + (k ? 1); x^(to + (k ? 1); to; '))j
to +(k?1)? +tto +(k?1)? +2
?
"
e
sup
jx(t; to; ')j :
to ? +(k?1)tto ? +k
By induction
sup
to ? +ktto ? +(k+1)
jx(t; to; ')j e?"k
sup
to ? tto ? +
jx(t; to; ')j :
(2:9)
On the other hand, it is not dicult to show that there exists a C2 > 0 such that
sup
to ? tto ? +
? s0
Substituting this into (2.9) yields
sup
jx(t; to; ')j C2 sup j'(s)j :
to ? +ktto ? +(k+1)
jx(t; to; ')j C2 e?"k sup j'(s)j :
? s0
(2:10)
Now for any t > to ? + , one can nd a k such that to ? + k t to ? +(k +1)
and hence
jx(t; to; ')j C2 e"?"(t?to ) sup j'(s)j :
? s0
But this holds for any to t to ? + as well. In other words, we have shown that
equation (2.1) is exponentially stable. The proof is now complete.
In Theorem 2.1, is chosen such that e?0:9 = 0:9, and it is not optimal of
course. To get a better result, one may choose two free parameters 1 ; 2 2 (0; 1) and
let = log(=2 )=(1 ), i.e. e?1 = 2. In this case,
e?(? ) =
and C1 becomes
2 e
(=2 )(1?1)=1
2 e
]e2(1+2 )
C1 = (= )(1?1 )=1 + 2 [(1 + 2) + 2
2
h
i
exp 2 (1 + 2)e2(1+2 ) :
5
So if is chosen so small that C1 < 1 then equation (2.1) is exponentially stable. That
is, we have established the following more general result.
Theorem 2.2 Let (H1) and (H2) hold. Let 1; 2 2 (0; 1) be two free parameters. Then
equation (2.1) is exponentially stable provided < (=2)^ , where = log(=2)=(1 )
and > 0 is the unique root to the following equation
2 e 2(1 +2 )
+
[
(
+
)
+
]
e
1
2
2
(=2 )(1?h1)=1 2
i
exp 2 (1 + 2 )e2(1+2 ) = 1:
(2:11)
Obviously Theorem 2.1 is a special case of Theorem 2.2 when 1 = 2 = 0:9. It is
open what the optimal parameters 1 and 2 are.
The above results are established in the case when the time lag is a constant,
but it is easy to see that these results still hold when the time lag is a variable of
t. More precisely, let : R+ ! [0; ] be a Borel measurable function, where > 0.
Consider a dierential delay equation
(
x_ (t) = f (t; x(t); x(t ? (t))); t > to ;
x(t) = '(t ? to ); to ? t to ;
(2:12)
where f is the same as before and ' = f'(s) : ? s 0g 2 C ([?; 0]; Rn). For this
delay equation we have the following result.
Theorem 2.3 Let (H1) and (H2) hold. Let 1 ; 2 2 (0; 1) be two free parameters.
Then equation (2.12) is exponentially stable provided
sup (t) < (=2) ^ ;
t0
where and are the same as dened in Theorem 2.2.
3. Examples
We give here two simple examples for illustration.
Example 3.1 Consider a one-dimensional dierential delay equation
(
x_ (t) = sin(x(t)) ? 2x(t ? ); t > to ;
(3:1)
x(t) = '(t ? to ); to ? t to ;
where ' = f'(s) : s 0g 2 C ([?; 0]; R). The corresponding dierential equation
has the form
(
y_ (t) = sin(y(t)) ? 2y(t);
y(to) = '(0):
t > to ;
(3:2)
It is easy to check that the solution of equation (3.2), denoted by y(t; to; '(0)), satises
jy(t; to; '(0))j j'(0)je?(t?to ) ; t to :
6
Hence one sees easily that the standing hypotheses (H1) and (H2) are satised with
1 = 1; 2 = 2; = 1; = 1:
To apply Theorem 2.1 we calculate = log(1=0:9)=0:9 = 0:117. Also equation (2.3)
becomes
?
0:89e + 8:072 (0:1755 + ) e1:417 = 1
(3:3)
which has the root = 0:028. Therefore, by Theorem 2.1, the delay equation (3.1) is
exponentially stable provided < 0:028.
Example 3.2 Consider a two-dimensional dierential delay equation
(
x_ 1 (t) = ?2x1 (t) + sin(x2 (t ? ))
x_ 2 (t) = ? sin(x1 (t)) ? 2x2(t ? )
(3:4)
on t > to with initial data x(t) = (x1(t); x2(t))T = '(t ? to ) on to ? t to , where
' = f'(s) : s 0g 2 C ([?; 0]; R2). The corresponding dierential equation has
the form
(
y_1 (t) = ?2y1 (t) + sin(y2(t))
(3:5)
y_2 (t) = ? sin(y1(t)) ? 2y2(t)
on t > to with initial value y(to) = (y1(to ); y2(to ))T = '(0). It is easy to check that
the solution of equation (3.5), denoted by y(t; to; '(0)), satises
jy(t; to; '(0))j j'(0)je?(t?to ) ;
t to :
Hence one sees easily that the standing hypotheses (H1) and (H2) are satised with
p
1 = 2 = 5; = 1; = 1:
To apply Theorem 2.1, compute = 0:117. Also equation (2.3) becomes
?
0:9995e + 14:238 (0:234 + ) e3:33 = 1
(3:6)
and it has the root = 0:0143. By Theorem 2.1, one sees that the delay equation
(3.4) is exponentially stable provided < 0:0143. We now show that one can apply
Theorem 2.2 to improve this result. To do so, choose 1 = 0:8 and 2 = 0:95. Then
= 0:0641 and equation (2.11) becomes
?
0:938e + 8:871 (0:1282 + ) e1:138 = 1;
(3:7)
which has the root = 0:0184. Thus, by Theorem 2.2, one can now conclude that the
delay equation (3.4) is exponentially stable provided < 0:0184. This improves the
previous result by 28.7%.
Acknowledgement
The author would like to thank the anonymous referee for his helpful suggestions.
7
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