Pergamon
Appl. Math. Left. Vol. 10, No. 3, pp. 111-115, 1997
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N e c e s s a r y a n d Sufficient C o n d i t i o n s
of Asymptotic Mean Square Stability
for S t o c h a s t i c L i n e a r D i f f e r e n c e E q u a t i o n s
L. SHAIKHET
Department of Mathematics, Informatics and Computering
Donetsk State Academy of Management, Chelyuskintsev str., 163-A
Donetsk 340015, Ukraine
leonid@dsam, donetsk, ua
(Received May 1996; accepted August 1996)
Communicated by R. P. Agarwal
Abstract--Many
processes in automatic regulation, physics, mechanics, biology, economy, ecology, etc. can be modelled by hereditary systems (see, e.g., [1-4]). One of the main problems for the
theory of such systems and their applications is connected with stability (see, e.g., [2-4]). M a n y stability results were obtained by the construction of appropriate Lyapunov functionals. At present, the
method is proposed which allows us, in some sense, to formalize the procedure of the corresponding
Lyapunov functionals construction [5-10]. In this work, by virtue of the proposed procedure, the
necessary and sufficientconditions of asymptotic mean square stabilityfor stochastic lineardifference
equations are obtained.
1. M A I N R E S U L T
Consider the scalar difference equation
k
Xi+I = E a j x i - j "~ °'xi-l~i'
j=o
xi = ~oi,
iEZ,
(1)
i E Zo.
Here i is the discrete time, i E Z U Zo, Zo = { - h , . . . , 0}, Z = {0, 1 . . . . }, h = max(k, l).
Let {12, a, P } be a probability space, {fi e a}, i E Z, be a sequence of a-algebras, ~0, ~1,...
be mutually independent random values, ~i be f i + r a d a p t e d and independent from fi, E~i = 0,
Eel = 1.
DEFINITION. Zero solution of equation (I) is cMled mean square stable if for every e > 0 there
exists a 5 > 0 such that Ex~ < e, i 6 Z, if [[~o[[2 = supiez o E~o2 < 5. If, besides, l i m i _ ~ E x 2 = 0,
then equation (1) zero solution is called asymptotic mean square stable.
THEOREM 1. [9]. Let there exist the nonnegative functional V~ = V(i, X - h , . . . , xi), i E Z, for
which the conditions
EV(0,x_h,... ,x0) _< ciH~lT2,
EAVi
_~ - c 2 E x 2,
iEZ,
Typeset by AA,~S-TEX
IIi
112
L. SHAIKHET
where AVi = V/.{_1
square stable.
-- Vi, c1 > 0, c2 >
0 hold. Then equation (1) zero solution is asymptotic mean
( 10 0/
Consider the vectors x(i) = ( x i - k , . . . , xi-1, x~) ~ and b = ( 0 , . . . , a) ~ of dimension k + 1 and the
square matrix
0
A=
1
...
0
. . . . . . . . . . . . . . .
ak
0
0
ak-1
ak-2
...
1
...
ao
Then equation (1) can be described in the form
x(i + 1) = Ax(i) -t- bxi-~i.
(2)
A'DA - D = -U,
(3)
Consider the matrix equation
in which the square matrix U = [[uij [[ of dimension k + 1 has all zero elements except for
?~k-t-l,k%l :
1.
THEOREM 2. Let equation (3) have a positive semidefinite solution D. Then, for asymptotic
mean square stability of equation (1) zero solution, it is necessary and sufficient that the inequality
(4)
cT2dk+l,k+l < 1
hold.
PROOF. Consider the functional
l
Vi = x'(i)Dx(i) + 0 " 2 d k + l , k + l ~
X S--j"
2. •
(5)
j--1
Calculating EAVi by virtue of (5),(2), we obtain
[
EAV~ = E ~'(i + 1)D~(i + 1) + ~2dk+~,k+~ ~
' ~+l-j -- ~'(i)D~(i)
X2
_
~r2d,~+~,k+i ~
j----1
= ~,[(Ax(i) +
, ~_j]
X2
j----I
bx~_~,)'D(Ax(i) + bx~_~) - x'(i)Dx(i)]+ a2dk+l,k+W, (X~ -- X2i-b)
= E [x'(i)(A'DA - D)x(i) + b'Dbx2_l] + a2dk+l,k+lE (x~ -- x~_t)
= (a2dk+l,k+l -- I) Ex~.
Let condition (4) hold. Then the functional (5) satisfiesthe conditions of Theorem 1. It means
that equation (1) zero solution is asymptotic mean square stable. It follows that condition (4) is
sufficient for asymptotic mean square stability of equation (1) zero solution.
Let condition (4) not hold, i.e.,a2dk+1,k+1 > 1. Then E A V ~ _> 0. From here it follows that
i-1
E~Vj = EV~ - Ego > 0,
j=O
i.e., E ~ _> EV0 > 0. It means that equation (1) zero solution cannot be mean square stable.
Therefore, condition (4) is necessary for asymptotic mean square stability of equation (1) zero
solution. Theorem 2 is proved.
113
Necessary and Sufficient Conditions
2. E X A M P L E S
R e m a r k t h a t for e v e r y k, e q u a t i o n (3) is t h e s y s t e m of (k + 1)(k + 2 ) / 2 e q u a t i o n s . C o n s i d e r
t h e different p a r t i c u l a r cases o f e q u a t i o n (3).
EXAMPLE 1. L e t k = 0, a0 = a. I n t h i s case, e q u a t i o n (3) has t h e form d l l ( a 2 - 1) = - 1 . T h e
n e c e s s a r y a n d sufficient c o n d i t i o n of a s y m p t o t i c m e a n s q u a r e s t a b i l i t y is
1
- a1- _- -~-
0<dll
<
or_ 2
or a 2 + ~2 < 1. If a = 0, t h i s c o n d i t i o n t a k e s t h e form lal < 1.
EXAMPLE 2. L e t k = 1, a0 = a, al = b. I n this case, e q u a t i o n (3) is t h e s y s t e m of e q u a t i o n s
b2d22 - d l l = 0,
(b - 1)d12 + abd22 = O,
d l l + 2ad12 + (a 2 - 1)d22 = - 1 .
Solving t h i s s y s t e m , we o b t a i n t h e n e c e s s a r y a n d sufficient c o n d i t i o n of a s y m p t o t i c m e a n s q u a r e
stability
0 < d22 =
1- b
< 0._2.
(1 + b)((1 - b) 2 - a 2)
I f a = 0, t h i s c o n d i t i o n h a s t h e form Ibl < 1, lal < 1 - b. In F i g u r e 1, t h e region o f s t a b i l i t y is
s h o w n b y (1) a 2 = 0, (2) a 2 = 0.4, (3) a 2 = 0.8.
b*
I
I
I
I
I
p
1
.....
I
i
I
i
I
i
I
I
I
!
Figure 1.
EXAMPLE 3. Let k = 2, ao = a, a l = 0, a2 = b. I n t h i s case, e q u a t i o n (3) is t h e s y s t e m of
equations
b2d33 - d l l -- O,
bdx3 - d12 -- O,
d12 + ad13
-
d23 = O,
d l l - d22 -- O,
bd~3 + abd33 - d13 = O,
d22 + 2ad23 + (a 2 - 1)dz3 = - 1 .
Solving t h i s s y s t e m , we o b t a i n t h e n e c e s s a r y a n d sufficient c o n d i t i o n o f a s y m p t o t i c m e a n s q u a r e
stability
(
0<das=
-1
1-b2-a
-1--(a+b)bJ
<a-2"
I n F i g u r e 2, t h e region o f s t a b i l i t y is shown b y (1) a 2 = 0, (2) a z = 0.4, (3) a 2 = 0.8.
114
L. SHAIKHET
b$
I
I
I
I
I
i
$
i
I
I
I
I
(
i
l
I
Figure 2.
b~
I
i
t
i
J
I
I
I
t
i
I
i
I
i
I
I
Figure 3.
EXAMPLE 4. Let k = 2, a0 = a, al -- b, a2 ~- b2. In this case, equation (3) is the s y s t e m of
equations
b4d33 - dll = 0,
b2d13+b3dz3-d12=O,
b2d23 ~- ab~d33 - d13 ~- 0,
dll+2bd13+b2dzz-d22--O,
d12 -~ ad13 + abd33 + (b - 1)d23 = 0,
d22+2ad23+(a2-1)d33 = -1.
Solving this system, we obtain the necessary and sufficient condition of asymptotic mean square
Necessary and Sufficient Conditions
115
stability
I n F i g u r e 3, t h e r e g i o n o f s t a b i l i t y is s h o w n b y (1) a 2 = 0, (2) 0.2 _ 0.4, (3) 0.2 : 0.8.
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