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Conditions for the moment stability of linear stochastic systems
Systems & Control Letters 20 (1993) 227-232 North-Holland
227
Conditions for the moment stability of linear stochastic systems R o m a n V. B o b r y k Institute for Applied Problems of Mechanics and Mathematics of Ukrainian Academy of Sciences, Lviv, Ukraine Received 23 May 1992 Revised 24 September 1992
Abstract: Sufficient conditions for the moment stability of linear differential stochastic equations with Ornstein-Uhlenbeck parameters are obtained on the base of moment stability theory with white noise parameters.
Keywords: Stochastic differential equations; moment stability; Ornstein-Uhlenbeck process.
I. Introduction
Consider linear dynamic systems with stochastic parameters described by the linear differential equation in R n, dx(/) dt = A x ( / )
rn + E i=1
~i(t)Cix(t),
t > 0,
(1)
where A, CI .... , C m are constant square matrices of order n and coloured noises ~l(t) . . . . . ~,,(t) are independent zero-mean Ornstein-Uhlenbeck processes with correlation function
B(t) =E[~i(tq-s)~i(s)]
=~"
exp{-~-It[},
~->0,
(2)
where E denotes the expected value. It is known [2,10] that the moments of the solution of equation (1) converge to the solution of the It6 equation while ~-~ 0, dx(t) = (A+
~ Ci2)x(t)dt+ i=l
v~- ~ Cix(t ) dwi(t ),
(3)
i=1
where Wl(t) .... , Win(t) are independent Wiener processes. The stability investigation for equation (3) is more easy to work with than for equation (1). In this paper we deal with the following stability definitions [8,9]. Definitions. The null solution of a stochastic dynamic system is called p-th moment stable, where p is a positive integer, if for all e > 0 there exists a/~ > 0 such that II x(0)II < ~ implies E [ X l ( t ) pl " " x n ( t ) p"] < e
for t > 0 and for any set of nonnegative integers Pl . . . . . p, such that Pl + "'" +Pn =P. Here LI IL denotes Euclidean norm. Correspondence to: R.V. Bobryk, Institute for Applied Problems of Mechanics and Mathematics of Ukrainian Academy of Sciences, 3b Naukova St., Lviv, Ukraine. 0167-6911/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
R.V. Bobryk / Moment stability of linear stochastic systems
228
Note that the mean-square stability (p = 2) is most often used. If the moments E[xl(t)m ... xn(t)~'o] converge exponentially to zero while t ~ oo, then the null solution of a stochastic dynamic system is called p-th moment exponentially stable. Necessary and sufficient conditions for the moment stability of equation (3) were obtained [3,4,6,8] since the moments of the solution of equation (3) satisfy a linear ordinary differential equation [3,4]. In the scalar case (n = 1) one can obtain explicit formulas for all moments of the solution of equation (1). It follows from these formulas that the p-th moment exponential stability of the null solution of equation (1) is caused by the p-th moment stability of the null solution of equation (3). This result is true when the Lie algebra generated by the matrices A, C 1.... , C m is solvable [9]. In this paper we shall show that this fact has place when the matrices A, C l . . . . . C m are symmetric. Here this problem is studied also for the mean-square stability of the random harmonic oscillator. It is one of the prototypical systems of stochastic stability and it was treated by numerous people [1,6-8].
2. Main result
Theorem. Assume the matrices A, C~ .... , C,~ in equation (1) are symmetric, and the Ornstein-Uhlenbeck processes ~l(t),..., ~,n(t ) are independent. I f the null solution of equation (3)/s p-th moment exponentially stable, then the null solution of equation (1)/s also p-th moment exponentially stable. Proof. For notational simplicity, we shall only treat the problem for m = 1. Let ~l(t) = so(t), C~ =- C and consider the case p = 1. We write equation (1) in the integral form (4)
x ( t ) = fotAX(t,) dt, + f o C ~ ( t l ) X ( t l ) dt 1 + x ( 0 ) .
The solution of equation (4) is a functional with respect to process ~:(t). It follows from convergence of the iterative method for the solution of equation (4) that the variational derivatives
6kx(t) ~(Sl)
"'" ~ ( S k )
exist, and also
~kx(t)
-0
6~(Sl) ''' ~(Sk)
if there exists an i = 1. . . . . k, with s i > i, k = 1, 2, 3 , . . . . By (4), these variational derivatives satisfy the equations
6kx(t) t t~kx(tl) rt . 6kx(tl) 3~(Sl) ... 3~(Sk) = fo A 6sO(S1)''- 6~¢(Sk)dtl + JoC~(tl) 6~(sl) . . . 3~(sk) dtl k
+ Y'. CO(t --Si) i=1
k = l , 2, 3 . . . . ~(S1)
"'" ~(Si-l)~(Si-1)
"'" ~ ( S k )
'
'
(5) where O(t) is the Heaviside unit function.
R.V. Bobryk / Moment stability of linear stochastic systems
229
The formulas
E[ ~( t ) x( t ) ] = f/B( t - s ) E-~-~ds,6tX()
E ~(t)
~(S1)
~k+lx(t)
t~kx( t ) .. " ~(Sk)
= B(t-Sk+l)E(sl)
~(Sk+I)
...
ds~+ 1,
k = 1, 2, 3,.. ",
follow from [5] because the process ~:(t) is Gaussian. Applying these formulas to equation (1) we obtain under (5) the infinite chain of integro-differential equations
dEx(t) dt
AEx(t) + f/B(t -s)CE ~x(t) ~--~ds,
6xk(t)
=A fiE
dq
~kx(tl)
E ~ ( s 1 ) "'" ~ ( S k )
~ ( s 1 ) "'" ~ ( S k )
+
~xk(tl) ft ~tjoJoln(tx _ S k + l ) f E
dSk+ldtl
(S1) "'" ~(Sk+l)
k
~k-lx(si )
+ E O(t-si)Ce i=1
k = l , 2, 3,..
~ ( S 1 ) "'" ~ ( S i - 1 ) ~ ( S i + l )
"'" ~ ( S k )
'
Let Yo( t ) = Ex( t ),
l, Yk(t)= f / ' ' ' f/exp '
E
)
~kx(t) e
k = 1, 2, 3 , . . . . 8
(s1)
• • •
'
Then one can reduce this chain to the chain of the linear ordinary differential equations dYo(t)
dt
Ay°(t) + ~'CYl(t)'
dYk(t)
dt
Y°(O) =x(O),
kzyk+l(t)+AYk(t)+~'CYk+s(t)+kCYk_l(t),
Yk(0)= 0,
k=1,2,3
.....
(6)
We show that chain (6) has a unique solution. Let us introduce a Banach space F of infinite sequences of functions
g= (gk(t),
k=0,1 ....
:supsup{exp{--trt}dkl[gk(t) ll
t>0
with the norm II g II = sup sup {exp{ -trt}d k
k II gk(t)II},
t>0
where d, o- are positive parameters to be determined. After multiplying the equations for Yk(t), k = 1, 2, 3 , . . . , with exp{k~-t} and transfering them to the integral form we can rewrite the chain (6) in space F as the linear equation
y=Gy+f, where the operator G and f are determined by the right-hand side and the initial values of the chain.
R.V. Bobryk / Moment stability of linear stochastic systems
230
Then for the norm of operator G we have IIGII _<
(d,lA,[+(z+dk),[C,[ d(cr+k)
sup
k =0,1 ....
) "
It follows from this that there exist d > 0, ~r > 0 such that II G II < 1 and so chain (6) has a unique solution in F. It is well-known that there exists an orthogonal matrix U such that D = U(A + C2)U ~ is diagonal because matrix A + C 2 is symmetric. Introducing the change
Vk(t ) = ~ U y ~ ( t ) ,
k=0,
1. . . . .
into the chain (6), we can reduce it to the form dvo(t)
---Dvo(t dt
dvk(t)
) + ~ U C U - l v , ( t ) - VC2U-'vo(t), Dvk(t ) - k r u k ( t ) + ((k + I)zUCU
dt
(7a)
lvk+,(t ) - U C 2 U - ' v k ( t ) + ~ / ~ r U C U - l v k _ , ( t ) , (7b)
Vo(0) =
Ux(O),
vk(0)=0,
k=1,2,3,....
(7c)
Let ( , ) denote the scalar product in Euclidean space R n. Using the orthogonality of U and the symmetry of C, from (7) we obtain 1 d [I Vo(t) 11z 2
(Dvo(t), Vo(t)) - II CU-1uo(t)I12+ v/~(cU-lvo(l), U-iv,(/)),
dt
1 d II Vk(t)I[
2
(Dvk(t) , v k ( t ) )
dt
+ ({~
- kz [I Vk(t)II 2 - II
1)r ( U - l c k + l ( t ) ,
(8a)
cv-lUk(t)I1 2
CU-~vk(t))
+~(CU-IUk_l(t), U-Iuk(t)),
k = l , 2, 3 . . . . .
(Sb)
The mean of the solution of equation (3) satisfies the equation dz(t)
dt
- ( A + C2)z( t)"
Then the diagonal elements of matrix D are negative because the null solution of this equation is exponentially stable. So there exists a oh > 0 such that (DVk(t). v k(t))_< --crI I[vk(t)[[ 2. Taking into consideration the orthogonality of U we have [[U-lVk(t)[[ 2= [[vk(t)[[ 2, k = 0 . 1 . . . . . The series k=0 I[ v~(t)l[ converges because the solutions of chain (6) belong to F and so Tk
T k C 'rt
II Vk(t)II 2 = ~.T I[ Yk(t) [I 2 --
dkk------~..
Therefore by (6), the series ET=0 d II vk(t)II From (8) one can obtain oo
k =o
d II Vk(t) II 2 dt
2/dt
oo
converges as well. oo
2 E (Dvk(t),
Vk(t)) -- 2 E
k=0
k=0 c~
_<-2~r 1 ~ k--O
IIv/,(t) I12
IhCU-lvk(t)
- ~(k + 1)~ U-lVk+l(t)[]2
R.14.Bobryk / Momentstabilityof linearstochasticsystems
231
and we conclude under the Lyapunov theory of stability that the mean of the solution of equation (1) is exponentially decreasing. For the case p > 1 we introduce under [2,3] the binomial coefficient q(n, p) = (n +~-l) and the vector X[P](t) E R q(n'p) of all p-th forms of the components of the vector x(t):
x[P]( t) = col( xl( t) p, ,~PXI( t ) P - I x 2 ( t) . . . . . xn( t)P), and the scale parameters Ai are chosen so as to obtain II xtp](t)II = II x(t)II P. For vector xtPl(t) we have under (1) the equation
dxtPl( t ) A[p]Xtpl( t ) + ~( t )C[p]Xtp]( t )'
dt
where the matrices A[R], C[p] are determined from the matrices A, C [2,3]. The equation
dxtpl( t ) = (AIR] + C~p])xtpl( t ) at + ~f~ Ctp]xtpl( t ) dwtpl( t ) is associated with the equation (3) [2,3]. For the full proof of the theorem it is enough to note that the matrices A[p], C[p] are symmetric as well [2,3].
3. The harmonic oscillator with random damping
Let us consider the equation of the harmonic oscillator, d2v(t)
dv(t)
-------7~+(k+/3£(t)) dt
dt
+av(t)=O,
k,a>0.
(9)
Introducing the notation d v ( t ) ]2, ~-x(t)=col((
dt
]
dv(t)
~/2av(t)--~,av(
t)2) '
we can obtain for equation (1) vector x(t), where m = 1 and -2k A = 0
- 2v/2-a-
0
-k
-2¢~
~
0
c=
-2/3
0
o
-/3
0
0
i)
.
Let us assume that ~:(t) is the white noise in (9) and interpret this equation as a Stratonovich stochastic differential equation. Then writing the equation for the second moment of the solution of this equation one can show [1,6] under Hurwitz conditions that 2/32< k is the necessary and sufficient condition for mean-square stability of the null solution of this equation. Let ~(t) be the Ornstein-Uhlenbeck process in (9) with the correlation function (2). Then we show that this condition is sufficient for mean-square stability of the null solution of equation (9) for all ~- > 0, a>0.
232
R.V. Bobryk / Moment stability of linear stochastic systems
After substitution of zk(t) = ~~-" r k / k ! yk(t), k = 0, 1 . . . . . for yk(t) in chain (6) we obtain analogously to (8), 1 d [I Zo(t)II 2
2
dt
1 all z ~ ( / ) II 2 2 dt
- ( D , z o ( t ) , z 0 ( t ) ) - II Czo(t)II
~(Czo(t), z,(t)),
(Dlzk(t)' Zk(t))--krllzk(t)ll2--Ilfzk(t)ll2 + (~
D = diag(-2k
2+
+ 1)~" ( Z k + t ( t ) , C Z k ( t ) ) + f k ~ ( C z k _ , ( t ) ,
+ 4132 , - k
zk(t)),
k=1,2,3
.....
+ / 3 2 , 0),
b e c a u s e C is s y m m e t r i c a n d A is s k e w - s y m m e t r i c . If 2132 < k, t h e n
E k =0
dllzk(t)ll = dt -< E 2 ( O l Z k ( t ) , z k ( t ) ) - - <0 k=0
and under the L y a p u n o v theory of stability we conclude that the null solution of equation (9) is m e a n - s q u a r e stable.
References [1] L. Arnold, Stochastic Differential Equations." Theory and Applications (Wiley, New York, 1974). [2] G.L. Blankenship and G.C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbance, SIAM J. Appl. Math. 34 (1978) 437-476. [3] R.W. Brockett, Lie algebras and Lie groups in control theory, in: D.O. Mayne and R.W. Brockett, Eds., Geometric Methods in System Theory (Reidel, Dordrecht-Boston, 1973) 43-82. [4] R.W. Brockett, Parametrically stochastic differential equations, Math. Programming Study 5 (1976) 8-21. [5] M.D. Donsker, On function space integrals, in: Proc. Conf. Theory and Applic. Analysis in Funct. Space (MTI Press, Cambridge, MA, 1964) 17-30. [6] I.I. Gichman and A.V. Skorokhod, Stochastic Differential Equations (Springer-Verlag, Berlin, 1972). [7] E.R. Infante, On the stability of some linear nonautonomous random systems, ASME J. Appl. Mech. 35 (1968) 7-12. [8] F. Kozin, A survey of stability of stochastic systems, Automatica 5 (1969) 95-112. [9] J.L. Willems and D. Aeyels, An equivalence result for moment stability criteria for parametric stochastic systems and It6 equations, lnternat. J. Systems Sci. 7 (1976) 577-590. [10] E. Wong and M. Zakai, On the relation between ordinary and stochastic equations, Internat. Z Engng. Sci. 2 (1965) 213-229.
227
Conditions for the moment stability of linear stochastic systems R o m a n V. B o b r y k Institute for Applied Problems of Mechanics and Mathematics of Ukrainian Academy of Sciences, Lviv, Ukraine Received 23 May 1992 Revised 24 September 1992
Abstract: Sufficient conditions for the moment stability of linear differential stochastic equations with Ornstein-Uhlenbeck parameters are obtained on the base of moment stability theory with white noise parameters.
Keywords: Stochastic differential equations; moment stability; Ornstein-Uhlenbeck process.
I. Introduction
Consider linear dynamic systems with stochastic parameters described by the linear differential equation in R n, dx(/) dt = A x ( / )
rn + E i=1
~i(t)Cix(t),
t > 0,
(1)
where A, CI .... , C m are constant square matrices of order n and coloured noises ~l(t) . . . . . ~,,(t) are independent zero-mean Ornstein-Uhlenbeck processes with correlation function
B(t) =E[~i(tq-s)~i(s)]
=~"
exp{-~-It[},
~->0,
(2)
where E denotes the expected value. It is known [2,10] that the moments of the solution of equation (1) converge to the solution of the It6 equation while ~-~ 0, dx(t) = (A+
~ Ci2)x(t)dt+ i=l
v~- ~ Cix(t ) dwi(t ),
(3)
i=1
where Wl(t) .... , Win(t) are independent Wiener processes. The stability investigation for equation (3) is more easy to work with than for equation (1). In this paper we deal with the following stability definitions [8,9]. Definitions. The null solution of a stochastic dynamic system is called p-th moment stable, where p is a positive integer, if for all e > 0 there exists a/~ > 0 such that II x(0)II < ~ implies E [ X l ( t ) pl " " x n ( t ) p"] < e
for t > 0 and for any set of nonnegative integers Pl . . . . . p, such that Pl + "'" +Pn =P. Here LI IL denotes Euclidean norm. Correspondence to: R.V. Bobryk, Institute for Applied Problems of Mechanics and Mathematics of Ukrainian Academy of Sciences, 3b Naukova St., Lviv, Ukraine. 0167-6911/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
R.V. Bobryk / Moment stability of linear stochastic systems
228
Note that the mean-square stability (p = 2) is most often used. If the moments E[xl(t)m ... xn(t)~'o] converge exponentially to zero while t ~ oo, then the null solution of a stochastic dynamic system is called p-th moment exponentially stable. Necessary and sufficient conditions for the moment stability of equation (3) were obtained [3,4,6,8] since the moments of the solution of equation (3) satisfy a linear ordinary differential equation [3,4]. In the scalar case (n = 1) one can obtain explicit formulas for all moments of the solution of equation (1). It follows from these formulas that the p-th moment exponential stability of the null solution of equation (1) is caused by the p-th moment stability of the null solution of equation (3). This result is true when the Lie algebra generated by the matrices A, C 1.... , C m is solvable [9]. In this paper we shall show that this fact has place when the matrices A, C l . . . . . C m are symmetric. Here this problem is studied also for the mean-square stability of the random harmonic oscillator. It is one of the prototypical systems of stochastic stability and it was treated by numerous people [1,6-8].
2. Main result
Theorem. Assume the matrices A, C~ .... , C,~ in equation (1) are symmetric, and the Ornstein-Uhlenbeck processes ~l(t),..., ~,n(t ) are independent. I f the null solution of equation (3)/s p-th moment exponentially stable, then the null solution of equation (1)/s also p-th moment exponentially stable. Proof. For notational simplicity, we shall only treat the problem for m = 1. Let ~l(t) = so(t), C~ =- C and consider the case p = 1. We write equation (1) in the integral form (4)
x ( t ) = fotAX(t,) dt, + f o C ~ ( t l ) X ( t l ) dt 1 + x ( 0 ) .
The solution of equation (4) is a functional with respect to process ~:(t). It follows from convergence of the iterative method for the solution of equation (4) that the variational derivatives
6kx(t) ~(Sl)
"'" ~ ( S k )
exist, and also
~kx(t)
-0
6~(Sl) ''' ~(Sk)
if there exists an i = 1. . . . . k, with s i > i, k = 1, 2, 3 , . . . . By (4), these variational derivatives satisfy the equations
6kx(t) t t~kx(tl) rt . 6kx(tl) 3~(Sl) ... 3~(Sk) = fo A 6sO(S1)''- 6~¢(Sk)dtl + JoC~(tl) 6~(sl) . . . 3~(sk) dtl k
+ Y'. CO(t --Si) i=1
k = l , 2, 3 . . . . ~(S1)
"'" ~(Si-l)~(Si-1)
"'" ~ ( S k )
'
'
(5) where O(t) is the Heaviside unit function.
R.V. Bobryk / Moment stability of linear stochastic systems
229
The formulas
E[ ~( t ) x( t ) ] = f/B( t - s ) E-~-~ds,6tX()
E ~(t)
~(S1)
~k+lx(t)
t~kx( t ) .. " ~(Sk)
= B(t-Sk+l)E(sl)
~(Sk+I)
...
ds~+ 1,
k = 1, 2, 3,.. ",
follow from [5] because the process ~:(t) is Gaussian. Applying these formulas to equation (1) we obtain under (5) the infinite chain of integro-differential equations
dEx(t) dt
AEx(t) + f/B(t -s)CE ~x(t) ~--~ds,
6xk(t)
=A fiE
dq
~kx(tl)
E ~ ( s 1 ) "'" ~ ( S k )
~ ( s 1 ) "'" ~ ( S k )
+
~xk(tl) ft ~tjoJoln(tx _ S k + l ) f E
dSk+ldtl
(S1) "'" ~(Sk+l)
k
~k-lx(si )
+ E O(t-si)Ce i=1
k = l , 2, 3,..
~ ( S 1 ) "'" ~ ( S i - 1 ) ~ ( S i + l )
"'" ~ ( S k )
'
Let Yo( t ) = Ex( t ),
l, Yk(t)= f / ' ' ' f/exp '
E
)
~kx(t) e
k = 1, 2, 3 , . . . . 8
(s1)
• • •
'
Then one can reduce this chain to the chain of the linear ordinary differential equations dYo(t)
dt
Ay°(t) + ~'CYl(t)'
dYk(t)
dt
Y°(O) =x(O),
kzyk+l(t)+AYk(t)+~'CYk+s(t)+kCYk_l(t),
Yk(0)= 0,
k=1,2,3
.....
(6)
We show that chain (6) has a unique solution. Let us introduce a Banach space F of infinite sequences of functions
g= (gk(t),
k=0,1 ....
:supsup{exp{--trt}dkl[gk(t) ll
t>0
with the norm II g II = sup sup {exp{ -trt}d k
k II gk(t)II},
t>0
where d, o- are positive parameters to be determined. After multiplying the equations for Yk(t), k = 1, 2, 3 , . . . , with exp{k~-t} and transfering them to the integral form we can rewrite the chain (6) in space F as the linear equation
y=Gy+f, where the operator G and f are determined by the right-hand side and the initial values of the chain.
R.V. Bobryk / Moment stability of linear stochastic systems
230
Then for the norm of operator G we have IIGII _<
(d,lA,[+(z+dk),[C,[ d(cr+k)
sup
k =0,1 ....
) "
It follows from this that there exist d > 0, ~r > 0 such that II G II < 1 and so chain (6) has a unique solution in F. It is well-known that there exists an orthogonal matrix U such that D = U(A + C2)U ~ is diagonal because matrix A + C 2 is symmetric. Introducing the change
Vk(t ) = ~ U y ~ ( t ) ,
k=0,
1. . . . .
into the chain (6), we can reduce it to the form dvo(t)
---Dvo(t dt
dvk(t)
) + ~ U C U - l v , ( t ) - VC2U-'vo(t), Dvk(t ) - k r u k ( t ) + ((k + I)zUCU
dt
(7a)
lvk+,(t ) - U C 2 U - ' v k ( t ) + ~ / ~ r U C U - l v k _ , ( t ) , (7b)
Vo(0) =
Ux(O),
vk(0)=0,
k=1,2,3,....
(7c)
Let ( , ) denote the scalar product in Euclidean space R n. Using the orthogonality of U and the symmetry of C, from (7) we obtain 1 d [I Vo(t) 11z 2
(Dvo(t), Vo(t)) - II CU-1uo(t)I12+ v/~(cU-lvo(l), U-iv,(/)),
dt
1 d II Vk(t)I[
2
(Dvk(t) , v k ( t ) )
dt
+ ({~
- kz [I Vk(t)II 2 - II
1)r ( U - l c k + l ( t ) ,
(8a)
cv-lUk(t)I1 2
CU-~vk(t))
+~(CU-IUk_l(t), U-Iuk(t)),
k = l , 2, 3 . . . . .
(Sb)
The mean of the solution of equation (3) satisfies the equation dz(t)
dt
- ( A + C2)z( t)"
Then the diagonal elements of matrix D are negative because the null solution of this equation is exponentially stable. So there exists a oh > 0 such that (DVk(t). v k(t))_< --crI I[vk(t)[[ 2. Taking into consideration the orthogonality of U we have [[U-lVk(t)[[ 2= [[vk(t)[[ 2, k = 0 . 1 . . . . . The series k=0 I[ v~(t)l[ converges because the solutions of chain (6) belong to F and so Tk
T k C 'rt
II Vk(t)II 2 = ~.T I[ Yk(t) [I 2 --
dkk------~..
Therefore by (6), the series ET=0 d II vk(t)II From (8) one can obtain oo
k =o
d II Vk(t) II 2 dt
2/dt
oo
converges as well. oo
2 E (Dvk(t),
Vk(t)) -- 2 E
k=0
k=0 c~
_<-2~r 1 ~ k--O
IIv/,(t) I12
IhCU-lvk(t)
- ~(k + 1)~ U-lVk+l(t)[]2
R.14.Bobryk / Momentstabilityof linearstochasticsystems
231
and we conclude under the Lyapunov theory of stability that the mean of the solution of equation (1) is exponentially decreasing. For the case p > 1 we introduce under [2,3] the binomial coefficient q(n, p) = (n +~-l) and the vector X[P](t) E R q(n'p) of all p-th forms of the components of the vector x(t):
x[P]( t) = col( xl( t) p, ,~PXI( t ) P - I x 2 ( t) . . . . . xn( t)P), and the scale parameters Ai are chosen so as to obtain II xtp](t)II = II x(t)II P. For vector xtPl(t) we have under (1) the equation
dxtPl( t ) A[p]Xtpl( t ) + ~( t )C[p]Xtp]( t )'
dt
where the matrices A[R], C[p] are determined from the matrices A, C [2,3]. The equation
dxtpl( t ) = (AIR] + C~p])xtpl( t ) at + ~f~ Ctp]xtpl( t ) dwtpl( t ) is associated with the equation (3) [2,3]. For the full proof of the theorem it is enough to note that the matrices A[p], C[p] are symmetric as well [2,3].
3. The harmonic oscillator with random damping
Let us consider the equation of the harmonic oscillator, d2v(t)
dv(t)
-------7~+(k+/3£(t)) dt
dt
+av(t)=O,
k,a>0.
(9)
Introducing the notation d v ( t ) ]2, ~-x(t)=col((
dt
]
dv(t)
~/2av(t)--~,av(
t)2) '
we can obtain for equation (1) vector x(t), where m = 1 and -2k A = 0
- 2v/2-a-
0
-k
-2¢~
~
0
c=
-2/3
0
o
-/3
0
0
i)
.
Let us assume that ~:(t) is the white noise in (9) and interpret this equation as a Stratonovich stochastic differential equation. Then writing the equation for the second moment of the solution of this equation one can show [1,6] under Hurwitz conditions that 2/32< k is the necessary and sufficient condition for mean-square stability of the null solution of this equation. Let ~(t) be the Ornstein-Uhlenbeck process in (9) with the correlation function (2). Then we show that this condition is sufficient for mean-square stability of the null solution of equation (9) for all ~- > 0, a>0.
232
R.V. Bobryk / Moment stability of linear stochastic systems
After substitution of zk(t) = ~~-" r k / k ! yk(t), k = 0, 1 . . . . . for yk(t) in chain (6) we obtain analogously to (8), 1 d [I Zo(t)II 2
2
dt
1 all z ~ ( / ) II 2 2 dt
- ( D , z o ( t ) , z 0 ( t ) ) - II Czo(t)II
~(Czo(t), z,(t)),
(Dlzk(t)' Zk(t))--krllzk(t)ll2--Ilfzk(t)ll2 + (~
D = diag(-2k
2+
+ 1)~" ( Z k + t ( t ) , C Z k ( t ) ) + f k ~ ( C z k _ , ( t ) ,
+ 4132 , - k
zk(t)),
k=1,2,3
.....
+ / 3 2 , 0),
b e c a u s e C is s y m m e t r i c a n d A is s k e w - s y m m e t r i c . If 2132 < k, t h e n
E k =0
dllzk(t)ll = dt -< E 2 ( O l Z k ( t ) , z k ( t ) ) - - <0 k=0
and under the L y a p u n o v theory of stability we conclude that the null solution of equation (9) is m e a n - s q u a r e stable.
References [1] L. Arnold, Stochastic Differential Equations." Theory and Applications (Wiley, New York, 1974). [2] G.L. Blankenship and G.C. Papanicolaou, Stability and control of stochastic systems with wide-band noise disturbance, SIAM J. Appl. Math. 34 (1978) 437-476. [3] R.W. Brockett, Lie algebras and Lie groups in control theory, in: D.O. Mayne and R.W. Brockett, Eds., Geometric Methods in System Theory (Reidel, Dordrecht-Boston, 1973) 43-82. [4] R.W. Brockett, Parametrically stochastic differential equations, Math. Programming Study 5 (1976) 8-21. [5] M.D. Donsker, On function space integrals, in: Proc. Conf. Theory and Applic. Analysis in Funct. Space (MTI Press, Cambridge, MA, 1964) 17-30. [6] I.I. Gichman and A.V. Skorokhod, Stochastic Differential Equations (Springer-Verlag, Berlin, 1972). [7] E.R. Infante, On the stability of some linear nonautonomous random systems, ASME J. Appl. Mech. 35 (1968) 7-12. [8] F. Kozin, A survey of stability of stochastic systems, Automatica 5 (1969) 95-112. [9] J.L. Willems and D. Aeyels, An equivalence result for moment stability criteria for parametric stochastic systems and It6 equations, lnternat. J. Systems Sci. 7 (1976) 577-590. [10] E. Wong and M. Zakai, On the relation between ordinary and stochastic equations, Internat. Z Engng. Sci. 2 (1965) 213-229.