Some results on stability of stochastic dynamical systems

Some results on stability of stochastic dynamical systems F. Kozin Professor Brooklyn,

of Svstems, he& York

Polvtechnic _

Institute

of New York,

INTRODUCTION The general topic of stochastic differential equations, their properties and their applications, has been a source of a great many problems of fascination, of depth, of interest with the potential for significantly useful results. The development of the subject from the point of view of applications has been due to the studies of a great many engineers, scientists and mathematicians throughout the world. Those who have become students of stochastic differential equations are very familiar with the names of the people who have played a central role in developing the topic. Although the study of stochastic differential equations essentially began in the 1930s with the work of Pontryagin, Andronow and Witt’, as well as the works of S. N. Bernstein, it was not until the 1940s that the foundation for the great leap forward in the topic took place. In particular, the fundamental papers of K. Ito’ constitute the beginning of major development of stochastic equations and their applications to the Engineering Sciences. Motivated by a desire to construct explicit representations of Markov diffusion processes, he developed a stochastic integral through which he studied what are now referred to as Ito stochastic differential equations. The applications of these equations are to structures or systems subjected to gaussian white noise excitations. The development of the subject rapidly increased during the late 1950s and 1960s not only due to the increasing interest on the part of probabilists around the world, but to a great extent, the development took place because of the needs generated by advanced technological applications and especially the needs of the space programmes. Communications and control of large systems subjected to random inputs, random excitations, random environments, as well as randomly varying system components required that stochastic models were needed to study, in a more realistic fashion, the effects of these random quantities. Since dynamical systems are generally described by differential equations, then the stochastic models naturally were best described by differential equations with stochastic process coefficients and/or stochastic process inputs. The ideas of optimization, optimal filtering, identification, parameter estimation, optimal control and stability of systems are among the problem areas that gained great interest and importance during that period. Thus in these topics the stochastic differential equation AcceptedJanuary 1985. Discussion

026~8920/86/010013-10$2.00 0 1986 CML Publications

closes March

1986.

was the fundamental object of study. In particular, the properties of the solutions of the equations, both the sample properties as well as the average properties of the solution processes, became the fundamental problems to study. The study of the behaviour of the solution processes of stochastic differential equations in general requires rather advanced methods of analysis as well as stochastic processes. As these methods became known to the many researchers who were motivated by systems problems, further results and applications foliowed. In this manuscript we shall discuss only a few topics since the literature has grown by vast amounts, and covering all its many directions wouid certainly be difficult at the very least. We distinguish between (Ito) stochastic equations with white noise coefficients and stochastic equations with (non-white) physical noise coefficients. We shall discuss a few questions concerning problems that have occurred to many of us in the field for which results are available, but complete answers have not as yet been forthcoming. Surveys of the topic of stability for stochastic systems can be found in Refs. 3, 4 and 5.

STABILITY PROPERTIES FOR STOCHASTIC EQUATIONS WITH PHYSICAL NOISE COEFFICIENTS The general linear homogeneous stochastic differential equation with physical noise coefficients may be written as T=

(A + F(t))x(t),

(1)

where x(t) is an n-vector, A is a constant n x n matrix and F(t) is an n x n matrix whose non-identically zero elements Ifif( are zero-mean stochastic processes. The problem of stability has generally been concerned with the asymptotic properties of the equilibrium solution x(t)rO.

Although stability of the solutions of stochastic differential equations may be defined in a number of modes corresponding to the various forms of convergence of probability theory, the most often studied stability concepts are stability of the mean, in particular, asymptotic stability of the second moments, and almost sure sample asymptotic stability. For asymptotic stability of the second moments, one

rrobabilistic

Engineering

Mechanics,

1986, Vol. 1, No. 1

13

Stability of stochastic dynamical systems: F. Kozin requires lim E{ II x(t)112} = 0.

(2)

tTm

If the second moments approach zero exponentially, then the second moments are said to possess exponential stability. For almost sure sample asymptotic stability, one requires lira IIx (t)l[ = 0

(3)

tTm

with probability one (almost surely). We should note that (3) implies stability (Lyapunov) for linear systems. For non-linear systems, we also require lira sup [[x(t:t0,xo)]t = 0

(4)

llxoll&O t>to

with probability one, where xo denotes the initial value of x(t) at t = to. In general, conditions for stability of the second moments cannot be obtained exactly for systems subjected to physical noise excitations. The basic reason for this is that the exact moments cannot, in general, be obtained. The fact that for the system (1) the vector x(~) is correlated with the stochastic coefficient F(t) does not allow us to obtain an exact equation satisfied by the moments of x(t). That is, we cannot write

e{ F(~)x (t)} = E{ F(t)}e{x (t)}

(5)

for the linear system (1), where the matrix F(t) contains components which are physical random processes. [If F(t) matrix contains only white noise components, then (5) does hold.] The major approaches that are available presently to study the moment behaviour of systems with physical random coefficients are approximate methods. These are based upon the fact that the coefficient process is a small perturbation, the coefficient process is narrowband or, finally, that the process is wideband so that the Markov diffusion approximations may be applied via ideas of Stratonovich 6, Wong-ZakaiT, KhazminskiiS, in which the original physical coefficient equation may be substituted by an associated white noise coefficient equation, whose moments can be studied exactly in the linear case, but still require approximations in the general non-linear case. In particular, it is known that for the general non-linear stochastic differential equation dx(t) - f(x(t)) + G(x(t))-~ dt

(6)

where ~ yt is a wideband physical noise, the equivalent tto differential equation is of the form, for dW denoting the differential of Brownlan motion, dx(t) = (f(x(t)) +½G~(x(t))G(x(t)))dt + G(x(t))d W(t)

(7)

which can then be studied via the tools of diffusion process theory. Here, G~ denotes the partial derivatives of G with respect to x. These ideas have been recently applied by Lin 9, Ariaratnam ~°. The approach was used for the first

14

Prvbabilistic Engineering Mechanics, ]986, VoL t, No. i

time x1 to study the stability of 2nd order linear systems. Exact results were obtained there. F o r the linear case. the associated Ito equation is a~so linear° and thus all moments can be obtained, thereby allowing moment stability prope~ies to be studied in the wideband case. For the small perturbation problem, early works ef Stratonovichl 2, Weidenhammer t 3. and more recently by Ariaratnam ~4 and Wedig is find mean square stability conditions in terms of the values of the spectra1 density a~ twice the undamped naturai frequency of the linear oscillator. For the slowly varying case. Samuets ~6 obtainee early results. There is. however, another relatively recent directio~ for studying the moment (as we1! as the sample) properties of linear systems with arbitrary physical noise coeffÉcients. This approach involves the ideas of Lie Algebras° first motivated by Brockett ~v. and studied by Wi!lems ~s Willsky eral. ~9, as well as Kistner 20. The idea is basically as follows. We consider the general linear system (1} written in the form dx(t) -- Axlt t + 2 f~(t)Bmg), (8} dt i= ~. where {~(t)} are physical noise coefficients and {A,B 1.... ~B~} are known constant matrices° If the matrices {A,B~ .... ,B~} are upper triangular (that is. all elements below the main diagonals are identically zero), then the system (8) can he integrated exactly. indeed, the asymptotic momem properties can be de~ termined as, for example in Ref. 18o In particular, if the matrices are upper triangular and if a = [a 1. . . . a~] is the main diagonal of A, and b s = [ b ~ bnj]T is the main diagonal of Bj then the necessary and su~cient condition, s for exponential stability of the ptb mean are given as .....

RAaj) < - ~R~(br)SsI(0)R~(bg,

,= 1..... n°

{9)

where R~ denotes real part. and $¢/0) is the sporran_ matrix for the vector process f(t ~= {fj. (t), r f~, r luated at 09=0. The question then is, under what conditions can we find a single transformation matrix R. ~non-singular) such that R A R - ~, R B i R - ~, j = 1..... n are all upper triangular? The answer lies with Lie Aigebra theory, tn particular, a Lie Algebra,L, generated by mutates {A,B~ ..... B~ } is the smallest se~ of matrices for which C, D~L implies that [C,D] =-CD-DCeL. One defines the derived series as L(°) = L ~L<'~= [L,L] .... L(n + ~' = [L(%Lq ..... The L algebra is said to be solvable ff there exists an n such that L("~=[0], that is alt matrices in L ("} are cormmulativeo The necessary and sufficient condition for this to occur is that there exists a non-singular matrix R such that BBR -x is upper triangular for atl BeL. Hence, the con° dition that will allow us to integrate the system equa~tions (8) exactly is that the algebra generated by {A,B1 ..... B,} is solvable. This, of course, allows the exact soiufions and stability conditions m be obtained for many dynamical systems. The onty problem is that for the cases most directly o1" concern to the structural engineer, even for the simple second order oscillator, the Lie Algebras generated by the matrices are not solvable. Thus, no !ight is shed in this most important application.

Stability of stochastic dynamical systems: F. Kozin Let us now tum to the case of sample properties and conditions that will guarantee asymptotic stability almost surely. It is clear from the description above for solvable Lie Algebras that conditions can be obtained for which the desired asymptotic almost sure stability holds. However, the results are limited and may not be applicable for structural applications. Instead, we shall take another approach that we have suggested in the past x~. Consider the system (1) written as the general linear homogeneous differential equation

dd(tt) = A(t)x(t ).

(10)

It can easily be shown that for the norm Ilxllp=xrpx, the equality I

logllx(t)llp-logllx(O)l[e

t

In particular, for the classical second order oscillator wRh zero mean, stationary ergodic physical noise coefficient, f(t),

5i(t)+2~m2(t)-~(m2 + f(t))x(t)=O he found for co = 1,

~} = E{f2(t)} < 4( 2

(18)

as the sufficient condition extending the results in Refs. 22 and 23. These results are based only upon knowledge of the second moment properties of the f process. Later, in 1972 Wu and Kozin 24, based upon the Infante approach, presented an extension to allow use of the properties of the probability density for the f-process, again applied to second order oscillators. Indeed, if we define P, = Prob{ f (t) >~l}, one can show that the sufficient condition becomes,

1 ~xr(s)[Ar(s)P +PA(s)]x(s) •

=;j

os

t°

(11)

holds, where 'T' denotes transpose. If the quotient on the left hand side of (11) remains negative as t approaches zero, it must follow that limllx(t)llp=0. t?ce

Therefore, the system will be asymptotically stable. Thus, the time average of the quotient within the integral if it exists, must determine the asymptotic properties of x(t). We notice that the transformation ,~(t)= x(t) (12) IIx(t)llp transforms the trajectories onto the surface of an ellipse, since P is positive definite. From the properties of pencils of quadratic forms, it is known that in terms of ,l(t) the quotient in the integral of (11) satisfies

co(s) <.2r(s)[Ar(s)P + PA(s)] ~.(s) 4 f~(s)

1

where D = (l+ 1 - (2)1/2. As in Infante's procedure, I must be chosen to maximize the sufficiency region. Numerical procedures must be applied to obtain the region. But as shown in Figure 1, the region of sufficiency is very greatly increased. A similar result holds for the randomly damped second order oscillator

5i(t)+[2~co+ f(t)]~(t)+co2x(t)=O

[Ar(s)P + P A ( s ) ] P - t.

Figure 2. In an interesting paper 2~, Blankenship introduced a more general norm into this problem. If the norm of a matrix is denoted by '11 [1', then the measure of a matrix A, denoted as #(A) is defined as

(13)

!1

(14) yz

Clearly, these are real functions since the matrix is symmetric. Thus, it will follow that if the elements (aij(t)) of the Amatrix are stationary, ergodic random processes, then we can write from (1), with probability one,

A ~C

t

t

0

2C t t~

~
e{£~(.)} < 0

(I

(15)

Thus, we obtain a sufficient condition for almost sure sample stability, as

i0

[2_4]

, . "i .2 ,, • "~.~--zl¢ tzl]

/

/ / . . 551i i ('

(16)

The condition in this form for physical noise processes was first obtained by Infante2L

fZ---

/

E{co(')} =lim 1- Ico(s)ds ~
(20)

and the sufficient condition obtained again leads to a greatly increased region of sufficiency, for almost sure sample asymptotic stability. These results are shown in

where co(s), f~(s) are, respectively, the smallest, and largest characteristic values of the matrix

tTo~ t , J

(17)

I

I '

t

G.g Figure 1 Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 1 15

Stability of stochastic dynamical systems: F. Kozin
i7_ _....

// ~'"Jt

0

0?

:?J !5_,4]

/1

t

_4

i c,

24

~4 :

Figure 2 Thus for example, in the simple case, lixil =(Xx2, ) "2,

IIAI! = m a x R , ( A r A ) ,

where 2i(M) denotes the ith eigenvalue of the matrix M, one finds that #2(A) = ½max ~i(A + AT). i

The basic result is the following lemma,

L E M M A (Coppe126) :~(t) = A(t)x(t),

If then t

Ilxotjexp -

/~[-A(s)]ds

t

)

;

~<[Ix(t)ll ~
to

to

(22) Blankenship shows that the Kozin-Wu approach is an application of Coppel's lemma. Thus, it may be possible through this general measure/4 to extend the sufficiency condition of Ref. 24 further for systems of the form (10). Almost sure sample stability problems that remain to be studied further for applications in the physical noise parameter case are: (1) determine the exact stability boundaries for the second order oscillator case with random varying spring and/or damping terms. (2) determine the stability properties of the undamped linear oscillator with randomly varying spring terms. (3) study the regions of stability for higher order linear systems of the form (10). tn connection with these three problems, there are a number of comments that can be made. In problem (1), for exact stability boundaries in general, if we denote the integrand in the right hand integral of(1 I) as Q(s), then if suitable ergodic properties hold, it follows from the discussion related to (11) that we will have t

tim i f Q(s)ds n~

~ < 0 , stability ( > 0, instability

(23)

0

16

Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 1

Therefore. evaluation of this integral is the key factor t:a determining the stability properties for linear systems with random coefficients. This approach was taken in Refl 27 to attemp~ to establish it as a viable approach for studying stability properties statistically and emmer~calty. However, many statistical questions remain relative to ~he convergence rate and asymptotic variance of the iimk i~v {23). This numerical approach was studied later by Amoi~_ and his co-workers. In particular 28. rather smooth hour: daries are presented for the case of the second order oscillator (17), where fig) is the Ornstein-Uhlenbeck gaussian process. Their boundary resuk is a!so shown on Figure L We have no details, however° on how such a smooth boundary was achieved by simulation. Our past experience has always led to numerical d~Ticukies for determining sample behawour near ~he smbhity boundary. Of course, we know that the limit in (23) is maoh smoother than the sample behaviour. Bu~, further darifi.. cation of the result in Ref. 28 would ce~ainly be desirable. Relative to the result of problem (2)° it has been know~ at least since 197129 that if the random spring parameter is gaussiav, white noise, then the undamped second orde~ oscillator is unstable with probability one. [n a recent paper Kotani has shown tha~ if the randomly varying spring term is stationary~ ergodic and bounded {in general the range of values belong m a compact state space) then the solution is unstabie almos~ surely. Again, in this problem the unbounded {for exao mple, gaussian) case is still of interest. Relative to problem 3. although much of the theory for sample stability is for arbitrary nth order coupled linear systems, the sample stability regions have rmt been easity obtained for higher order systems° In a certain ~ n s e this has been due to the complexity of obtaining useft~l approximations and numerical studies of the sufficient conditions However in Ref. 30, an extension of the results of Ref. 24 were obtained and applied to coupled systems of the ~ .

.

+f1224_(c2+f2(t))Xl + [C3 + f t ([))X2 : @ (24)

where f~ (0,f2(0 are independent Gaussiae, processes. A sufficient condition of the form. __ ~ .~_[ ( K ! ! )1/2 or- (K22)1/2][(I1 .~_ ~2) "+"2[ ~ (E 1 -~- ~ }~@

was obtained, where K!

i, K 22 are functions of {cl,c>cs,9,l~,t2) W/= Prob{f/(t) ~>l~}, Ei = E{ f,(t)lZ(~)~> i~),

i = 1 2 "~ i = :<2.

~26}

and the region of sufficiency is maximized by choosing the constants i~,12 optimally. The numerical procedures to obtain these regmns are quite complex, however, requiring so-called simplex moo thods and the Fletcher-Powe!l technique 3~. The problem was motivated by study of the stability of moving elastic strips in the presence of in-plane loads. Another approach to such problems with physical noise

Stability of stochastic dynamical systems: F. Kozin The important difference between the physical noise case discussed earlier and the white noise case is that for integer p, the pth moments of the process generated by the linear equation (27) can be obtained explicitly. This is due to the fact that the following relation holds,

~2 u

d E{g(x(t))} = E{ff~9(x(t))} /

/ / / ' ~

/

where ~ex is the backward diffusion operator C which we discuss in more detail below) generated by the system (27). It is a second order operator on the components of the xvector. Indeed for integral powers of the components of the vector x, the right hand side will also yield the same integral powers which lead to a closed set of equations for solution of the moments. This idea was applied at least as early as 196236 , and has been used many times since then. In particular for the second order oscillator with white noise spring term,

f i

\ \

/ /

/

\.

(29)

Y(t) + 2 ~ + (o~2 + l~V)x= 0,

Figure 3

(30)

(Here l;Vdenotes the gaussian white noise), we can write it in Ito linear system form as coefficients can be found in the dissertation by Ly a2, written under the direction of Professor Ariaratnam. STABILITY P R O P E R T I E S F O R STOCHASTIC D I F F E R E N T I A L EQUATIONS WITH GAUSSIAN WHITE NOISE C O E F F I C I E N T S - I T O EQUATIONS The linear Ito equation is certainly the most widely studied and understood, due basically to the many analytical tools that are available for determining the properties of its solutions. The general linear homogeneous Ito equation may be written as,

dxl = x2dt dx2 = - (2(x2 +

(27)

r=l

where x is an n-vector, A, B,, r = 1..... l, are constant n × n matrices, and the W~(t), r = 1..... l are independent Wiener processes. That is, they are zero mean stationary, independent increment processes with E{W~(t)}=a2,t. The increments dW,(t) are rigorous interpretations of the socalled gaussian white noise. The solution processes of (27) are Markov diffusion processes and thus there is quite a large body of analyses available to study them. As in the case of the physical noise coefficient system we are concerned with asymptotic properties of the moments, for example,

(31)

In this case the backward diffusion operator (also referred to as the generator of the solution process) is of the form

L#x=xz~_~_(2(x2+co2x . O a2 2 02__2_ 1)~x2+-~Xl Ox~ (32) --x

Thus, for O(x) given respectively by x~,x 1Xz,X~ we find the second moment equations as

d: E{x~} , 0 tE{xlx22})=(-co2 E{xz} az

l

dx(t)=Ax(t)dt + ~ B,x(t)dW,(t),

~Zxl)dt +xldW

2 0 \ / E{x~} , -2( 1 )[E{xlx2}) -2co 2 - 4 ( / \ E{x~} /

(33) and whose solution can easily be seen to be asymptotically stable (that is, mean square stability) if36 4~o2(> a 2

(34)

Recently, the general equations for the arbitrary pth order moments have been presented. These results are due mainly to Brockett and his co-workers 37,3s. Motivated by the algebraic theory of linear differential equations, one can define for a g i v e n n-vector x and a given positive integer p, the associated vector x [p] whose components are

lim E{xr(t)x(t)} t'foo

•\

XlPtx2Pa

PP

(28)

or, generally,

(35) for

lim E{ Ilx(t)[lf,} tTcc

~ pi=p,

where

pi>~O.

i=1

IIx(t)/l~= ~ x/'(t),

Xn p"

p>0.

i=1

Of course for p = 2, this is the second moment stability problem, studied for many years by Refs. 33, 34, 35, 36 as well as many others from many countries.

The components ofx ~1 are ordered lexicographically. For example, for n = p =3, the components of x t3], for X1

X3

Probabilistic Engineerin9 Mechanics, 1986, Vol. 1, No. 1 i7

Stabiiity of stochastic dynamical systems: F. Kozin are, in order,

X[3]=

N•XIX2X3 (36)

xl

xi The vector x M satisfies/xL~?I = Ilxl v, where Ilxl[: = ( ~ ) and more generally, (x,yF= (xL~I,yM). Furthermore, this concept extends to matrices through the definition y = Ax defines A [p], or that matrix that yields

y[;] -- A[P]X[p].

(37)

Finally, for linear systems

~(t) = A(t)x(t).

(38)

the differential equality x(t + h) = (] = hA(t))x(t) + 0(h 2) holds, yielding from the definition (37) xM(t + h ) = ( I +hA(t))MxM(t)+O(h2).

(39)

thus be determined. In Ref. 18, for example, Wilems notes that if the eigenvalues of the matrices B, are not a l imaginary, then there is no value of the noise intensity that would yield exponentially stable pth moments, for a! p. tndeed, if B, has an eigenvalue with non-zero teat pa~% then Kp must have at least one positive real eigenvalue for p large enough. He considers other examples as weiL We might mention that the results of Brockett, Wtlems~ have been derived from algebraic considerations and more particularly by consideration of the Lie algebras generated by the matrices (A, B~, B2 .... N~). The explicit determination of the asymptotic behaviour of the sample solutions of (27) in the Ito case was studied by Khasminskii 39, who derived the fundamental resuit which can yield a necessary and sufficient c~oadition for sample stability° This allows, in principle, the dete_~mination of the exact boundary of the region of; stabil~y ~br the linear 1to equation. We can motivate Khasminskifs approach very simply by looking at ordinary differer~tiai equations. We have already seen from equations (11), (12) that fo-r the Q function defined in (23), the exact stability bout> dafies may be obtained in theory. The point is, however, that the limit in (23) must be eva!unfed. For the general physicai noise parameter case this is, as yet, ~ot possible analyficaly. However, for the gaussian white noise (I~o) case, this is possible as first noted by Khazminskii. Let us consider the linear Ito system (27) with A = (a@ B , - (b[j), and E{ W~(t)VC)(t)}= 6ij rain(t4), we ca~ write the backward operator as

Upon defining the limit

( - ~ !¢0~ 5Y= z~ = u ' ~ J [ - ~ i z~ bu(x)ixfi~,ci~

1 lim ~ ((i + hA(t)) [p]- Ilv]))= A~],

i,j= 1

~J~i

"~i,j = I

(13)

where

h+O

we determine the associated differential equation for x ~A,

~["] (t) = Al~r~[~] (t).

tn order to relate these ideas back to the original equation (1), one can apply the 1to calculus directly or study the Stratonovich equivalent to obtain the ][to differential equation for x [~] as

dx~I(t)=

[(" A-=

bu(x )= ~

(40)

E B,2~ Z' B~L,Il xM(t)dt z,=l /M+,= ! _j (41)

g

+ ~ B,~,~r~(t) dW,(~) r=l

where the W~-processes are independent standard Brownian motions. Applying expectations to the Equation (41) immediately yields the linear differential equation for the pth order moments as

r=l

b~ b= k,rn= l

and we define

t ( x ) = (bu(x)). Making the change in variables /~(t) = x(0 /x(01]'

(43a)

where /'/ denotes Euclidean norm, Khazminskii ~ecognized that the Markov process generated b y (43N is mapped onto the surface of the n-dimensional unit sphere and is a Markov process, satisfying an !to differential equation. Furthermore, if we define p = l o g / x t , the ~to differential formula yields dp = Q(a)dt + 2 2rg,3"dN , r =

(44)

Z

where t Q(~.) = ~r A2 +i~rB(2 ) - irB(2), = KpE{x[P?(O}.

(42)

Although, in principle, the equations for the pth moments for linear homogeneous Ito differential equations have been known for many years, the general form (42) can be quite useful. These ideas extend quite simply to the nonhomogeneous case as well. The asymptotic properties of the solution processes can

18

Probabitistic Engineering Mechanics, 1986, Vol. 1~ No. 1

(45)

where tr denotes trace. Indicating the dependence of the ~(~) process e~ its initial condition ,~0, by a~-°(O = x=o(O/l x~o(Ol, where

~o=Xo//Xol,

Stability of stochastic dynamical systems: F. Kozin we have

Theorem (Khasminskii39). If there exists n linearly independent vectors 21 .... ¢~. in E. such that with probability one t

lim t - 1 { Q (2~'(z))dz < 0 d

(46)

t-~OO

0

for i--1 ..... n, then the system (27) is almost surely asymptotically sample stable. If, however, for any 2 o in E. t

lim t- 1 t Q (2z°(z))dz > 0 d

(47)

t~Ct3

0

then

the

system

(27)

is

unstable

in

the

sense

P(lim[t~ ~ [[x~°(t)'] = oo}= 1. In general, (46) and (47) are quite difficult to apply without an ergodic property to enable their evaluation. However, as can be shown by examples, they are very useful in specific instances. It should be pointed out that (46) provides a method for computing the sample stability boundaries for linear stochastic differential equations of order higher than two. One procedure would be to generate solutions 2(t) on a digital computer for a linearly independent set of 2i and construct the time averages indicated in (46) for each 2~. Then statistically test the signs of these time averages for large t. This was initially done in Ref. 27. The ergodic properties of the 2(t) process enables us to obtain an analytical expression for the almost sure sample stability boundaries. If the diffusion process on the sphere is nonsingular, then as shown by Khasminskii a9, it is ergodic on the entire sphere. For the stochastic differential equations occurring most often in engineering problems, the diffusion process 2(0 is singular, in which case it may not be ergodic on the entire sphere, S~,but rather on specific components A~ on the surface. An ergodic component may take the form of the entire sphere, as in the nonsingular case, or a region of the sphere, or the locus of points formed by the intersection of specific hyperplanes and the sphere. Depending on the initial condition, a trajectory of the 2(0 process may start within an ergodic component and, with probability one, remain there for all time, or initially belong to a non-ergodic component and after some time enter an ergodic component to remain or it may never belong to an ergodic component except perhaps as a limit point as t - * ~ . For an initial condition 20 that results in a particular trajectory of ;txo(t) eventually reaching an ergodic component A~ the strong law of large numbers can be applied. Thus from (46) the condition for asymptotic stability of the corresponding sample solution x(t) is that

(" a, = E{Q (2~°(t))} = / Q (2)H,(2)d2 < 0

A necessary condition for the system (27) to be asymptotically stable is that (48) be satisfied for all ergodic components. If with probability one, all trajectories of 2x°(t) reach ergodic components for all ;to in E., then this is also a sufficient condition. If, however, for some ;to, trajectories of ;tzo(t) can remain in a nonergodic component, then a sufficient condition for asymptotic sample stability is the satisfaction of (46) for all ;to in E.. This is expressed in the following corollary to the previous theorem.

Corollary [KhasminskiiS9]. If there exists n distinct at values, (48) then for asymptotic sample stability of (27), it is necessary and sufficient that ai < 0 for i = 1..... n. These results have been applied 11'29 to determine the exact sample stability boundaries for the general second order linear Ito differential equation, dxl =xzdt dxz = - (oZdt + aldWa (t))xl - (2~o)dt +azdW2(t))x~49)t For the second order equations (49) the R-process is a one dimensional diffusion defined on the boundary of the unit circle. Indeed for any second order system this will hold. Upon setting 21 = c o s ~b,

22 =sin q~,

the backward operator for the ~b process is of the form 1 tY2

dE

d

(q~)d~ + m(~b)~-

(50)

From diffusion process theory 4°, we have

Definition A singular point of the ~b-proeess is any point satisfying

~2(q~)=0.

If a2(~b)=0, m(~b)#0, then ~b is called a shunt. If az(~b)=0, m(~b)30, then ~b is called a trap. At a shunt, the process can only pass through in one direction. At a trap the process will remain there forever. For the second order linear equation, it can be shown that there are at most four singular points. The ergodic properties of the q%process are completely determined by the properties of these singular points. For example, for the second order oscillator dxl (t) = x2 (t)dt

(5t)

dx2(t) = - (2¢c0x2 (t) +c02xl (t))dt-xl (t)dW(t) the only singular points are shunts located11,29 at + zr/2 as shown in Figure 3. Thus the ~b-process moves in a clockwise direction around the unit circle and is an ergodic (recurrent) process on the unit circle. Hence, the entire circle is essentially the only ergodic component for the ~b-process. For c~ = 1, the oscillator (51) yields

(48)

A~

where #~(2) denotes the stationary probability density function corresponding to the ergodic component A~. If the initial condition 20 is such that the trajectory 2~o(t) never~ belongs to an ergodic component, then a sufficient condition for asymptotic stability of xXo(t) is (46).

G2

E{Q (2(q~))} = E{ ~-- cos 2 q~cos 2~b - 2(sin 2 ~b} (52)

f

-n/2

Probabilistic Engineerin9 Mechanics, 1986, Vol. 1, No. 1 19

Stability o f stochastic dynamical systems: F. Kozin ~.z i

3O

/

0 Figure 4

where dS(¢) . 2 d¢ = c-~s27 exp [3_~_tan¢ (3 + 3(tan ~b+ tan 2 q~)

(53) dM(~b) d4

exp [ 3- @ a2cos2¢ 2

tan¢ (3 + 3~ tan ¢ + tan 2 ¢ ) ?

The functions S, M are the so-called speed and scale measures of the diffusion process. Upon evaluating the integral (52) where S, M are given by (53), the exact sample stability region for (52) is shown in Figure 4. For the undamped oscillator, dxl (t) = x 2(t)dt dx 2 (t) = - ¢o2x1(t)dt - xl ( t ) d W ( t )

(54)

we find that the evaluation of (52) for (53) with ( = 0 yields E{Q(2(¢))}>~0 for any e 2 > 0 . Hence, the undamped oscillator with white noise coefficients is always unstable. Many examples with the exact stability boundaries are contained in Ref. 11. A study of the general second order system case can be found in Ref. 41. One point of importance relative to the results in Ref. 11. The idea was to study the stability of second order linear oscillators with wide band excitations in the coefficients. Hence, the Wong-Zakai, or Stratonovich correction terms were included in the study of the Ire equivalent equations (7). The interesting point is that the Ire equivalent equations are exactly the same form as the original physical noise equations for systems (54). That is, the correction terms are zero. Thus, the stability boundaries for the wide band gaussian coefficient process in (54) will be the same as for the gaussian white noise (Ito) case. In fact it can be shown that there are non-zero corrections only when the random noise coefficient multiplies terms of order one tess than the order of the differential equation. What about higher order linear Ito equations. In theory the technique that we have described can be used to obtain exact stability boundaries for the higher order case°

20

Probabilistic Engineering Mechanics, 1986, VoL ], No. 1

Unfortunately, there exist basic proNems in app1~ng this approach. Again noting that the 2oprocess defined by (43a) is a unit norm vector defined on the surface of the n dimensional unit sphere. We must study the expected value of Q(I) on the sphere. But. for higher dimensional spherical suffaces~ ::e cannot classify the singularities suffÉciently well to determine the associated ergodic properties of the diffusmn process. The singularities on the n-dimensional spherical surface are whole curves or boundaries instead of simple points. Also the diffusion can behave in many differen~ ways near these boundaneso More study will have to be made of the singularities and ergodic properties of diffusions defined on general manifolds, in order for us to be able to apply the techniques above. Of course if the diffusion has no singuiarities on the surface of the unit n-sphere, then we know that the process is ergodic. However. most physical systems of interest generate diffusion processes that have singularities. Therefore. the question of exact stability regions for higher order linear Ire equations remains an impo~:an~ open question for applications. The connection between the asymptotic propert.ies of moments and of samples of the solution processes to stochastic differential equations has been of interest; for many years. A partial answer to this problem was established quite recently.2 for linear 1to stochastic differential equations. We shall motivate the results by studying the simpie first order stochastic differential equation with a white noise coefficient. We consider the first order 1to equation dx(t) + (a& + d W { t ) x ( t ) ) = 0

{55}

where a ~s constant and W(t) represents the Browniae Motion process with zero mean and variance v2t. It is well known that the solution process of (55) is

since it is weli known tha~ since W(t) grows like x / ~ g log t with probability one° then the stability pro~ perties of the sample solutions are determined by the deterministic term in the exponent of (56). Hence~ the " region of sample stability is dearly glven by a +- a 2 /=9 ; > r ~ ~. i.e. ~rz > -- 2 a .

(5 7 )

We now consider the moments of the solution process (56). The absotute pth moment is simply found to be E{lx(t)l ~} = lxoiPe-m+~tm~,E{e-~'w:"~} = txe[p exp

,5~,

- i)e 2 - 2 a

where x(0)= xo with probability one. The region of stability for the integer moment p := n( > ; } is given by (58) as 2 F r o m f59) it is obvious that the constant a must be greater than zero for stability of the nth moment. This is not necessary for sample stability as given by the region (57). As n increases, the stability region decreases and lies

Stability o f stochastic dynamical systems: F. Kozin ffz

CONCLUSION

P~3 D~/a

I

" 24

We have in this brief survey described a n u m b e r of results concerning the stability of stochastic systems. We have concentrated mainly on the linear system. Although certain results are available for the non-linear stochastic system, the results do not contain as large a b o d y of stability conditions as are k n o w n for the linear case. N o n - l i n e a r systems have been studied by m a n y researchers, for example, K l i e m a n n 44, as well as S u n a h a r a and his co-workers 45. But, m o r e remains to be developed for the stability properties of the non-linear stochastic system. O u r objective has been to state certain results that are available, and certain p r o b l e m s that remain. We hope that the interested reader will look further into these questions, and, perhaps, provide the needed answers.

Figure 5

ACKNOWLEDGEMENT 2 below the line 0.2= n--~a, (a > 0) in the (a,tr 2) plane.

This p a p e r was supported by N S F G r a n t CEE-8311190.

This illustrates the k n o w n facts a b o u t stability for the first order Ito differential equation. The regions of stability for higher integer m o m e n t s are included in the regions of stability of lower integer m o m e n t s , and all integer m o m e n t stability regions are included in the region of sample stability. G o i n g back to (58) we notice that the equality holds for all p ~>0. Therefore, let us break from traditional studies of integer m o m e n t s and consider the case 0 < p < 1. F o r this range of values of p, the stability region obtained from (58) is

REFERENCES

42>

2 1a , p-

( 0 < p < 1).

(60)

2. 3. 4. 5. 6 7

A rather curious and interesting result follows from (60). We see that as p a p p r o a c h e s zero, the half line that bounds the stability region for the pth m o m e n t (the other b o u n d a r y is the positive a-axis) rotates counter-clockwise and a p p r o a c h e s the half line that bounds the region of sample stability. It appears, at least for the simple stochastic equation (55), that the region of sample stability can be characterized as the union of all pth m o m e n t stability regions for p > 0. This is illustrated in Figure 5. The next obvious question is, does this p r o p e r t y hold more generally? F o r the linear Ito system, the answer is yes, as shown in Ref. 1,-423. Very recently, Arnold 43, has extended the results in Ref. 1,-423 to include linear systems of the form ~(t)=A(t)x(t),

1.

(61)

where the stochastic elements of the matrix A(t) are stationary ergodic processes. T h e one dimensional case of the system (61) was also treated in Ref. [42]. Unfortunately, this explicit connection does not lead us directly to the sample stability b o u n d a r y , since the limiting m o m e n t stability boundaries are not directly obtainable as p~0 for higher order systems. In any case the exact connection between m o m e n t stability and sample stability does exist and m a y ultimately lead to a means of determining the exact stability boundaries or, at least, good a p p r o x i m a t i o n s to the exact boundaries. These questions are still open for study.

8 9 10

Andronow, A. A., Pontryagin, L. S. and Witt, A. A. On a statistical study of dynamic systems, J. Exp. Theor. Phys. 1933, 33, 165-180 Ito, K. On stochastic differential equations memoirs, Am. Math. Soc. 1951, No. 4 Kozin, F. A survey of stability of stochastic systems, Automatica 1969, 5, 95-112 Kozin, F. Stability of the linear stochastic system, Lecture Notes in Math. 1972, 294, 186-229 (Springer Vedag) Arnold, L., Kliemann, W. 'Qualitative theory of stochastic systems', Prob. Analysis and Related Topics(Ed. A. H. BharuchaReid), Academic Press, Vol. 3 (1981) Stmtonovich, R. L. A new representation for stochastic integrals and equations, SIAM Journ. Control 1966, 4, 362-371 Wong, E., Zakai, M. On the convergence of ordinary integrals to stochastic integrals, Ann. Math. Statist. 1965, 36, 1560--1564 Khazminskii, R. Z. On stochastic processes defined by differential equations with a small parameter, Theory Prob. and Appls. 1966, 11, 211-228 Lin, Y. K. 'Structural response under turbulent flow excitations, CISM Lectures (Ed. H. Parkus), Springer-Verlag, No. 225 (1977) Ariaratnam, S. T. and Tam, D. S. F. Moment Stability of Coupled Linear Systems Under Combined Harmonic and Stochastic Excitation, Clarkson, Pitman: London, pp. 90-105 (1977)

11 12 13 14 15 16 17 18 19 20

Mitchell, R. R. and Kozin, F. Sample stability of second order linear differential equations with wide band noise coefficients, SIAM J. Appl. Math. 1974, 27, 571-605 Stratonovich, R. L. Topics on the Theory of Random Noise, Gordon and Breach, New York, Vol. 1 (1963) Weidenhammer, F. Stabilit/itsbedingungen filr Schwinger mit zufalligen parametererregungen, Ingenieur Archiv 1964, 33, 404415 Ariaratnam, S. T. 'Dynamic stability of a column under random loading', Proc. Int. Conf. on Dynamic stability of Structures, Pergamon, New York, p. 267 (1967) Wedig,W. Regions of instability for a linear system with random parametric excitation, Lecture Notes in Math 1972, 294, 160-172 (Springer Verlag, New York) Samuels, J. C. On the mean square stability of random systems, Trans. IRE PGIT-5 1959, 248 Brockett, R. W. Lie theory and control systems defined on spheres, SlAM J. Appl. Math. 1973, 25, 213-225 Willems,J. L. 'Moment stability of linear white noise and colored noise systems', Stochastic Problems in Dynamics (Ed. B. L. Clarkson), Pitman, London, pp. 67-89 (1977) Willsky,A. S., Marcus, S. I. and Martin, D. N. On the stochastic stability of linear systems containing colored multiplicative noise, IEEE Trans. AC, 1975, 20, 711-713 Kistner, A. 'On the moments of linear systems excited by a colored noise process, Stochastic Problemsin Dynamics(Ed. B. L. Clarkson), Pitmann Press, pp. 36-53 (1977)

Probabilistic Engineering Mechanics, 1986, Vol. 1, No. 1 : 21

Stability o f stochastic d y n a m i c a l s y s t e m s : F. K o z i n 21 22 23 24 25 26 27 28 29 30 31 32 33

22

Infante, E. R. On the stability of some linear nonautonomous random systems, ASME J. App. Mech. 1968, 35, 7-!2 Kozin, F. On almost sure stability of linear systems with random coefficients, J. Math. and Phys. i963, 42, 59-67 Caughey, T. K. and Gray, A° H., Jr, On the almost sure stability of linear dynamic systems with stochastic coefficients, ASME J. Appl. Mech. i965, 32, 365 Kozin, F. and Wu, C. M. On the stability of linear stochastic differential equations, J. AppL Mech. 1973, 46, 87-92 Blankenship, G. Stability of linear differential equations with random coefficients, IEEE Trans. AC-22 1977, 834-838 Coppel, W. A. Stability and Asymptotic Behavior of Differential Equations, Heath, Boston (1965) Kozin, F. and Sugimoto, S. 'Decision criteria for stability ef stochastic systems from observed data', Stochastic Problems in Dynamics (Ed. B. L. Clarkson), Pitman, London, pp. 8-35 (1977) Kliemann, W. and R/imelin, W. 'On the growth of linear systems parametrically disturbed by a diffusion process', Report No. 27, Department of Mathematks, Un. of Bremen (July 1981) Kozin, F. and Prodromou, S. Necessary and sufficient conditions for almost sure sample stability of linear Ira-equations, SIAM Y. Appl. Math. 1971, 21, 413-424 Milstead, R. M. and Kozin, F. The stability of a moving elastic strip subjected to random parametric excitation, ASME K~pt. Mech. !979, 46, 404-410 Fletcher, R. and PoweU, M. J. D. A rapidly convergent descent method for minimization, Computer J. 1963, 6, 163 Ly, B. L. Topics in the stability of stochastic systems, Ph.D. Dissertation t974, Dept. of Civil Eng., Un. of Waterloo, Ontario, Canada Samuels, J. C. On the stability of random systems and the

Probabilistic Engineering M e c h a n i c s , 1986, Vol. 1, N o . i

34 35 36 37 38 39 40 41 42

43 44 45

stabilization of deterministic systems with random no~se~ 5 Acoust. Sac. Am 1960. 32. 594~01 Wedig, W. Stabilit/it stochastischer systeme. Z A M M !975. 554 185-187 Nevelson, M. B. Some remarks on the stability cf a ~ioeaz" stochastic system, Y. AppL Math. Mech. 1966. 30..;332-!335 Bogdanoff, J. L. and Kozin. F. Moments of the outpu~ of iinear random systems. Y. Acoust. 8oc. Am. t962. M, i063-!066 Brockett, R. W. Parametrically stochastic linear differentia1 equations, Math. Programming Study I976o 5. 8-2~ Brockett, R. W. 'Lie algebras and Lie groups in control t h e o ~ ° Geometrk Methods in System Theory (Eds. Brockett and Mayne), Reidel, Boston (1973) Khasminskii, R. Z. Necessa~ and sufficient conditions for the asymptotic stability of linear stochastic systems. Theor. Prob. end Appls° !967, 12. 144-147 1to, K. and McKean° H. P. Jr, Diffusion Processes an~ their Sample Paths, Springer-Verlag, New York (1965) Nishioka, K. On the stability of two-dimensions1 linear stochas~ tic systems, Kodai Math. Sere. Rap. 1965.27. 21!-230 Kozin, F. and Sugimoto, S. °Re!aliens between sampie anc~ moment stability for linear stochastic differential equations° Proceedings of the Conf. on Stochastic Diff. Eq, ~Ed. D. Mason}~ Academic Press, pp. 145-162 (1977) Arnold, L. 'A formula connecting sample ana moment stability ~ linear stochastic systems, Repor~ No. 92. Dept. of Math.. U~. e)r Bremen (May 19831 K!iemann, W. ~Quatitative theorie nichtlinearen stochas~ischer systems', Repor~ No. 32. Dept. of Math.. Un. of Breme~ (Aug. I98G} Sunahara, Y. et aL. Noise stabilization of non-linear dynam~caR systems, Trans. Japan Soco Mech. End. !974, 4~, 338