WA3 - 10:45 *

\'IBRATIOSAL COSTROL OF NONLIXEAR SYSTEblS

Proceedings of 23rd Conference on Decision and Control Las Vegas, NV, December 1984

Richard Bellman * *

Joseph Bentsman and Semyon bl. bleerkov

Departments of Mathematics Electrical Engineering 6 Medicine University of Southern California

Los hgeles, CA 3 O O O i

ABSTRACT

The purpose of this paper is to establish the basis of vibrational control theory for nonlinear systems. The notions of vibrational stabilizability and vibra- tional controllability of nonlinear finite-dimensional systems are introduced and analyzed. Calculation formu- lae are derived. Transient behavior of vibrationally controlled nonlinear systems is studied. Several exam- ples, including forced Duffing, Rayleigh and Van der Pol equations are discussed.

INTRODUCTION

This paper is concerned h-ith vibrational control of nonlinear systems. Linear systems have been studied in [I] \<here the criteria of vibrational stabilizability and.vibrationa1 controllability have been formulated. A few facts concerning vibrational control of nonlinear systems have also been discussed in the literature (see, for instance, [ 2 ] , [3]), However, general theory of vibrational control for nonlinear systems is unknown.

Since the method of vibrational control has found several important applications (see [4] - [ 6 ] and also [ 3 ] ) , de\.elopment of such a theory is desirable. In the present paper this theory is developed and applied to a number of dynamical systems.

The class of systems considered here is:

dx/dt = Y(x, A) , X : Rn x Rm + Rn, (1)

where xeRn is the state, t is dimensionless time and X E R ~ is a parameter. Assume that for any fixed i = ), €[a, b] , (1) has an equilibrium point x, = xs(A0) Eyal, bl] , where a, bERm and al, blcRn are constant vectors.

Introduce in (1) parameEric vibrations according to the law

x = + f(t), ( 2 1

* This work was performed under the auspices of the U.S. Department of Energy under Grant DE-FGOZ- 84ER13205.

** On March 19, 1984, Professor Bellman passed away. His death has ended a life of enormous intellectual intensity and achievements. His influence on modern culture is difficult to elraluate. Historians of science will do that. For us, his students and friends, he was an ultimate example of courage, strength and optimism, devotion and support.

CH2093-318410000-0084 $ 1 .OO 0 1984 IEEE 84

Department of Electrical & Computer Engineering

Illinois Institute of Technology Chicago, IL 60616

where i o = const and f(t) is a T-periodic zero average vector. As a result we obtain

dx/dt = X(x, A. + f(t)) . (3)

Definition 1. An equilibrium point xS(i.oj of (1) is said to be vibrationally stabilizable (v-stabilizablej if for any E > 0 there exists a T-periodic zero mean vector f(t) such that (3) has an asymptotically stable T-periodic solution xS(t), - m < t < m , characterized by

where 7

ys = 1' xs(t)dt. 0

Definition 2. An equilibrium point x s ( l 0 ) of (1) is said to be totally vibrationally stabilizable (t-stabi- lizable) if it is v-stabilizable and xS(t) = const = xs(A0) , - m < t < =,

Frequently, however, only one component of xs re- quires o r permits vibrational control. This is the case for chemical reactors where only one of two, rate of conversion or the temperature, can be vibrationally controlled. Having in mind such a situation, we intro- duce Definition 3. An equilibrium point xS(i.oj of (1) is said to be partially vibrationally stabilizable (p- stabilizablej with respect to the component xsi(Xo) of xs(Xo) if for any 6 > 0 there exist T-periodic zero mean vector f(t) and X1 = const €[a, b] such that sys- t em

dx/dt = X(x, rl + f(t)) (5)

has an asymptotically stable T-periodic solution, xS(t), - m < t < =, the i-th component of which is character- ized by

Assume that ( 3 ) has the form

dx/dt = X(x, A ) t pX (t, x), X1 : R+ X Rn + Rn, (7) 0 1 where X1 (t, - ) is a periodic function. Equation ( 7 ) describes a large class of nonlinear systems with para- metric oscillations. Indeed, several examples of Xl(t, x) can be given as:

(i). Xl(t, x ) = L(t), where L(t) is a periodic zero mean vector. In this case the vibrations will be referred to as vector additive. In a specific case where all but the last components of L(t) are zero, the vibrations will be referred to as periodic forcing;

(ii). Xi(t, x) = B(t)x, where B(t) is a periodic zero mean matrlx. In this case the vibrations will be

called linear multiplicative;

(iii). Xl(t, x) = B(t)T(x), : : Rn + Rn, Here the vibrations are nonlinear multiplicative.

In the present paper the theory of vibrational con- trol is developed for the cases of periodic forcing, linear multiplicative and vector additive vibrations. The case cf nonlinear multiplicative vibrations requires further investigation.

The paper consists of three parts. In Part I exis- tence of stabilizing vibrations is discussed. In Part I1 design formulae are given and vibrational controlla- bility is analyzed. In Part I11 the transient behavior of vibrationally controlled systems is studied. The Appendix contains mathematical techniques developed for analysis of the problem at hand. Due to the space limi- tation, all proofs are omitted.

P.-\RT I , VIBR.\TIO?\'.-\L ST.ABILIZ.4BILITY

1. Linear 3lultiplicative Vibrations

Theorem 1. Assume that 1). X ( 0 , i o ) = 0;

2). there exists a sufficiently large set Cc Rn(xS~C.) such that X(x, is continuously differentiable for all xcC;

5 ) . matrix G 2 ;);(x, j o j / ; x ~ is nonderoga- , x=o

tory.

Then a sufficient condition for t-stabilizability of 0 of (1) by linear multiplicative vibrations is

Tr G < 0. (8 1 If, in addition, matrix G + IB(t) has at least one char- acteristic exponent with nonzero real part, condition (8) is also necessary.

Example. Duffing equation,

+ ax - bx + cx3 = 0, a,b,c, > 0. (9)

The linearization around xs = 0 is

x + ax - bx = 0.

Since a > 0, 0 of the Duffing equation is t-stabilizable by linear multiplicative vibrations.

Theorem 1 is applicable only t o trivial equilibria since for xg # 0 the usual equilibrium transfer to the origin cannot be employed.

Theorem 2 . Nonzero equilibria of (1) are not t-stabi- lizable by linear multiplicative vibrations.

As far as the v-stabilizability of nontrivial equi- libria is concerned, the trace condition is not valid any more. However, the following necessary (Theorem 3 j and sufficient (Theorem 4 ) conditions can be formulated.

Theorem 3. Suppose that

1). X(X,, r,) = 0, xs + 0; 2). assumption 2 of Theorem 1 holds.

Then xs # 0 of (1) is v-stabilizable by linear mul- tiplicative vibrations only if there exists a T-periodic zero mean vector $(t)E?, VtE[O, =) and 0 < s o << 1 such that for any 0 < 1 f 5 f o the following is true:

P(t) = ;x(x, . ) / a x I ',O , x = x + 5 + 2(t)

and 1

* T - p(t) = lim AI ,' P(t)dt.

T ' w T 0

Examule. Consider

x - x + x 2 + 2 x + x 2 + 1 = o ,

or in terms of x1 E x and x2 E x, , .

x1 = x2

Here xs = [-1 0 I T and xs(t) = [l+ilfil(t), E2+q2(t)] T . Matrix P(t) is given, then, by

P(t)=1 I i /-2-2x 1-2x ' 1 I-2t1-2C1(t) 1-222-2r L 2J x=xs (t)

1

- Since for 5 2 < 0.5, Tr P(t) > 0, xs of (loj is not v- stabi1i;able by linear multiplicative vibrations.

Theorem 4 . Let assumptions 1 and 2 of Theorem 3 hold. Then the equilibrium ~~(i.0) of (1) is v-stabilizable by linear multiplicatixre vlbrations if there exists T-per- iodic zero mean matrix :B(t) such that a). the state-transition matrix $(t, 0), tE(-m, a), of

x = pB(t)x is almost periodic; I -

3 ) . matrix S(t, 0) = lim --! i C(t, 0)dt is nonsingu- 1 '

lar; 0

y) , the equation z = Z ( z ) ,

where Z ( z ) = s -1 (t, O)X(C(t, O j z , j 'ol ,

has an equilibrium point z s characterized by

and Q = ; Z ( z j / 3 z , is a Hurwitz matrix. z = z

S

Remark. If for any ;IB(t) matrix Q is noncritical, i.e., has no pure imaginary eigenvalues, and is not Hurwitz and/or (12) does not hold, assertion of Lemma 2.q im- plies that x~().~! is not v-stabilizable by fast linear multiplicative vlbrations.

The situation when condition ( 1 2 ) holds is rather an exception than a rule, since vibrations usually in- duce a transition of the nonzero steady state, i.e., the resulting periodic solution, xs(t), has an a priori unknom average value, x*, such that x* # xs ( i o ) . Con- sequently, equation (X6) has equilibrium z s character- ized by

* h(r, z s ) = x # x ().o). (13)

F o r example, in the case of linear multiplicative vibra- tions (13) becomes -

Tr P(t) - < 0, where O(t, 0 ) z s = x* # xs("o).

85

Khen the transition described by (13) occurs, we will say that xS(Xo) is not preserved. If, however, ( A B ) and, hence, ( 1 2 ) hold, we will say that x ~ ( ? . ~ ) is preserved. Sufficient condition for preservation of nonzero equilibria is given by Lemma 1.

Lemma 1. Suppose that all assumptions of Theorem 4 hold. Then all equilibria xs(Xo) of (1) are preserved and no new equilibria x* # xS(Ao) are created by linear multiplicative vibrations if there exists a T-periodic zero mean matrix uB(t) such that conditions a) and E ) of theorem 4 hold and the following is true:

y ) d ( t , O)X(@(t, 0)z) = 8 ( t , O)X(@(t, O)z), vz.

In the case when the preservation of equilibria does not occur, introduction of vibrations can result, at most, in p-stabilizability, A sufficient condition for p-stabilizability is given by

Theorem 5. Let all assumptions of Theorem 4 hold. Then the equilibrium xS(Xo) of (1) is p-stabilizable by lin- ear multiplicative vibrations with respect to the com- ponent xsi(A0) if there exist vector Xl&[a, b] and T- periodic zero mean matrix B(t) such that conditions a ) and 9) of Theorem 4 hold and y) equation (11) with

Z ( z , A1) = Z ( z ) = O-l(t, O)X(@(t, O ) Z , X1) ( 1 4 )

has an equilibrium point zs(A1) characterized by

@(t, 0li Zs(hl) = Xsi(X0)’ (15)

where O(t, 0)i is the i-th row of matrix D(t, 0), and

Q = a z ( z ) / a z is a Hurwitz matrix.

Remark. If Q is noncritical and not Hurwitz and/or (15) does not hold for any uB(t), then, by virtue of Lema SA, xS(Xo) is not p-stabilizable by fast linear multiplica- tive vibrations.

2. Periodic Forcing

In this Section we study the stabilizing effect of periodic forcing on a system of the form

where x is a scalar, is the i-th time derivative of x, ai are constant coefficients, and f(x, e . . , x(n-1)) is an analytic around xs function such that f(0, I . . , 0) = 0 if (16) has an equilibrium xs = 0 and f(xs,O,.. .,O) + anxs = 0 if xs # 0; grad f(0, e . . , 0) = 0 in both cases. Lemma 2. An equilibrium xs = 0 of (16) is preserved under sufficiently fast periodic forcing if f(x, e - - , x(~-I)) has no terms of the form

g2 , g2(*) is not odd; (17)

an equilibrium xs # 0 of (16) is preserved if f(x, .., has neither ( 1 7 ) nor terms of the form

where g1 and g2 are analytic functions of their argu- ments.

Theorem 6. An unstable equilibrum xs of (16) is v-sta- bilizable by periodic forcing only if f(x, . .., x(n-1)) has terms of the form

where grad g1(0 ) # 0 and g2 ( e ) is not odd.

3 . Vector Additive Vibrations. In this Section we restrict the discussion to a

class of odd functions, X, khich guarantee the preserva- tion of equilibria, xs, under vector additive vibrations.

Consider the Taylor expansion of X(x) X(x, Xo) of (1) around an equilibrium point xs:

le will say that X(x) in the vicinity of xs is an odd r-algebraic function if:

1 ) . expansion (20) has r < - terms; 2 ) . expansion (20) has no terms with i = 2k, k =

0, 1, e . . , [r/2] t 1.

Without loss of generality ve assume that xs = 0. The i-th term of (20) with xs = 0 and x = y + u can be represented as:

where the elements of vector vi are algebraic forms Of order i with respect to the components of vector u; the elements of matrix Pi are algebraic forms of order i-1 with respect to u , and H.O.T.(y) stands for higher order terms in y.

Define u in (21) via vector additive vibrations i;L(t) as

u(t) = ulL(t)dt, h ( 2 2 )

and introduce a matrix

+ ... t - (S-1) ! Ps(u(t))ldt, (23)

where S = r-1 if r is even and S = r if r is odd.

Theorem 7. Assume that X(x, Xo) in the vicinity of 0 is an odd r-algebraic function. Then, 0 of (1) with )~ = X0 is v-stabilizable by vector additive vibrations if there exists uL(t) such that A is a Hurwitz matrix.

Remark. If for any pL(t) matrix x is noncritical and not Hurwitz, by virtue of Lema 2A, 0 of (1) is not v-stabilizable by fast vector additive vibrations.

PART 11. VIBRATIONAL CONTROLLABILITY

be the linearization of (1) with A = Xg at a steady state xS. Assume that after introduction of vibrations equilibrium xs is preserved and the linearization of the averaged equation around its equilibrium Zs, corre- sponding to xs, is represented as

= (G + BIZ, where x = (E..) is a constant matrix. Definition 4. An element gij of matrix G is said to be

1 J

v i b r a t i o n a l l y c o n t r o l l a b l e i f t h e r e e x i s t s a p e r i o d i c ze ro mean o s c i l l a t i o n o f t h e f o r m ( 2 ) s u c h t h a t b i , f 0 .

by v a r i o u s t y p e s o f v i b r a t i o n s a r e g i v e n b e l o w .

1. L i n e a r M u l t i p l i c a t i v e V i b r a t i o n s .

Theorem 8 . Let assumptions 1 and 2 o f Theorem 1 h o l d . Then t h e r e e x i s t s '0 s u c h t h a t f o r a l l E 5 and l i n - e a r m u l t i p l i c a t i v e v i b r a t i o n s ( a / E ) B ( t / E ) ,

C o n d i t i o n s o f v i b r a t i o n a l c o n t r o l l a b i l i t y o f g i j ' s

- B z R - G ,

where

R = lim - I C ( T , O ) G C(T, O)dT , T = t / E (25) 1 T 1 -1

0 and ' C ( T , 0 ) i s a n a l m o s t p e r i o d i c s t a t e t r a n s i t i o n matr ix of dx/d- i = ~ ? B ( T ) x .

2 . P e r i o d i c F o r c i n g

Theorem 9 . Assume t h a t e q u a t i o n ( 1 6 ) s a t i s f i e s c o n d i - t i o n s o f Lemma 2 . Then c o e f f i c i e n t a i o f ( 1 6 ) i s v i b r a - t i o n a l l y c o n t r o l l a b l e b y p e r i o d i c f o r c i n g i f f ( x , e . . , x ( n - 1 ) ) h a s t e r m s o f t h e f o r m ( 1 9 ) . I f t h e p e r i o d i c f o r c i n g i s ( x / E ) i , ( t / E ) , t h e r e e x i s t s EO such t h a t f o r a l l E 5

where U(T) = cxjZ(T)dT and gi, i = 1, 2 , a r e d e f i n e d i n (19) *

3. V e c t o r A d d i t i v e V i b r a t i o n s

Theorem 10 . Coef f ic ien t a i , i = 1, * * * , n of (16) i s v i b r a t i o n a l l y c o n t r o l l a b l e b v v e c t o r a d d i t i v e v i b r a - t i o n s form

where odd. t h e r e

- b . =

i f f u n c t i o n f ( x , . . , k ( n - l ) ) h a s t e r m s o f t h e

g l ( * ) i s odd , g rad g l (0 ) f 0 , and g 2 ( - ) i s n o t I f t h e v i b r a t i o n s h a v e t h e f o r m ( a / E ) L ( t / E ) , e x i s t s EO s u c h t h a t f o r a l l E 2 -c0,

1 ( k - l ) ! d z k 1 1 I----- J

i z=o k= 1

where V ( T ) = d L ( T ) d T , T = t / E

PART 111. TRAVSIENT BEHAVIOR

I n P a r t I , t h e e x i s t e n c e o f s t a b i l i z i n g v i b r a t i o n s has been ana lyzed . In Pa r t 11, t h e e f f e c t s o f p a r a - m e t r i c o s c i l l a t i o n s o n l i n e a r i z e d ( l o c a l ) b e h a y i o r o f n o n l i n e a r s y s t e m s h a v e b e e n s t u d i e d a n d a p p r o p r i a t e ca l cu la t ion fo rmulae have been deve loped . Pa r t I11 i s d e v o t e d t o t h e e f f e c t s o f p a r a m e t r i c o s c i l l a t i o n s o n g l o b a l b e h a v i o r o f n o n l i n e a r s y s t e m s . N a m e l y , t h e o s c i l l a t i o n s i n d u c e d t r a n s i t i o n s i n t h e t r a j e c t o r i e s o f n o n l i n e a r s y s t e m s are ana lyzed . The b a s i s f o r t h i s a n a l y s i s i s g iven by

Theorem 11. S u p p o s e t h a t

1 ) . f o r s y s t e m (AS) of t he Append ix , a s sumpt ion 2 o f Lemma 1 A h o l d s ;

2 ) . t h e g e n e r a l s o l u t i o n h ( T , c ) o f (A3) i s bounded f o r a l l T C [ O , a) and a l l C E G ;

3 ) . u n i f o r m l y w i t h r e s p e c t t o YET: t h e r e e x i s t s limit (Ai ) .

Then f o r e v e r y p o s i t i v e 2 and c a s s m a l l a s d e - s i r e d and K as l a r g e as d e s i r e d t h e r e e x i s t s ~ ~ ( 5 , K , o) s u c h t h a t f o r e a c h 0 < E - < io t h e f o l l o w i n g i s t r u e :

< 5, T = t / E , T E [ T 0' T o + K I > (29)

where - 1

T ' u T 0

TI x ( y ( t ) ) lim - I 1 h ( s , y ( t , yo , ~ ~ ) ) d s , (30)

and y ( t , y o , TO) and z ( t , y o , :0) a r e s o l u t i o n s o f t h e i n i t i a l v a l u e p r o b l e m s f o r (A5) and (Ab) , respec t ive ly , p r o v i d e d z ( t , y o , T ~ ) t o g e t h e r w i t h i t s - - v i c i n i t y b e - l o n g s t o ? f o r a l l T E [ T O , =).

( i i ) . Khenever (A6) h a s g l o b a l l y a s y m p t o t i c a l l y s t a b l e e q u i l i b r i u m zs, i n e q u a l i t y ( 2 9 ) h o l d s f o r a l l t c [ t o , m) *

( i i i ) . h h e n e v e r (X6) h a s l o c a l l y a s y m p t o t i c a l l y s t a b l e e q u i l i b r i u m z s , t h e r e e x i s t s a domain Cl, z s ~ C l , s u c h t h a t ( 2 9 ) h o l d s f o r a l l t E [ t O , =) i f zo = y E -

The proof of Theorem 11 i s based on [i] and [8 ] .

Examples. 1. D u f f i n g e q u a t i o n w i t h l i n e a r m u l t i p l i c a - t i v e v i b r a t i o n s ,

o *

x + ax - ( b + ( a / E ) s i n ( t / E ) ) x + cx = 0 . 3 (32)

To o b t a i n e q u a t i o n (A6) c o r r e s p o n d i n g t o ( 3 2 ) , i n - t r o d u c e t h e g e n e r a t i n g e q u a t i o n

dxl/dT = 0 ,

dx /dT = as in ( . r )x l ; (T = t / c , x1 = x, x = dx/d-c) 2

and t h e r e s u l t i n g s u b s t i t u t i o n

(33) 2

X1(T) = y1 (T) ; X,(.) = -ucOs(T)y1 (T) + Y2(T). (34)

Equat ion (A6) then becomes

dy /dT = E { - ~ c o s ( T ) ~ ~ + y21 1 dy2/dT = c I - a ( y - s c o s ( ~ ) y ) + byl - cy1 3

2 1 + aces(:) (y, - crcos(T)yl),

a n d c o n s e q u e n t l y , t h e a v e r a g e d e q u a t i o n ( i n t h e o r i g i - n a l t i m e t ) i s :

2 + az - (b - J / 2 ) i + c i = 0 , 2 3 (35)

From (31) and (33) it f o l l o w s t h a t - x l ( z ( t ) ) = Z l ( t ) J x 2 ( z ( t ) ) =

- (36)

Thus , t he ave rage behav io r o f (32 ) i s d e s c r i b e d b y (35 ) . F igs . l a , b show t h e c h a n g e s i n t h e g l o b a l b e - h a v i o r o f D u f f i n g e q u a t i o n , i n d u c e d b y l i n e a r m u l t i p l i - c a t i v e v i b r a t i o n s . F i g . IC gives a -compar ison of the s o l u t i o n s x l ( t ) o f ( 3 2 ) a l o n g w i t h x l ( z ( t ) ) d e f i n e d by (36) . Obvious ly , E 2 0 . 1 e n s u r e s a n a c c e p t a b l e p r e c i - s i o n o f t h e a v e r a g e d d e s c r i p t i o n .

2 . R a y l e i g h e q u a t i o n w i t h p e r i o d i c f o r c i n g

x + 'A1 ( 5 - 1 ) x + x = ( a / E ) s i n t / E . - 2 .

I n t r o d u c i n g t h e g e n e r a t i n g e q u a t i o n

1 1 - 2 dx /d.r = 0; dx2/d-i = I s i n T ; ( T = t / c , x ' = r , x =dx/d:),

a 7

the resulting substitution,

2 X1(T) = y,(T) ; X ( T ) = -&COST + y2(T), (37)

yields the equation in standard form dy /d.r = E(-CXCOST + y ) 1 2

(Y2-CXCOST) 3 dy /dT = - E P ~ [ 2 - (Y2-OCOST)] - E y 1

and the averaged equation (in time t)

Due to (37) relation ( 3 6 ) holds here as well. The change in the global behavior of Rayleigh equation due to periodic forcing is presented in Figs. 2a,b. Tra- jectories x2(z(t)) vs. xl(z(t)) and x2!t) 1's. xl(t) are given in Fig. 2c by dashed and solid llnes, respectively (a = 2, E = 0 . 1 3 ) . An analogous situation is also ob- served in the case of Van der Pol equation. Namely, the introduction of vector additive vibrations results in a destruction of the limit cycle and in the v-stabi- lization of the trivial solution.

APPESDIX - LEMMATA

Assume that (3 ) has the form dx/dt = X(x, X o ) + (a/E)Xl(t/E, X), 2 7 ~ ~ = T, ( A l )

where E and 3 are positive constants and E << 1. This does not necessarily imply that ;I of (3 ) is large, since a can be small; u, however, is not an asymptotic parame- ter whereas E is.

Introducing fast time T = t/E, equation (Al) be- comes

dx/d-c = EX(X) t a X 1 ( ~ , X ) , X(X) 5 X(X, hO).(A2)

Consider an equation

dx/dr = uX ( T , x), 1

which is further referred to as the generating equation, and assume that there exists a unique solution of (A3) defined by every initial condition XOEQCR" for all T - > 0. Denote the general solution of (A3) as

X(T) = h(T, c), C = COnSt. Introduce in (A2) a substitution:

X(.) = h(T, Yb)) 9 YER". (A4 1 Assuming that X1(T, x) is differentiable with respect to x for all T 0 and recognizing that

det[ah(T, y)/ay] = det[ah(O, y)/ay]

expfi tr[aXl(r, y)/ay]d~l, T

0 we obtain the following equation in the standard form [9] for Y(T):

dy/d.r = E [ ah/ay]-'X(h(~, y)) = E Y ( T , y) . 0 (A5 1

Thus, substitution (A4) reduces (A2) to a standard form. Introduce, finally, the averaged equation:

dz/dr = EZ(Z), (A61

Z(y) = lim - I i [ah/ay]-'X(h(~, y))dT. (A7 1 where 1

T wT 0

Lemma 1A. Assume that

T 1

1). h(T, c) is almost periodic with respect to T ;

2 ) . 1 ~ Y ( T , y)1 I 5 N, VTE[O, -1, VYE~CR",

I IY(T, Y')-Y(T, y") i i 5 K I i y ' - y ' ' i I , ~ ~ € 1 0 , -1, Vycsl CRn;

3 ) . Z(y) is continuously differentiable for all y&.

Then the equilibrium point ~ ~ ( 1 . ~ ) of (1) is v-sta- bilizable if there exists T-periodic zero mean vibration f(t) about 1.0 such that (A6) has an equilibrium point zs characterized by:

1 T' lim - 1 J h(7, z ) = x

S S ' (A8 1 0

Q = 3Z/az is a Hurwitz matrix, (A9 1 l z = z S

Lemma 2A. Let assumptions 1 - 3 of Lemma 1A hold and matrix Q in (A9) be noncritical. Assume that for any 2n~-periodic zero mean f(t) (A6) has no equilibrium point, which satisfies both (A8) and (A9). Then there exists such that xs of (1) is not v-stabilizable by fast ( E < E ) oscillations.

Lemma 3A. Consider an almost periodic system

- 0

EY(t, y), ~ E R ~ , Y : R + x R ~ + R ~ , Y(t, 0) = 0. (.410)

Assume that Y(t, e ) is continuously differentiable with respect to y for all ye% R~(~EQ), Y(t, a ) has the linearization about zero,

(A1 1) Y = 0 ,

VtE[O, m) . (A121 The linear system

= EF(t)y (A13)

is reducible in Liapunov's sense. The time-invariant system

1 = EZ(Z) , Z(z) = lim 1 Y(t, z)dt (A14)

TI- 0

T'

has an equilibrium z s = 0. Matrix F of the time invariant system

* - - 1 T' z = Fz , F = lim J F(t)dt, (A151

0

is noncritical. Then there exists such that for any 0 < E 2 the trivial solution y(t) = 0 of (A10) is

(i). uniformly asymptotically stable if 0 of (A15) is asymptotically stable;

(ii). unstable, if 0 of (A15) is unstable.

Lemma 4.4. Let assumptions 1 - 3 of Lemma 1A hold. Then the equilibrium point x s ( X 0 ) of (1) is p-stabilizable with respect to component xsi(X0) of x S ( X o ) if there exist hlE[a, b] and T-periodic zero mean vector f(t) such that ( A 6 ) , corresponding to (1) with A = AI, has an equilibrium z s , characterized by

lim - I i h. ( T , z ) d ~ = x .(A ), 1 T I s1 0 (A161

T1-raT 0

Q = az(xl, z)/azj is a Hurwitz matrix. (A17) z = z

88

Lemma 5A. Let assumpt ions 1 -3 o f Lemma 1.1. hold and ma- t r i x Q i n (A9) b e n o n c r i t i c a l . Assume t h a t f o r any 27E-pe r iod ic ze ro mean f ( t ) a n d a n y ) . l ~ [ a , b ] (A6) h a s n o e q u i l i b r i u m p o i n t , w h i c h s a t i s f i e s b o t h (A16) and (A17). Then xS(J.o) of (1) i s n o t p - s t a b i l i z a b l e w i t h r e s p e c t t o i t s component xsi(Xo) by fas t (E 5 E O )

o s c i l l a t i o n s .

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x 1 "I -5

b C

a b F ig . 2

c