TECHNICAL STABILITY OF MOTION WITH

RESPECT TO SEPARATE SPECIFIC COORDINATES

A. A. Martynyuk UDC 517. 912

The n e c e s s i t y of inves t iga t ing the technica l s tabi l i ty of mot ion with r e s p e c t to s e p a r a t e speci f ied coo rd ina t e s a r i s e s in many applied p r o b l e m s . An example is the s tudy of mul t id imens iona l s y s t e m s and s y s t e m s with cyc l i c coo rd ina t e s , nonholonomic s y s t e m s , e tc .

We s ta te the ge ne ra l method of inves t iga t ing p r o b l e m s of such kind, based on the second method of A. M. Lyapunov [1] and s o m e in teg ra l inequal i t ies [3].

1. We c o n s i d e r the equat ions of p e r t u r b e d mot ion of the s y s t e m

dx~ , x . ) (s = 1, 2, . , n) , ( 1 . 1 ) d--7- ----" X~ (t , x , . . . . . x k, xk+ l . . . . . .

w h e r e the Xs a r e r e a l v a r i a b l e s that c h a r a c t e r i z e the devia t ion of the s y s t e m f r o m the unpe r tu rbed mot ion ; the Xs(Y , x i , . . , Xn) a r e r ea l funct ions defined and cont inuous fo r all va lues tE Jt in s o m e reg ion G of the space {Xs} containing the point x s = 0; J t = [to, to + T].

Having denoted the v a r i a b l e s x t , . . . , x k by Yi (i = 1, 2 . . . . . k) and the r e m a i n i n g m = n - k v a r i a b l e s by zj (j = k + 1 . . . . . m), we r e wr i t e s y s t e m (1.1) in the f o r m

-@; . . . . g~ (t, g, . . . . . gk, z, . . . . . Zm) (i ---- 1, 2 . . . . . ~);

(I .2) = z j ( t , z , . . . . . zm, y , . . . . . v ) .

We a s s u m e that the solut ion xs( t ) of s y s t e m (1.1) ex i s t s in the r eg ion Gp(y)and z fi D, and can be continued along zj E D for t s Jt" This condi t ion e l imina te s the poss ib i l i ty of the coo rd ina t e s zj going off to infinity in a f ini te t ime .

The r eg ions G p ( y ) a n d D have the f o r m

D = Z: {X \ / = l / )

In o r d e r to apply the Lyapunov funct ions cons ide red in [2] to the inves t iga t ion of technica l s tab i l i ty based on coord ina t e s Yi, we d e t e r m i n e an aux i l i a ry function, loca l ly l a rge with r e s p e c t to this v a r i a b l e .

The funct ion V(t, y, z) is ca l led loca l ly l a rge with r e s p e c t to the v a r i a b l e s y ( local ly y - l a r g e ) if f o r a spec i f ied e s t i m a t e A fo r any va lue 0 < e* < A and t o ~ 0 t h e r e ex i s t s a pos i t ive n u m b e r o" such that outs ide the sphe re Zy~ -- cr the inequal i ty V(t , y , z) > c* holds fo r t ~ t o and z E D.

Let c(t, ~) be s o m e cont inuous funct ion on the (t, ~) plane on the se t E. We c o n s i d e r the m a x i m u m solu t ion of the Cauehy p r o b l e m

dr d--i- = c [ t , r + ~l it)l; r (t o) = r 0. (1.3)

Inst i tute of Math6mat ics , A c a d e m y of Sc iences of the Ukra in ian SSR, Kiev, T r a n s l a t e d f r o m P r i k - l adnaya Mekhanika, Vol. 8, No. 3, p p . 87 '91 , March , 1972. Or ig ina l a r t i c l e submi t ted May 31, 1971.

�9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. I0011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

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We note that by the m a x i m u m so lu t ion r(t) = r( t ; r0, to) of the p r o b l e m (1.3) we m e a n a so lu t ion on the m a x i m u m r a n g e of e x i s t e n c e such tha t if ~(t) i s any o the r so lu t ion of p r o b l e m (1.3), then the in- equa l i ty 7(t) _< r(t) holds fo r all va lues of t be longing to t he i r g e n e r a l i n t e r v a l .

LEMMA 1. Le t the funct ion c(t, u) be cont inuous and n o n d e c r e a s i n g with r e s p e c t to u fo r t E Jt, and a lso , on this i n t e r v a l le t the fol lowing inequal i ty be sa t i s f i ed :

t

u (t) ~< q (t) -i- ~ c [s, u (s)] ds + r (/0), (1.4) _ to,

w h e r e u(t) and ~(t) a r e cont inuous funct ions of the t i m e t E Jt .

Then

u (t) ~ r (t) -~- ~1 (t), (1.5)

w h e r e r(t) is the m a x i m u m so lu t ion of the o n e - d i m e n s i o n a l p r o b l e m (1.3), ~](t) is a cont inuous funct ion of

t ~ a t.

The a s s e r t i o n of L e m m a 1 is not d i f f icul t to obta in f r o m the c o m p a r i s o n p r i n c i p l e , p r e s e n t e d in the m o n o g r a p h [3]. We now denote

dV r = og + d---f ~ grad VY (t, g, z); (1.6)

t

w(t) =M~" IIo(s)llas, vt~7,. (1.7) to

H e r e ~t = [to, to + T); M is a cons t an t def ined by the condi t ion

M ---- sup (ll grad V I1 g E r A \ F a , z E D) .

The funct ion o-(t) i s such that

llZ(t, z, Y)I/~; II ~(0 I1- (1.8)

LEMMA 2. F o r s y s t e m (1.2) let t h e r e ex i s t a loca l ly y - l a r g e funct ion V(t, y, z) and a cont inuous funct ion c(t, V) that is n o n d e c r e a s i n g with r e s p e c t to V fo r V t E Jr; f u r t h e r m o r e , fo r in i t ia l va lues (tl, r l t t h e r e e x i s t s a m a x i m u m so lu t ion r(t , tl, rl) of the equa t ion

dr - - c It, r + W (t)], (1 .9 ) dt

w h e r e t ie J t ; r l = Vl~,21 ( t l ) - *( t t ) ; Y I ( f A \ r x .

Then, if the condi t ion AgE] ~c(t , v) (1.10) dt [y

i s s a t i s f i e d fo r all tE J t and y E FXFX; z E D, t h e n a t o n g the so lu t ionx( t , Y0, z0), z0 E D the fol lowing inequal i ty hoids :

V (t, t t. x0) ~ r (t, t 1, rt) + tF ( t )

P r o o f . F r o m condi t ions (1.6), (1.7), .(1.8), (1.10) it i s not d i f f icul t to find

dV 7 / - < ~ (t, v) + M II {, (t) It.

I n t e g r a t i n g inequa l i ty (1.12) and apply ing L e m m a 1, we obta in the e s t i m a t e (1.11).

2. We c o n s i d e r the ques t ion of the t echn ica l s t ab i l i ty of mo t ion with r e s p e c t to s e p a r a t e c o o r d i - n a t e s . We in t roduce the nota t ion:

(1.11)

(1.12)

V~n(t) = minlV(t, g, z), Ilg.ll = s zCDI;

V~ax (t) = max IV (t, g, z), ]l Y I[ = A, z E DI;

V ~ (t) --- max [V (t, g, z), II VII < ~, 'z E Dl;

V~ (t) = max IV (t, g, z), ~P~\r~, z 6DI.

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Definition 1. For specified es t imates (X, A, gt) sys tem (1.2) is technical ly y-s tab le if for any solu- tion y(t), beginning in the region

the following inequality holds:

Ily(to)l] <~, zoED,

IIv(t, to, xo)ll < A, v tEJt .

THEOREM 1. The sys tem (1.2) is technical ly y-s tab le if there exist functions V(t, y, z) and c(t, V), indicated in Lemma 2, and the following conditions are satisfied:

1) the inequality (1.10) is valid in the region r A ~ r x, z E D, t E Jr;

2) the maximum solution of the Cauchy 'problem (1.9) with initial value r(t0) = Vk(t0)-~(t0) sat isf ies the inequality

r(t, t o, r o) < V~i, (t) - - ~F (t) (2.1)

for all t E Jr-

Proof . We consider the t r a j ec to ry xs(t), beginning in the region I]Y(t0)II < k, z 0 fi D, and we assume tha t at some moment ti E gt the norm of the solution y(t) attains the value A, i . e . , Ily(tl)II = A. We follow, for change in the function V(t, y, z), along the solution xs(t ). On the basis of Lemma 1 we obtain

V[t. y(O,z(t)] < r ( t , t o, r o) + ~FCt), v t E d t.

Hence, by v i r tue of condition (2.1) we obtain

A t V~,, (tt) < V [ t , y(t), z(t)l ~< r(t, to, ro) < Y.~, (,).

The inequality obtained is cont rad ic tory and hence the assumption that t 1 E Jt is untrue. theorem.

Definition 2. For specif ied es t imates (X, A, Jr), fo rmly if for any solution xs(t), beginning in the region

I{ v (6) II < z,

the following inequality is sat isfied:

(2.2)

This proves the

(X - A) the sys tem (1.2) is technical ly y-s tab le uni-

z(6) ED,

lly(t)]J < A, v t > t 1, (tl, t )6Jf . (2.3)

THEOREM 2. The sys tem (1.2) is technically y-s tab le uniformly if there exist functions V(t, y, z) and c(t, V), indicated in Lemma 2, and the following conditions are sat isfied:

1) inequality (1.10) holds in the region rA , z ~ D for all t E Jt;

2) the maximum solution of the Cauchy problem (1.9) with initial condition r(tl) = VXmax(tl)-~I,(ti) for any value t 1E Jt sat isf ies the inequality

r it, tl, r (t31 < VAin (t) - - ~ (t) (2.4)

for all t > tl, (tl, t) E fit.

The proof is ca r r i ed out in a manner s imi la r to the preceding theorem. The purpose of the following definitions is to more fully take into account the p roper t i es of technical ly y-s tab le t r a j ec to r i e s .

(X > B) the sys tem (1.2) is technical ly quasicom- zj(t)], beginning in the region

z(to) ~ D,

Definition 3. For specified es t imates (k, B, Jr), p re s s ive y-s tab le when for any t r a j ec to ry xs(t ) = [Yi(t),

we find t 1 E gt such that

IIv(t0)ll < ~,

l lg( t ) l l<B, g tE( t l , to+Tl .

The exis tence of the noted proper ty of motion of the sys tem (1.2) is given by the following theorem.

THEOREM 3. The sys tem (1.2) is technically quas ieompress ive y-s tab le if for it there exist func- tions V(t, y, z) and c(t, V), indicated in Lemma 2, and the following conditions a re satisfied:

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1) the inequal i ty (1.107 holds in the r eg ion F x fo r al l t E J t ;

2) the m a x i m u m so lu t ion r[t , tl, r(t t)] of the Cauchy p r o b l e m (1.9) with in i t ia l condi t ion r ( t 0 = VBmin(tl) - ~I, (tit fo r any t E J t s a t i s f i e s the condi t ions

B r [ t o + T, t 1, r(tl) ] < Vmln (t o + T ) - - W(t o -}-7); (2.5)

t r [t o + T, t o, V~ (to) - - �9 (to) ] < V ~ ( o + T) --~F (t o + T), (2.6)

and fo r v a l u e s y E F B the fol lowing e s t i m a t e is s a t i s f i ed :

B V [t o -~ T, y (t o -{- T), z ( t o -{- T)] ~ Vmln (to -~- T). (2.7)

P r o o f . We c o n s i d e r the t r a j e c t o r y x(t) = [y(t), z(t)], beginning in the r e g i o n Hy(t0)H< B, z 0 ~ D, and we a s s u m e that [[ y(t 0 + T)l] - B. Then t h e r e ex i s t s a m o m e n t t 1E J t such that

II v (tO II -- B, 11 v (0 tt > B, v t E(t=, to + T] (t, > to).

Accord ing to the f i r s t condi t ion of T h e o r e m 3 and a c c o r d i n g to L e m m a 1 we have

V [t, x (t)l ~ r [t, t v r (tl)l + tF (0-

Tak ing into account condi t ions (2.5) and (2.6) we obta in B B V~,in (to + T) ~< V [to + T, x (t o + 7)1 ~< r [t o + T, t x, r (tl)l + tit (to + T) < Vmin (to 2f_ T), (2.8)

w h e r e the inequa l i ty (2.8) i m p l i e s

lie(to + 7")11 < B.

Now le t the t r a j e c t o r y x(t) beg in in the r e g i o n rX'x, x r B , i .e . ,

yoEP~\FB; zo6O; l l y ( t o + T ) l i • B .

F o r t --- t o + T we will have

B Vm~n (to + T) ~< V [to + T, x (t o + T)l.

Then f r o m condi t ion 1 of T h e o r e m 3, and (2.5) and (2.6), a c c o r d i n g to L e m m a 2 we obta in

B Vmin (to + T) < V [t o + T, X (to + 7)1 ~< r [t o + T, tl, r (t0l - - ~F (to + T) < vBIn (to + T).

The con t r ad i c t i on that is ob ta ined p r o v e s the t h e o r e m .

R e m a r k . F o r the i nves t i ga t i on of the s t ab i l i ty of the type of s y s t e m s unde r c o n s i d e r a t i o n

dg~ dt = Y~ (t, Yl . . . . . Yk, zl . . . . . Zm) + eR~ (t, y, z);

(2.9)

dZ--L = Z i (t, z t . . . . Zrn' Yl . . . . . gk ) ' dt

w h e r e e > 0 is a s m a l l p a r a m e t e r ; the Ri(t , y, z) a ' re cons t an t ly ac t ing p e r t u r b a t i o n s , which a r e c o m - p l e t e ly app l i cab le to the me thod deve loped in w

It fo l lows that we can use a loca l ly l a r g e funct ion V(t, y, z), c o n s t r u c t e d f o r the s y s t e m (2.9) with = 0, and a l so we can d e t e r m i n e the m e a n

to+T t*

"T1 ~ -~OV O, (to, xo) = E R, (t, y, z) dt

t, f=l

on a f ini te i n t e r v a l Jt , a long the i n t eg ra l c u r v e s of the u n p e r t u r b e d s y s t e m (2.9).

L I T E R A T U R E C I T E D

1. A . M . Lyapunov, G e n e r a l P r o b l e m of S tabi l i ty of Motion, Col lec ted Works , Vol . 2 [in Russ ian] , I zd -vo AN SSSR, Moscow (1956).

2. A . A . Mar tynyuk , "Method of a v e r a g i n g and p r i n c i p l e of c o m p a r i s o n in t echn ica l t h e o r y of s t ab i l i ty of mot ion , " P r i k l . Mekhan. , 7, No. 9 (1971).

3. V. L a k s h m i k a n t h a m and S. Lee la , D i f f e r en t i a l and I n t e g r a l Inequa l i t i es , T h e o r y and Appl ica t ions , Vol. 1, A c a d e m i c P r e s s (1969).

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