Chapter III Stability Theory

Chapter I11 Stability Theory 1.

Basic Concepts

Under the composition of a differential equation, describing some real phenomenon, it is always necessary to simplify and idealize this phenomenon, isolating only some basic factors and neglecting the others. The initial conditions in real problems are usually the result of measurements and, consequently, of necessity are determined with some error. In order that this conditional differential equation even approximately describe the studied phenomenon, it is necessary that a small change of the initial function and small, in some sense, changes of the differential equation result in only small changes in the solution determined by this initial function. Stability theory studies conditions under which small, in some sense, changes or, as is often said, small perturbations of the differ ential equations and initial conditions result in small changes in the solutions. Let us introduce the basic definitions with regard to a differential equation with retarded arguments: %(t) = f (t,X(t) ,X(t-Tl(t))

,...,X(t-Tm(t))).

(1)

I. Definition of Stability. The solution x (t) of Eq. (1) is called stable, if for any E > O , 4 there exists a 6 ( ~ ) > 0 ,such that from the inequality I $ (t)- $ (t)I < 6 ( € 1 on the initial set, there follows Ix4 (t)-x$ (t) I < E for all tLto, where JI (t) is any continuous initial €unction.

Solutions not possessing this property are called unstable.

119

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

11. Definition of Asymptotic Stability. A stable solution x (t) is called asymptotically 4 stable, if & i g Ix (t1-x (t)I = 0 for any continuous 4 JI initial function $(t), satisfying for sufficiently small 6 1> O the condition 1 (I (t)-$(t) 1<61111. Definition of Uniform Asymptotic Stability. The solution x (t) of Eq. (1) is called uniformly 4 asymptotically stable if there exists 6 > 0 , such that for each E > O there exists a T(E) such that for for any continuous init>tl+T(E), Ix4 (t)-xJ,(t) tial function $(t) satisfying the inequality 14 (t)-J,(t)I < 6 on the initial set Et , where 6 does 1 >t0' not depend on the choice of t1IV. Definition of Exponential Asymptotic Stability. The solution x (t) of Eq. (1) is called 4 exponentially asymptotically stable if there exist constants 6>0, u > O , B > 1 such that from the inequality 14 (t)-J, (t)1 < 6 , there follows Ix (t)-x (t)I < 4 J, B max I + - ~ , l-a(t-tO) e for t>T. tEEt

I
0

V. Definition of Asymptotic Stability in the Large. The solution of Eq. (1) x (t) is called 4 asymptotically stable in the large, if it is stable and'

for all continuous initial functions $(t). All definitions remain unchanged if, in Eq. (11, x(t) is an n-vector or even an element of a Banach space. In this case, only the modulus sign I I need be changed to the norm sign 1 1 I I.

120

Ill. STABILITY THEORY

Sometimes in the definitions of stability and asymptotic stability it is expedient to use not the metric space C but some other (cf. 11.131 1. 0' For an equation of neutral type, 9 (t) = f (t,X(t),X (t'T1

.

(t)) I . . ,X (t-T (t)) m

all definitions given above remain unchanged, but in place of 'the requirement 1 4 (t)-J, (t)I < 6 it is usually necessary to require nearness in the space C1 : I+(t)-J,(t)1.6

and I@'(t)- ~l'(t)1.6.

For investigations into the stability of some solution x (t) of Eq. (1) or (21, it is possible by the change4 of variable y(t)=x(t)-x (t) to transform 4 the discussion to the stability of the solution x (t) into that of y(tIr0. Therefore, in the fol4 lowing discussion on stability, we study only the trivial solution.

2.

The Stability of Solutions of Stationary Linear Equations

All solutions of the linear equations with a deviating argument L(x(t)) = f(t)

(3)

with fixed initial point t , as for a linear equation without a deviating apgument, are stable or unstable simultaneously. In fact, any solution x (t) of Eq. (3) by the 4 change of variable y(t)=x(t)-x (t) turns into the 4 trivial solution of the corresponding homogeneous equation L(Y(t)) = 0.

121

(4 1

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Consequently, all solutions of Eq. (31, in the sense of stability, will lead in the same way to the trivial solution of Eq. ( 4 ) . In particular, for f (t)-0 all solutions of the homogeneous equation, in the sense of stability, will lead in the same way to the trivial solution of this equation. Particularly simple is the study of the stability of the solution of linear equations with constant coefficients and constant deviating arguments:

where apj and

T

j

are constants, T m > T ~ - ~ > . . . > T ~ > T ~ = ~ .

..

In Chapter II,§3, it was shown that if in Eq. # 0 and a = 0, j = O,l,. ,m-1 (in this nj case Eq. (5) is of advanced type), then there always exists a solution of the form Cekt, where c is an arbitrary constant with Re k>O. Consequently, all solutions of Eq. ( 5 ) in this case are unstable, since,for arbitrarily small IC1,the solution of the form Cekt is either unboundedly increasing in modulus as t- or for complex k , the solutions CRe ekt and C Im ekt oscillate with unboundedly increasing amplitudes. (51, anm

If Eq. ( 5 ) is an equation with a retarded argument (a # 0 and ani= 0, i=l,2,...,m~, then any nO solution x (t) for t +T(tT may be decomposed into 0 0 an absolutely and uniformly convergent series of basic solutions.

where p.(t) is a polynomial of degree less than or 3 equal to c1 -1, c1 is the multiplicity of the root j j k . of the characteristic quasipolynomial 7

122

Ill. STABILITY THEORY

-? .z m c a zpe 3 p=o j=O pj Re kl > - Re k&. Re knl..

n

(7)

1

..)

.

If all roots of the characteristic quasipolynomial have negative real parts, then the remainder in the series may be represented in the form -knt e Rn(t) , where IRn(t) I < € for n>N(E). Thus, there follows the asymptotic stability of solutions of Eq. (5) t'tO+'

1

Moreover, it is possible to prove that for (Re kn+l+E)t c pj(t)ekjt/ < BRe I j=n+l

where B and E are constants, E > O and arbitrarily small (see [1.17]). Consequently, for Re kld; conse3 j quently, max Re k = Re kl exists and, if Re kl O , such that Re kl+E
.

An palogous theorem holds for equations of neutral type but in this case, because of the possibility of an unbounded approach of the roots to the complex axis, the sufficient condition for exponential asymptotic stability will not be the requirement "all Re ki
123

DIFFERENTIAL EOUATIONS WITH DEVIATING ARGUMENTS

Because of the possibility of the approach of the roots to the imaginary axis for equations of neutral type, the proof of exponential asymptotic stability of the solution for equations of neutral type under the condition "all Re k.<-a
k.t Im e

I

a>l,

the solution is unstable. Analogous theorems, besides the last proposition, hold for a system of linear equations with constant coefficients and constant deviating arguments (in the last assertion, the multiplicity of

124

Ill. STABILITY THEORY

the root was necessary to note the magnitude of the analogous degree of the elementary divisor). Finally, we consider peculiarities connected with the possibility of the appearance, for the equations of neutral type, of asymptotic/critical and supercritical cases of the location of the roots of the characteristic quasipolynomial. For Eq. (5) of neutral type with one deviating argument (m=l, ano#O, anl#O) , W. Hahn r158.11 showed that under the condition "all Re k.
the multiplicity k, satisfying the inequality (k-1)x (e-l)>n, then the solution of Eq. (5) is unstable. Instability of the solutions of Eq. (5) of neutral type is possible also in the supercritical case. For example, as shown in [23.1], for the equation

125

DIFFERENTIAL EQUATIONS WITH OEVlATlNG ARGUMENTS

2(t)-?(t-l)

+ a(x(t)+x(t-1))

= 0,

for any a>O the characteristic quasipolynomial has the supercrionly simple, purely imaginary roots tical case. For this problem, on the half-axis (O,-) there is situated an everywhere dense set of values of the parameter a, for which the solution of the considered equation is unstable.

-

3.

Conditions for Negativity of the Real Parts of All Roots of the Characteristic Quasipolynomial

A necessary and sufficient condition for asymptotic stability of the solutions of stationary linear equations, with the exception of the consideration in the preceding paragraph of the basic critical cases, and a sufficient condition for asymtotic stability of the trivial solution of a wide class of equations, stationary in the first approximation (cf. Ch. 111, 17) is the negativity of the real parts of all roots of the characteristic quasipolynomial.

Since approximate calculation of all roots of the quasipolynomial is a problem of great difficulty, there is great value for investigations in stability, to obtaining different tests of negativity of the real parts of all roots of the quasipolynomial. Among such tests, most often the following are employed:

1) the amplitude-phase method and its modifications; 2) the method of D-partitions and its modifications; 3 ) the method of Meiman and Chebotarev.

Below we study the basic idea of the amplitudephase method, details of the development of Ya. 2. Tsypkin (cf. [127.1]) and the method of D-partitions (cf. Yu. I. Naimark 185.11, Pinney [I.17]).

126

1 1 1 . STABILITY THEORY

For readers desiring acquaintance with the method of Meiman, Chebotarev, we refer to i I . 5 1 . 1. The Amplitude-Phase Method. We call to mind that if the function f(z) is analytic and different from zero on some simple, closed contour C, and the interior of the contour has only a finiteset of polar singularities, then

where NC is the number of zeroes of f(z) in the interior of C, counted according to their multiplicities, and Pc is the number of poles in the interior of C, counted according to their multiplicities. The geometric interpretation of this theorem about the logarithmic residue leads to the "Argument Principle 2 n1 Ac Arg f(z) = NC I'

-

Pc.

(9)

Arg f(z) is the total increase of the argument of the function f(z) under a single circuit of the point z, in a positive direction, around the contour C. On the other hand, the difference N -P c c equals the number of complete revolutions which the vector w performs in the plane, going from the point w=O to the point w = f (z), when the point z describes the contour C in the positive direction (the number of revolutions are regarded as positive, if the vector is rotated counterclockwise, and is considered negative under a clockwise rotation). A

c

The reader may find the proof of this theorem in any book in the theory of analytic functions in the chapter dealing with logarithmic residues. For obtaining a condition €or the absence in the characteristic quasipolynomial (o(z) of roots with positive real parts, we apply the argument principle to the contour CR, consisting of the segment of the imaginary axis [-iR,iR] and the semicircle of radius R with center at the origin, lying 127

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

in the half-plane Re z>O (Fig. 7 ) ; as a preliminary, we check that the quasipolynomial does not have zeroes on the imaginary axis. We note that in this case Pc = 0 . Using the argument principle, we find from ( 9 ) NC and, if R lim NC = 0, then all roots zi of the quasipolynomial R+R satisfy the condition Re zi
For application of this general method to the quasipolynomial

corresponding to the nth order equation (and also to some system of n first order equations) with a s i n g l e retarded argument, where Pn(z) and Qn-,(z) are polynomials of degree n and not greater than n-I, respectively, it is possible to somewhat simplify the investigation. Instead of the function $ ( z ) , we consider the function

the zeroes of which coincide with the zeroes of the function $(z) (if Pn(z) and Qn-l(~)do not have common zeroes) and which has poles at the zeroes of the polynomial P (z). n

position as R- of the form of the contour CR under the mapping wT(z) is called the amplitude-phase characteristic. of the function

= 1

Since

@ correspond

which w T (z) = 1.

-

wT (z), the zeroes

to the points at

Therefore, applying the argument

128

Ill. STABILITY THEORY

principle to the function wT(z), it is necessary to calculate the number of circuits of the amplitudephase characteristic,not of the point z=O,but of the point z=1. The number of circuits of the amplitude-phase characteristic of the point z=l equals the difference NC-PC and, consequently, in order that NC = 0, it is necessary that the number of circuits of the amplitude-phase characteristic of the point z=l equal - P Again, we call to mind that, for this it is assu&ed that there are no zeroes of the function $(z) on the imaginary axis and that Pn(z) and Qn-l(~)do not have common zeroes, where both of these conditions are relatively easy to check.

.

We note that under the mapping wT(z), the semicircle entering into the structure of the contour contracts to a point (since the degree of as RcR’ Pn (z) is greater than the degree of Qn-1 (z)) and, consequently, it is necessary to construct only the form of the imaginary axis, transversed in the negative direction. For construction of the amplitude-phase characteristic, it is convenient at first to find the socalled limiting characteristic, appearing as the limiting form of the contour C R under the mapping

For construction of the form of the imaginary axis under the mapping

or

w (iy) = w (iy)e-Tiy I T 0

129

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

knowing already the limiting characteristic, it suffices to consider the influence of the factor e-'Iiy of rotation without change of modulus, the radius vector of the point of the limiting characteristic corresponding to the value y at the angle

-TY. For construction of the amplitude-phase characteristic, particular attention should be given to the points of the limiting characteristic lying on the circle 1z1=1 since these points, under rotation by an angle -ry, may find themselves at the point z = 1 .

As an example, we find the domain of asymptotic stability in the space of coefficients a and b of the trivial solution of the equation

+ aX(t) + bx(t-'I)

*(t) where a r b rand

'I

= 0,

(10)

are constants, r>O.

In the case under consideration, the characteristic equation has the form

+

be-'IZ

wo(z) =

-b z+a -

z

+

a

=

0,

The limiting characteristic is the form of the imaginary axis under the fractional linear transformation (12). Under this mapping, the imaginary axis transforms into the circle of radius lb/2a] with center at the point z=-b/2a, the equation of which has the form

Let a>O, then the function wT(z) has no poles in the half-plane Re z>O and, if Ibl
Ill. STABILITY THEORY

of the quasipolynomial z+a+be-" left half-plane Re z
are located in the

Thus, if a>O and IblO. For Ibl>a>O (Fig. 9), for some value of T the points of the limiting characteristic, lying simultaneously on the circle IzI=l, represented in the figure by the dotted points, may pass through the point z=1. The smallest such value T , for given a and b, will be the value which, when passed through, the solution of Eq. (10) loses stability since under passage through this value the amplitude phase characteristic begins to include the point z=1. Writing the point w (iy) = -b/(iy+a) of the limiting charac0 teristic in exponential form, we obtain

If this point lies on the circle ~ z = l ,then

-T iy I and in order that, after multiplication by e the point pass into the point z=l, the argument of -Tiy must be a multiple of 2n: wo (iy)e

arc tan(-y/a) = 2kr.

(16)

The smallest positive value of T, defined by (161, is this critical value T = T 0' beginning with which stability is lost. From (16) and (15), we obtain

- arccos (-a/b) 'O-

JPGT 131

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

If T is assumed fixed, then eliminating the parameter y from (15) and (16), we obtain an equation for the boundary curve of the domain of stability. Analogously developed are investigations for a
We shall divide the space of coefficients into regions by hypersurfaces, the points of which correspond to quasipolynomials having at least one zero on the imaginary axis (the case z=O is not excluded). Such a decompositioniscalled a Dpartition. The points of each region of such a D-partition obviously correspond to quasipolynomials with the same number of zeroes with positive real parts (by number of zeroes we mean the sum of their multiplicities), since under a continuous variation of the coefficients, the number of zeroes with positive real parts can change only if a zero passes across the imaginary axis, that is, if the point in the coefficient space passes across the boundary of a region of the D-partition. 132

Ill. STABILITY THEORY

Thus, t o e v e r y r e g i o n uk o f t h e D - p a r t i t i o n , i t i s p o s s i b l e t o a s s i g n a number k which i s t h e number of z e r o e s w i t h p o s i t i v e r e a l p a r t s o f t h e q u a s i p o l y nomial d e f i n e d by t h e p o i n t s o f t h i s r e g i o n . Among t h e r e g i o n s o f t h i s d e c o m p o s i t i o n are a l s o found r e g i o n s uo ( i f t h e y e x i s t ) c o r r e s p o n d i n g t o q u a s i p o l y n o m i a l s which do n o t have e v e n one r o o t w i t h a p o s i t i v e real p a r t . These r e g i o n s are r e g i o n s of asymptotic s t a b i l i t y of s o l u t i o n s corresponding t o t h e quasipolynomials of t h e s t a t i o n a r y l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h d e v i a t i n g arguments b e i n g studied. Thus, t h e i n v e s t i g a t i o n o f s t a b i l i t y by t h e method o f D - p a r t i t i o n i n t h e s p a c e o f c o e f f i c i e n t s ( o r o t h e r p a r a m e t e r s on which t h e c o e f f i c i e n t s and d e v i a t i o n s o f t h e argument depend) r e d u c e s t o t h e f o l l o w i n g scheme: w e f i n d t h e D - p a r t i t i o n and s i n g l e o u t t h e r e f r o m t h e r e g i o n uo. I f t h e r e g i o n uo i s c o n n e c t e d , t h e n it may be i d e n t i f i e d by v e r i f y i n g t h a t a t l e a s t one o f i t s p o i n t s c o r r e s p o n d s t o a q u a s i p o l y n o m i a l whose r o o t s a l l have n e g a t i v e r e a l parts. I n o r d e r t o c l a r i f y h o w t h e number of r o o t s w i t h p o s i t i v e r e a l p a r t s changes a s some boundary of t h e D-partition i s crossed, t h e d i f f e r e n t i a l of t h e r e a l p a r t of t h e r o o t i s computed, and t h e d e c r e a s e o r i n c r e a s e o f t h e number o f r o o t s w i t h p o s i t i v e r e a l p a r t s i s d e t e r m i n e d from i t s a l g e b r a i c sign.

I f + ( z , a l , . . . , a )=O i s a c h a r a c t e r i s t i c equaP t i o n containing t h e parameters al,a2,...,a then PI P a + dz = - C 2 dai, z = x + i y , az aai i=O dx = - R e

C

a 4 dai

i=O

i

133

aa

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Usually dx is computed on some boundary of the Dpartition for a change in only one parameter whose change guarantees passage across the boundary being examined. As an example, we consider the equation

+ ax(t) + bx(t-.r) = 0,

%(t)

(17)

already partially investigated by the amplitudephase method on pp. 12-13. The characteristic quasipolynomial 6 (z) = z+a+be-TZ,

(18)

corresponding to Eq. (17), has a zero root for a+b = 0.

(19)

This straight line is one of the lines forming the boundary of the D-partition. Now let the quasipolynomial have the purely imaginary root iy:

or

iy

+

a +b(cosry-isinTy) = 0.

Separating the real and imaginary parts, we obtain the equation of the D-partition boundaries in parametric form: a +bcos .cy = 0, y-bsin ry = 0, or

y COSTY sin.ry' slnry , a = 1 1 as y+O we obtain the cusp point ( - y , - ) . These lines and the straight line (19) form the D-partition shown in Fig. 11.

b = - y

134

Ill. S T A B I L I T Y T H E O R Y

For a>O and b=O the degenerate quasipolynomial has no roots with positive real parts. Consequently, region I is the region of asymptotic stability of solutions of Eq. (17) (as b+O the real parts of all roots of the quasipolynomial except one approach -1. Upon passing from region 1 into region 11 across the straight line a+b=O, one root with a positive real part appears, since from (18) we find that dx = da/(l-bT) on this straight line, and therefore for decreasing a and constant b, where b<(l/T), the real part of the root equal to zero on this straight line receives a positive increment. If b>(I/T), then dx>O also for da>O. Consequently, in region 111 two roots have positive real parts. This result may also be obtained in another way: on the boundary curve C1 of the D-partition (that is, on the curve a+bcosTy = 0 , y-bsinTy = 0; O
=

-

-Re

da l-bre-Tz

=

-Re

da 1 -b Te-TlY

da 1-bT (COSTy-iSinTy)

(1-bTCosTy)da (1-bTCOSTy)2 +b2 T 2 sin2 TY

On the curve C1 the roots z=iy are pure imaginary, and keeping in mind the equation of the curve C1, we find cos-ryl. Therefore, the sign of dx is again opposite to the sign of da. Upon crossing the boundary C1 from region I into region 111, the pair of complex conjugate roots gain positive real parts. The analysis on other boundaries of the Dpartition is completely analogous. Remark. Both in the present case and in other5 the isolation of the region u may be accomplished 0 with the help of Rouche's theorem, which asserts that if two functions $I (z) and $I (z analytic on and inside a simple closed contour do not reduce to zero at any point of C , and if 2) l
DIFFERENTIAL EQUATIONS W f T H DEVIATING ARGUMENTS

on C , t h e n t h e f u n c t i o n s $ ( z ) and $ ( z ) + $ ( z ) have t h e same number o f z e r o e s i n s i d e t h e r e g i o n bounded by C (see any c o u r s e o n a n a l y t i c f u n c t i o n theory). Applying R o u c h e ' s t h e o r e m t o t h e f u n c t i o n s $ ( z ) = z + a and $ ( z ) = b e - c z on t h e c o n t o u r C we note R' t h a t f o r ( a l > l b l and f o r a s u f f i c i e n t l y l a r g e r a d i u s R o f t h e semi-circle e n t e r i n g i n t o t h e c o m p o s i t i o n o f t h e c o n t o u r C , t h e f u n c t i o n s $ ( z ) and $ ( z ) s a t i s f y t h e c o n d i t i g n s o f Rouche's theorem: t h a t i s , o n t h e c o n t o u r C R w e have

where z=x+iy. I n f a c t , i n e q u a l i t y ( 2 0 ) i s o b v i o u s on t h e semi-circle € o r s u f f i c i e n t l y l a r g e R , s i n c e x>O, and

and, consequently,

for ( a ( > (bl. F o r a>O, t h e f u n c t i o n $ ( z ) h a s no z e r o e s i n s i d e t h e c o n t o u r CR; t h e r e f o r e , by R o u c h e ' s t h e o r e m t h e f u n c t i o n z+a+be-'z h a s no z e r o e s f o r any R . Thus, t h e component o f t h e D - p a r t i t i o n which c o n t a i n s t h e r e g i o n a > lbl (see F i g . 11) i s t h e r e g i o n o f asympt o t i c s t a b i l i t y uo. W e n o t e f u r t h e r t h a t f o r la1 > Ibl and a
Example 1. F i n d t h e r e g i o n o f s t a b i l i t y o f solutions of t h e equation x(t)

+

ak(t-1)

+

bx(t-1) = 0

i n t h e s p a c e o f t h e c o e f f i c i e n t s a and b .

136

Ill. S T A B I L I T Y T H E O R Y

The characteristic equation has the form z2

+ (az+b)e-'

= 0.

(21)

We shall find the boundary of the D-partition. For z=O,b=O. For z=iy, -y2+(aiy+b)(cosy-isiny)=O, and, separating this into real and imaginary parts, we obtain 2 -y + aysiny + bcosy = 0, aycosy

-

bsiny = 0,

yfO

or 2 a= ysiny, b=y cosy, OO and intive ab creasing b,the root with positive real part is lost when the axis b=O is crossed. Example 2 . Find the stability region in the space of the coefficients a and b for solutions of the equation x(t)

+

aZ(t-1)

+

bx(t) = 0.

The characteristic equation has the form

z2 + aze-'

+

b = 0.

Putting z=O we obtain b=O. For z=iy and O < y < = , we find -y2'+ aiy (cosy-isiny)+b=O, or

137

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

-y2

+

aysiny

+

b = 0,

aycos y = 0.

If afO, then from the second equation we find

therefore b = (-1)k+l (2k+l)7a 2 which are straight lines. The D-partition is given in Fig. 13, and the number p of roots with positive real parts is indicated. (For a=O, and b>O, az < 0 = -1/2 c o s a when z=+ Re - i&; hence Re aa aa

Example 3. Find the stability regions in the space of coefficients a and b f o r solutions of the equation x(t)

+ ajc(t) + bx(t-1)

= 0.

The characteristic equation has the form z2

+

az

+

be-'

= 0.

For z=O,b=O. For z=iy and O < y < w we obtain 2 -y +aiy+b(coSy-isiny)=O, whence b ~ 0 s y - y=~ 0, ay-bsiny = 0, or

The region of stability is shown in Fig. 14.

138

Ill. STABILITY THEORY

4.

The Case of Small Deviating Arguments If the retardations

T

m

in the equation (22)

(t-.c.)= 0, 3

where 0 = T < T <..<‘I are sufficiently small, then 0 1 m it is natural to expect that many properties of solu tions of Eq. ( 2 2 ) will be close to properties of solutions of the equation without deviating arguments

obtained from ( 2 2 ) for

.. .

= O (j=1,2,. ,m) j In particular, the following theorems hold: T

I. If the solutions of Eq. ( 2 3 ) are asymptotically stable, then for sufficiently small T 111 , the solutions of Eq. ( 2 2 ) are also asymptotically stable 11. If the characteristic equation for Eq. ( 2 3 ) has at least one root with a positive real part, and therefore the solutions of Eq. ( 2 3 ) are unstablg then for sufficiently small the solutions of Eq. ( 2 2 ) are also unstable.

-

111. If the characteristic eauation for Ea. I 2 3 1 has the simple root z=O, and the remaining roots have negative real parts, then for sufficiently small T the solutions of Eq. ( 2 2 ) are stable. -

A

,

In fact, for sufficiently small T~ all the multipliers e-Tj are as close as desired to unity for 1.1 < M. Therefore, the characteristic quasipolynomial $ ( z ) for Eq. ( 2 2 ) may be represented in the form $ ( Z )

= Ip(z) + r l ( z ) I

I39

.

DIFFERENTIAL EOUATIONS WITH DEVIATING ARGUMENTS

where + ( z ) is t h e c h a r a c t e r i s t i c polynomial f o r Eq. ( 2 3 ) and 1 n ( 2 ) I < E i n t h e r e g i o n IzI O i s a s s m a l l as d e s i r e d .

The r o o t s of t h e polynomial $(z) are a l l l o c a t e d w i t h i n some d i s c IzI
t h e quasipolynomial $(z) e x c e p t f o r one w i l l have n e g a t i v e r e a l p a r t s , s i n c e e i t h e r t h e y w i l l be close t o roots of t h e polynomial + ( z ) w i t h n e g a t i v e r e a l p a r t s , o r t h e i r r e a l p a r t s w i l l be n e g a t i v e and a r b i t r a r i l y l a r g e i n moduli. One r o o t of t h e q u a s i polynomial, which must be c l o s e t o t h e r o o t z=O o f t h e polynomial $(z), w i l l a l s o be e q u a l t o z e r o , since $ ( O ) = $ ( O ) = O . Everything s a i d here i s a l s o t r u e f o r a system of e q u a t i o n s w i t h r e t a r d e d arguments.

140

Ill. STABILITY THEORY

As an example, we shall study the equation k(t)

+

ax(t)

+ bx(t-T)

= 0,

(24 1

for which we shall estimate how small T must be in order that Theorems I,II, and I11 shall be valid. The stability region for solutions of Eq. ( 2 4 ) is depicted in Fig. 15. The stability region for solutions of the equation

is defined by the inequality a+b>O (see Fig. 15). From comparison of these regions it follows that: 1. If the solutions of Eq. ( 2 5 ) are unstable, then the solutions of Eq. ( 2 4 ) are also unstable for any T ; 2. If the solutions of Eq. ( 2 5 ) are asymptotically stable and lbl < a, then the solutions of Eq. ( 2 4 ) are asymptotically stable for any T ;

3. If the characteristic equation for Eq. ( 2 5 ) has the root z=O, i.e. if a+b=O, then the solutions of Eq. ( 2 4 ) are stable for any T for bO; 4. In the case of as m totic stability of solutions of Eq. ( 2 4 ) for YbP > a, the estimate for ‘I may be obtained from the equations of the boundary curve C1 (see Fig. 11);

a+bcosTy = 0, y-bsinTy = 0, O < y < s / ~ . Eliminating y and solving for T =

‘I, we

arccos (-a/b)

Jb2_,2

141

obtain

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Thus, in the present case the solutions of Eq. (24) are asymptotically stable for <

,arccos(-a/b)

K7-

For equations of neutral type, such a total correspondence with respect to stability, between solutions of the equations n

c

+ 0,

m

c

k=O j=O

a x(k) (t-T.1 = 0, a kj 7 nO

(26)

and n

m

c c k=O j=O

a x(k) (t) = kj

o

(27)

for small T~ is obtained without additional restrictions only in the case of instability, for the presence of at least one root of the characteristic polynomial $(z) of Eq. (27) with a positive real part. The difference between this case and the case of the retarded argument equation resides in the fact that the asymptotic roots of Eq. ( 2 6 ) may retain positive real parts for arbitrarily small T j‘ For example, solutions of the equation k(t)

+

ax(t)-bA(t-T)-abx(t-T)

= 0

(28)

for a>O, b>O, and arbitrarily small T > O are unstable, since the characteristic equation (z+a)(l-be-Tz) = 0

1 [In b+2k~il,with positive real has roots zk = T parts, while at the same time for T = O the solutions of Eq. (28) are asymptotically stable: x(t)=Ce-at. I42

Ill. STABILITY THEORY

This example again emphasizes that it is not always possible, even in qualitative investigations to ignore small, and even arbitrarily small deviations of an argument. 5.

The Case of Large Deviatinq Arquments Along with the stationary linear equation n m c c a x(k) (t-T.1 = 0, (29) 7 k=O j=O kj

where 0 = T 0< T 1<..
'

dk)

and their characteristic equations $ ( z ) = 0 and $(z) = 0, where k T.Z n m z a z e 3 $fz) = c k=O j=O kj n k O(z) = c akOz k=0

.

Theorem. If the polynomial $(z) has at least one root z D with a positive real part, then €or sufficientiy large T~ the solutions of Eq. (29) are unstable. Proof. In the vicinity of the root z=z the P' modulus of the difference $ ( z ) - $ (z) is arbitrarily small for sufficiently large T ~ ,since for Re z>O all terms containing the multipliers e-T jZ (-j=l12,...,m) are arbitrarily small. Therefore, by Rouche's theorem, for sufficiently large T ~ the , quasipolynomial $(z) has a root with a positive real part located in an arbitrarily small neighborhood of the zero z=z of the polynomial $ (z) P

.

I43

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

I n t h e case of t h e a s y m p t o t i c s t a b i l i t y of s o l u t i o n s o f Eq. (30), it i s i m p o s s i b l e t o a s s e r t t h a t t h e s o l u t i o n s of Eq. ( 2 9 ) w i l l b e asymptot i c a l l y s t a b l e f o r s u f f i c i e n t l y l a r g e -rl w i t h o u t a d d i t i o n a l a s s u m p t i o n s . N e v e r t h e l e s s , some r e s u l t s may b e o b t a i n e d i n t h i s d i r e c t i o n (see [ 1 7 4 . 2 ] ) . 6.

Lyapunov's Second Method

A s i s w e l l known, when a p p l i e d t o d i f f e r e n t i a l e q u a t i o n s w i t h o u t d e v i a t i n g arguments

,.. ., x n ( t ) 1 , ( i = 1 , 2 , . . ., n )

2 1. ( t ) = f1. ( t , x l ( t ) , x 2 ( t )

I

(31)

t h e i d e a of Lyapunov's s e c o n d method h a s t h e f o l l o w ,xn) ( o r V (xl ,x2 i n g form: a f u n c t i o n v ( x l ,

,. . .

,.. .

x

t ) ) i s s e l e c t e d which p l a y s t h e r o l e o f a g e n e r -

n' a l i z e d d i s t a n c e from t h e o r i g i n o f c o o r d i n a t e s t o t h e p o i n t ( X ~ ~ X ~ , . . . ~ Xi f~ )a;l o n g t r a j e c t o r i e s o f

Eq. ( 3 1 ) t h i s f u n c t i o n i s n o n - i n c r e a s i n g (dV/dt
I?yapunov's s t a b i l i t y t h e o r e m : The t r i v i a l s o l u t i o n of Eq. ( 3 1 ) i s s t a b l e i f , i n a neiqhborhood o f t h e o r i g i n f o r tit t h e r e e x i s t s a p o s i t i v e d e f i n i t e

0'-

d i f f e r e n t i a b l e function V (x1z2

..., x n ' t ) whose

,

d e r i v a t i v e a l o n g i n t e g r a l c u r v e s o f Eq. neighborhood i s n o n - p o s i t i v e .

144

(31) i n t h i s

Ill. S T A B I L I T Y T H E O R Y

W e r e c a l l t h a t a f u n c t i o n V(x1,x2,...,x

called positive definite i f

n'

t) i s

where t h e c o n t i n u o u s f u n c t i o n W i s e q u a l t o z e r o o n l y f o r x =x = . . . = x =O. 1 2 n &(E)

n

For a g i v e n E>O, I d e a of t h e p r o o f . sufficiently s m a l l that SUP

c xi i=l

V(x1,x2,

w e choose

... f x n , t O ) <

i n f W(xl,.. . , x n ) . c xi 2 =€ 2 i=l (33)

< S2(€)

Then c h o o s i n g a n a r b i t r a r y i n i t i a l p o i n t i n a E neighborhood o f t h e o r i g i n , by c o n d i t i o n s ( 3 2 ) and ( 3 3 ) , w e o b t a i n a t r a j e c t o r y which c a n n o t l e a v e t h e E-neighborhood o f t h e o r i g i n f o r t > t s i n c e t h e func0'

t i o n V cannot increase along t h e t r a j e c t o r y . Remark. T h i s t h e o r e m c a n b e i n v e r t e d , i . e . , if t h e t r i v i a l s o l u t i o n of (31) i s s t a b l e , t h e r e e x i s t s a f u n c t i o n v s a t i s f y i n g t h e c o n d i t i o n s o f Lyapunov's theorem. F o r s u c h a f u n c t i o n , i f smoothness i s n o t r e q u i r e d , it i s p o s s i b l e t o t a k e n 2 ~ ~ ~ 1 0 1 ~ 2 0 ' " ' ~ ~ n O= I SUP t 0 ~ 1 xi I i=l

tlto

n

xi2 i s t h e d i s t a n c e from t h e i n t e g r a l c u r v e i=l p a s s i n g t h r o u g h t h e p o i n t ( x 1 0 f x 2 0 ~ . . . , x nO ) f o r t=t0

where

to the origin. I t is a l s o n o t d i f f i c u l t t o c o n s t r u c t a smooth Lyapunov f u n c t i o n .

-

Lyapunov's A s y m p t o t i c S t a b i l i t y Theorem. The e q u i l i b r i u m p o i n t x . = O ( i = 1 , 2 , . . . f n ) of t h e s y s t e m 1

of equations (31) i s uniformly a s y m p t o t i c a l l y s t a b l e

145

DIFFERENTIAL EQUATIONS WITH D E V I A T I N G ARGUMENTS

i f t h e r e e x i s t s a function V(x1fi2,...,xn,t)

satisf y i n g t h e f o l l o w i n g c o n d i t i o n s i n t h e neighborhood o f t h e o r i g i n f o r t?tO: t h e function V i s p o s i t i v e d e f i n i t e ; t h e f u n c t i o n V ha.s an i n f i n i t e l y s m a l l n u p p e r l i m i t ( i . e . V+O f o r c xi2+o) ; i=l t h e d e r i v a t i v e of t h e f u n c t i o n V along i n t e g r a l curves i s negative d e f i n i t e , i.e.

1. 2.

3.

dv-av,

dt - at

n 1

i=l

... = *n-

av axi

fi

-w ( x l , .

..

,Xn)

5

0,

where t h e c o n t i n u o u s f u n c t i o n W e q u a l s z e r o o n l y f o r

5 6

3 (E

-

2 -

= 0.

Idea of t h e Proof. F o r a s i v e n E>O, w e c h o o s e sup V < i;f W , where U6 i s t h e U6 , t > t

> O such t h a t

- 0

SE

&-neighborhood o f t h e o r i g i n , and S E i s t h e s p h e r e o f r a d i u s E w i t h c e n t e r a t t h e o r i g i n . By t h e e x i s t e n c e o f t h e i n f i n i t e l y s m a l l u p p e r l i m i t , 6 ( ~ )may be chosen independently of to. T r a j e c t o r i e s beginning i n U

6

d o n o t r e a c h t h e l i m i t S E , and must p a s s

i n t o a n a r b i t r a r i l y s m a l l n-neighborhood of t h e origin. I n f a c t , i n t h e o p p o s i t e case, -dV < - a dt -

< 0

(34)

a l o n g a t r a j e c t o r y , where a i s a c o n s t a n t . Multip l y i n g ( 3 4 ) by d t and i n t e g r a t i n g from t o t o t > t O , we obtain V - V ~ 5 -a ( t - t O ) .

(35)

From ( 3 5 ) , f o r s u f f i c i e n t l y l a r g e t , w e o b t a i n V < O , which c o n t r a d i c t s t h e f i r s t c o n d i t i o n o f t h e theorem. A s a r e s u l t of t h e a r b i t r a r y c h o i c e of n , asymptotic s t a b i l i t y i s proved. As a r e s u l t o f t h e i n d e p e n d e n c e o f 6 ( ~ from ) t o , uniform asymptotic s t a b i l i t y i s a l s o proved.

146

I l l . STABILITY T H E O R Y

The theorem can be inverted. Chetayev‘s Instability Theorem. The trivial solution of the system (31) is unstable if,in an arbitrarily small neighborhood of the origin, for t>t there exists a neighborhood U, not depending on -0 t, in which the function V(xl, x-,t) satisfies the conditions:

...,

1. v>o; 2.

3 dt

> 0

dV > ’ dt-

3.

in such a way that

B > 0 in the region Vza>Q;

in the neighborhood of the origin for txt the function V is bounded.

Idea of the proof. We choose an initial point such that V ( ~ ~ ~ , . . . , x ~ ~ =, tal>O. ~) Then by condition 2, for t’tO the trajectory remains in the region V>al>O, and therefore along the trajectory, dV dt 2 B1

’ 0.

Multiplying (36) by dt and integrating from tO,to t>tO, we obtain

v-vo 2

B1 (t-tO).

(37)

From (37) we find that V-m for t-, which contradicts condition 3. Therefore, for t- the trajectory leaves the neighborhood of the origin. Since on the basis of condition 1 the initial point may be chosen arbitrarily close to the origin, the instability is proven. Obviously, the formulation and proof of these three theorems is almost unchanged if the system (31) is changed to the system of differential equations with deviating arguments,

147

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

jCi

...,xn (t),x1 (t-.cl(t)), ...,XI(t-Tm(t)1 ...,xn (t'Tl (t),...,xn (t"Trn(t)1 ,t) ,

(t)=fi (xl(t),

(i=1,2,. where

T

T

i (t)LO.

..,n),

I

(38)

In this case,

dV dt (xl,...,xn' t)

=

n av c i=l axi fi

av at +

becomes a function of the n(m+l)+l arguments t,Xl(t) ,...,xp ,x1(t-Tl(tH

r...rX

n (t-Tm(W

I

and its non-positiveness in the first theorem (or negative definiteness in the second theorem, and positiveness in the instability theorem) may be understood as non-positiveness for independently changing arguments, or else it may be borne in mind that xi(t-Tk(t)) is one of the previous values of the function xi (t). For example, the trivial solution of the system

?(t) = -x(t)-y(t)x

2

(t-T2(t)) ,

T . > O , ]=1,2,is asymptotically stable since for a 3Lyapunov function,it is possible to take V = x2+y 2 For this

.

However, for equations with a deviating argument such a direct transfer of Lyapunov's second method is impossible to consider as a general technique for stability investigations, since the theorems of this method do not admit conversion for equations with a deviating argument,

148

Ill. STABILITY THEORY

I n f a c t , if t h e s e theorems a d m i t t e d c o n v e r s i o n , then f o r the equations

2 ( t ) = f ( x ( t ) , x ( t - r ) ) and & ( t ) = kf ( x ( t ) , x ( t - T )

,

where k i s a p o s i t i v e c o n s t a n t , s t a b i l i t y must o c c u r f o r b o t h s i n c e , i f t h e t r i v i a l s o l u t i o n of t h e f i r s t e q u a t i o n i s s t a b l e , t h e n by t h e c o n v e r s e theorem, t h i s e q u a t i o n must have a Lyapunov f u n c t i o n V ( t , x ) , and t h e n i t i s p o s s i b l e t o choose V ( k t , x ) as a Lyapunov f u n c t i o n f o r t h e second e q u a t i o n . However, e l e m e n t a r y examples show t h a t t h i s may n o t be so. C o n s i d e r , f o r example, t h e e q u a t i o n s

and k(t)

+

k [ a x ( t ) + b x ( t - T ) l = 0.

I t i s e a s y t o show t h a t t h e s t a b i l i t y r e g i o n s f o r t h e s e e q u a t i o n s do n o t c o i n c i d e . On p. 1 6 t h e domain of s t a b i l i t y f o r Eq. ( 1 7 ) w a s c o n s t r u c t e d by t h e method o f D - p a r t i t i o n s ( t h e domain I, p=O - F i g . 11). For i t s upper boundary, upon e l i m i n a t i n g t h e parameter y, we o b t a i n t h e equation

=-T

arccos (-a/b)

.

Now it i s o b v i o u s t h a t t h e s u b s t i t u t i o n o f a and b

by ka and kb a l t e r s t h i s boundary.

N e v e r t h e l e s s , i n a series of cases, a p p l i c a t i o n o f Lyapunov f u n c t i o n s f o r s t u d y o f t h e s t a b i l i t y of s o l u t i o n s o f e q u a t i o n s w i t h a d e v i a t i n g argument have proved t o be e f f e c t i v e . Some m o d i f i c a t i o n s of t h i s method are g i v e n i n [ 1 0 1 . 2 1 , D01.41. An i d e a which h a s proven t o be much more f r u i t f u l i n t h e g e n e r a l case, i s t h a t o f N.N. K r a s o v s k i i ( c f . [ 5 2 . 1 ] , [ 1 . 3 ] ) , who has c o n s i d e r e d , i n p l a c e of Lyapunov f u n c t i o n s , f u n c t i o n a l s w i t h analogous properties.

149

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

...

We consider a vector function x(s) with compo,xn ( s ) I , defined on the internents {xl ( s ) ,x2 ( s ) , val - T < s < O (in what follows, s always varies within the inaicated limits). For each tLto, there is defined on the vector function x(s) a functional V[Xl(S) I . . . I Xn ( s ) ,tl = V[X(S) ,tl Definition I. The functional

is called positive-definite if there exists a continuous function +(r)>O such that for r # 0

A negative-definite functional is analogously defined.

The norm of the vector-function x(s) may be taken in various spaces. In what follows, we shall need norms in the spaces C0 and L2, and sometimes, particularly for equations of neutral type, in the space C1' We introduce the following notation:

11x1

l2

=I : i=l

150

Xi2]

i

I l l . STABILITY THEORY

U E w i l l mean t h e &-neighborhood o f t h e e q u i l i brium p o i n t x 1 =x 2= * * - = nx= O i n t h e m e t r i c o f C 0; S E i s t h e € - s p h e r e which i s t h e boundary o f UE.

D e f i n i t i o n 11. The f u n c t i o n a l V [ x ( s ) , t ] has an i n f i n i t e l y s m a l l upper l i m i t i f t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n c $ ( r~) ) O w i t h q1 (O)=O s u c h t h a t V[x(s),tl 5 41(llx(S)IIr)' S t a b i l i t y Theorem. The t r i v i a l s o l u t i o n o f t h e s y s t e m (38) i s s t a b l e i f t h e r e e x i s t s a c o n t i n u o u s vositive-definite functional

whose d e r i v a t i v e a l o n g i n t e g r a l c u r v e s i s non-positive.

where x ( t + s ) i s t h e s o l u t i o n o f Eq.

(38) defined by 4 t h e i n i t i a l v e c t o r f u n c t i o n $ ( t ) ,( t O - T & t z t 0 ) 2

Proof. such t h a t

For a given E > O

151

(E
w e choose ~ ( E ) > O

DIFFERENTIAL EOUATIONS WITH DEVIATING ARGUMENTS

(on t h e s t r e n g t h o f t h e p o s i t i v e - d e f i n i t e n e s s of t h e f u n c t i o n a l V [ x ( s ) , t ], and t h e c o n t i n u i t y of t h e f u n c t i o n a l V [ x ( s ) , t ] i n t h e neighborhood of x ( s ) =O). 0

For such a c h o i c e of & ( E ) , any i n i t i a l v e c t o r f u n c t i o n 4 ( t ) s a t i s f y i n g t h e c o n d i t i o n I 14 ( t O + sI )I T < & ( E ) d e t e r m i n e s a s o l u t i o n x (t) f o r t t o s u c h t h a t QI

I

s i n c e t h e f u n c t i o n i n t o which t h e f u n c t i o n a l V is converted along an i n t e g r a l curve does n o t i n c r e a s e a l o n g a t r a j e c t o r y , and t h e r e f o r e by i n e q u a l i t y (39) I I x 4 ( t ) I I Tc a n n o t e q u a l E . K r a s o v s k i i ' s Asymptotic S t a b i l i t y Theorem. The t r i v i a l s o l u t i o n o f Eq. (38) is u n i f o r m l y a s y m p t o t i c a l l y s t a b l e if f o r t2t0 and I Ix ( s ) I I T+ where H > O , t h e r e e x i s t s a c o n t i n u o u s p o s i t i v e - d e f i n i t e f u n c t i o n a l V [x ( s ) , t ] w i t h an i n f i n i t e l y s m a l l upper l i m i t such t h a t t h e d e r i v a t i v e w i t h r e s p e c t t o t of V [ x ( s ) , t ] i s n e g a t i v e - d e f i n i t e . Here x@!t+s) i s t h e s o l u t i o n o f Eq. ( 3 8 ) d e f i n e d b t h e i n i t i a l vector-function & Q.> .Io.t( is sufficiently s m a l l . I.

N.N.

U

Proof. For g i v e n € 7 0 , w e choose 6 ( ~ ) 7 Osuch that=>$* ( 6 ) (see t h e d e f i n i t i o n s on pp.32). Then as a consequence of t h e i n e q u a l i t i e s

I 10 ( t O + sI)I r < & , t h e

f u n c t i o n V ( t ) = V[x@( t + s ), t ] Therefore, f o r t > t t h e t r a j e c t o r y x = x ( t ) remains i n t h e

For

i s a monotone d e c r e a s i n g f u n c t i o n o f t. - 0

Q

region 1 Ix,(t) 1 I T < € , s i n c e i n t h e o p p o s i t e case i n e q u a l i t y ( 4 0 ) would be v i o l a t e d . By t h i s , t h e s t a b i l i t y of t h e s o l u t i o n x ( t ) = O is proved.

152

I l l . STABILITY THEORY

We choose an arbitrarily small n>OI and for such that this we choose S,(n)>O

1 Ix(s)

SUP

V[x(s),tI

<

I lT<61 ( n )

inf I Ix(s)

V[X(S) ,tl-

I I,=o

Assuming that A1(n) 5 I Ix,(t) I contradiction, since in this case dV

dt

H, we arrive at a

[x,(t) ,tl 2 - a < O

from which

for all t>tO which contradicts the non-negativeness of the function VIxQ (t),tl (by (41), V[x@ (t),tl < O for t > (l-a)[V[x@ (to),tO]+atOl). Therefore, there exists a point

such that

I Ix,(t*) 1 l T < A l ( r l ) , *

IlxQ(t)]l,t

and consequently,

+T.

Since TI is arbitrary, asymptotic stability is proved. Since 6 ( r l ) is independent of to and the estimate ( 4 2 ) , the asymptotic stability is uniform. Remark I. The proof is essentially unchanged if the derivative in condition 3 is replaced by the AV right-hand derivative lim sup (see lI.31). Such a t+O a change is also possible in the stability theorem. Remark 11. N.N. Krasovskii's asymptotic stability theorem admits inversion, i.e. if the system of equations ( 3 8 ) has a uniformly asymptotically stable trivial solution, then there exists a functional V[x(s),t] satisfying all the conditions of

153

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Krasovskii's asymptotic stability theorem, and a Lipschitz condition in the first argument: IV[X,(S) rtl-V[x2(s) It1

lJ'I

Ix2(s)-x1(s)

I IT.

Scheme of the proof. From the uniform asymptotic stability follows the existence of a continuous monotone-decreasing function $(t) such that lim $ (t)=O, and satisfying the inequality t-+m for any initial funcI Ixo(t+s)I 1 ' 1 ~ $(t-to) for t>t - 0 tion I I@(to+S) I I <6. There also exists a monotone'I increasing continuously-differentiable function g ( $ ) such that

4.

g'(J,(s))eLSO. -

The proof of the existence of such functions J, and g may be found in courses in stability theory for equations without deviating arguments, in the proof of the inversion theorem (for example, see I.G. Malkin, Theory of Stability of Motion, Moscow, 1952 and J.L. Massera, "Contributions to Stability Theory", Annals of Math. (2), 64, (1956), 182-206).

r

It is not difficult to verify that the functional V[x(s) ,tl

=

t

g(1 Ixa(u+s)I

SUP g ( I Ixo(o+s) I t2U < m

IT

du +

IT)

satisfies the conditions of the asymptotic stability theorem, taking account of Remark I.

154

Ill. STABILITY THEORY

For systems whose solutions are exponentially stable, it is possible to give a simpler construction of the functional V[x(s),t] which, moreover, has some important additional properties.

where t is the initial point, then there exists a functioaal V[x(s) ,t] satisfying the following conditions

where C1,C,,C3 and C, are positive constants. Idea of the proof. the functional

SUP t
It is easy to verify that

( I IxO(o+s) I I T )

t

1 where T = In 2B satisfies all the conditions of a the theorem (for more details see [1.31).

In the construction of functionals for concrete equations, it is sometimes useful to use the metric in the space L2:

155

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

-7

Under the assumption that the right-side of Eq. ( 3 8 ) is continuous and satisfies a Lipschitz condition in all arguments beginning with the second, it is possible to prove the following theorem: Krasovskii's AsvmDtotic Stabilitv Theorem The solution of Eq. ( 3 8 ) is asymptotically stable if there exists a functional V[x(s) ,tI which satisfies the conditions N.N.

11.

where W1(r) and W2(r) are monotone increasing functions for r20, W,(O)=W,(O)=O, and w(r) and Q(r) are continuous functions which are positive for r>O. Proof. W1(6)T26&

For given E > O , we choose 6>0 such that W(E). Then by ( 4 3 ) and ( 4 4 1 , v[@(s),tol

< W(E)

(46)

for I I@(t+s) I IT<6. Since the function V[x@(t+s) ,tl= V(t) does not increase along a trajectory, then from ( 4 6 ) it follows that V[x@(t+s) ,t] < W(E) for all tlto, In this way, and therefore I Ix,(t)I < E by ( 4 4 ) . the stability of the solution is proved, and the asymptotic stability is proved in the same way as in the asymptotic stability theorem I.

IT

156

Ill. STABILITY THEORY

We now give some examples. Example 1.

The solution xE0 of the equation

where a and T are constants, T > O , and b(t) is a continuous function, is asymptotically stable if I b (t)I
+

2a

(t+s)ds, a>O. -T

For a>O, this functional satisfies the first two conditions of Theorem 11, p. 3 8 . In fact,

The quadratic form in the square brackets is positive definite €or a= a/2. By this we obtain a>O The function W(r) may be (since a>O& and a>lb(t) 1 . taken as r

.

157

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

By the same method it is also possible to obtain sufficient conditions for the asymptotic stability of the system n m n ki(t) = z a x. (t) + c c a (t)x. 1 (t-Tk ) j=l ij I k=l j=1 ijk

...,n) .

(i=l,2,

Here the functional V may be sought in the form:

v

n c B..X.X i,j=l 11 = j

=

m n c c a k=l j=l ij

+

I

0

xj2 (t+s)ds.

' k Example 2. We investigate the stability of the trivial solution of the equation X(t)

+ $(k(t),t)

f(X(t'T(t))

= 0,

(47)

where f is a continuously differentiable function satisfying the conditions

0 (y,t) and T (t) are continuous periodic functions of t such that

For t>T (if tO=O), Eq. ( 4 7 ) may be changed to the system 0 k(t) = y rO

158

I l l . STABILITY THEORY

For -r
1

X

V[x(s),y(s)l

= 2

< io1: T

-T

f(s)ds + y2 +

0

y2(s) ds dsl 1

.

satisfies the conditions of the asymptotic stability theorem I1 (for more details see 11.31 ) In concluding this section, we remark that for an equation with retarded argument, S.M. Shimanov has proved [131.4] an instability theorem analogous to Chetayev's theorem. 7.

Stability in the First Approximation

In investigating the stability of the trivial solution of the system A . (t)=f.(t,Xl(t)#X2(t)I 1

1

Xn (t-Tl(t))

1 . .

.

- - ,Xn (t)8x1 (t-Tl(t)

#X1(t-Tm(t) )

I

*

I

n (t-Tm(t)) )

IX

1 . .

i = 1,2,...,n

(48)

where the right-hand sides are all assumed differentiable in all arguments beginning with the second in the neighborhood of the null values of those arguments for t>t0' it is often expedient to separate out the linear part and represent the system in the form n m (t)X (t-TQ(t) +Ri (t,xl(t)I I A+) = c 7 j=1 R = O aijll Xn (t),X1 (t-Tl(t))

1 . .

.I Xn (t-Tl(t)

I

n (t-Tm (t) I (49) n ) , ~ ~ = 0T,11-> 0 1

...,

(i=1,2,

159

IX

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

where Ri is of greater than first order in the set of all arguments beginning with the second. In many cases, the investigation of stability of the null solution of the system ( 4 9 ) is equivalent to the investigation of stability of the null solution of the simpler linear system n m k. (t) = c C aij,(t)x.(t-r,(t)), (i=1,2 n),(50) 1 7 j=1 k = O

,...,

which is called the first approximation for the system ( 4 9 ) . The case of variable coefficients and variable retardations ~,(t) in the linear part of the system ( 4 9 ) is still insufficiently worked out. Only systems ( 4 9 ) for which the system (50) has constant coefficients and constant retardations have been studied in much detail. Such systems are called stationary in the first approximation. The following theorems analogous to the corresponding theorems of Lyapunov have been proved: Theorem I.

The null solution of the system n

m

is asymptotically stable if: 1. all roots of the characteristic equation for the first approximation system for ( 4 9 1 ,

where A k are the matrices)kj&=aA which, for fixed 8 , have roots w i t h negative real parts:

160

111. STABILITY THEORY

.

where a>O is a sufficiently small constant, all Iui are sufficiently small, lui I
Theorem 11. If at least one root of the char& r e t c a and condition 2 of the previous theorem is satisfied, then the trivial solution of Eq. (51) is unstable. Remark I. In place of condition 2, it is possible to require that Ri be in some sense small in the mean (see iI.31, [17.1]). Remark 11. If some roots of the characteristic equation have null real parts, then there arises the so-called critical case in which the nonlinear terms Ri begin to influence the stability. The critical case has been studied by S . N . Shimanov (r131.51, l138.81, [131.91), who has proved theorems similar to the analogous theorems for equations without deviating arguments for the case of one null root or a pair of purely imaginary roots of the characteristic equation. The proof of Theorem I has been carried out by various methods. The most simple and natural idea forms the basis of the following method of N . N . Xrasovskii. Condition 1 of Theorem I guarantees the uniform exponential asymptotic stability of the trivial solution of the first approximation system of equations. Therefore, for the system (50) with constant coefficients and retardations.there exists a func-

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Here all C . > O . 1

It is not difficult to verify that €or sufficiently small c1 this functional will satisfy the conditions of the asymptotic stability theorem I ( p . 34) for the system (51) also. Example 1. Investigate the stability of the trivial solution of the equation ?(t)

+

+ 2x(t-.r) =

3 sinx(t)

0.

(56)

The first approximation equation has the form ?(t) + 3X(t)

+ 2X(t-T)

= 0.

Its trivial solution is asymptotically stable for any T>O. Therefore, the solution of Eq. (56) is asymptotically stable for any T , O < T < ~ . Example 2. Investigate the stability of the solution x-0 of the equation A(t) + 2x(t) -sinh x(t)-2x(t-~)+cos x(t-T)=l ( 5 7 ) for T > O . The first approximation equation A(t)

+ x(t)

-

Sx(t-r) = 0

has roots with a positive real part €or any r > Q ; consequently, the solution x-0 of Eq. (57) is unstable. 8.

Stability Under Constantly-Acting Disturbances

The solution xiEO (i=1,2,...,n) of the system of equations Ki(t)=fi(t,x

j

(t-Tk(t)),

(i,j=1,2

,...,n;k=1,2,...,m) (58)

162

Ill. STABILITY THEORY

is called stable under constantly acting disturbances if for every E > O I there exists C ~ ~ ( E ) > Oand 6 2 ( ~ ) > 0such that solutions of the perturbed system jri(t) = f. (tIX (t-~k(t)))+Ri(t,X.(t-T~(t)f)I I j 3

...,m)

(59)

(i,j=l,Z,...,n; k=1,2, satisfy the inequalities Ixi(t)

(i=l121...rn)Itlto

for

Theorem. If the solution x.: O (i=l12,...,n) of 1 the system ( 5 8 ) is uniformly asymptotically stable and the tunctions fi satisfy Lipschitz conditions in all arguments beginning with the second, then this solution is stable under constantly acting disturbances. Scheme of the proof. For the system ( 5 8 ) there exists a functional V satisfying the conditions of the asymptotic stability theorem. In an arbitrarily small neighborhood of the trivial solution, this functional satisfies the condition < - a < O I where a is a constant for the lim+ sup At At+O system (58), on the basis of the negative-definiteness of lim+ sup AV At At+O

a

-.

Examining this functional along solutions of the perturbed system ( 5 9 ) for sufficiently small A,, we obtain lim sup - < -a+KsuplRil , where K is a& + At At+O positive constant. I63

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Choosing 61

a 2K'

c

(60)

AV 2 - a / 2 < 0 . Therefore, by the we obtain lim+ sup At At+O L1V negative-definiteness of lim+ sup =,in an arbitrarAt+o ily small neighborhood of the trivial solution a trajectory may not pass outside the limits of a sufficiently small neighborhood of the trivial solution, which signifies stability under constantlyacting disturbances. 1 - 7

Remark I. If the function fi(t,x.) and ~,(t) 3 are periodic functions of t with some period T (or, in particular, independent of t), then the asymptotic stability is always uniform. Remark 11. Sometimes the concept of stability under constantly-acting disturbances is also taken to include stability with respect to disturbances of the deviations of the arguments; that is, small variations of the functions T k (t) are allowed in the perturbed system ( 5 9 ) . In this case, the system ( 5 9 ) assumes the form: Zi(t) = fi(t,xj(t-hk(t)))+Ri(t,xj (t-hk(t))) (i=1,2,...,n) where all the differences where 63>0.

,

all hk(t),O,

I hk ft)-

I ~6~

T ft) ~

(€1,

The formulation of the theorem and scheme of its proof remain unchanged in this case, the only difference being that in the estimate (601, new terms appear which are arbitrarily small for sufficiently small 1 5 ~ . For more details, see [I.3].

164

Ill. STABILITY THEORY

Is t h e t r i v i a l s o l u t i o n o f t h e

Example 1. equation

+

2 x ( t ) +x(t--r) = 0 , r > o ,

s t a b l e under c o n s t a n t l y a c t i n g d i s t u r b a n c e s ? The t r i v i a l s o l u t i o n of t h e c o n s i d e r e d e q u a t i o n i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e and t h e r e f o r e i s

s t a b l e under constantly a c t i n g dis t u r b a n c e s . 9.

Lyapunov's Second Method f o r E q u a t i o n s o f N e u t r a l Type

I n t h i s p a r a g r a p h i t w i l l b e shown how N . N . K r a s o v s k i i ' s form of t h e theorems o f Lyapunov's second method may b e g e n e r a l i z e d f o r t h e s y s t e m o f d i f f e r e n t i a l e q u a t i o n s w i t h a d e v i a t i n g argument o f neutral type

zi ( t ) = x

n

f i ( t r x l ( t ),

(t-T

( t ))

,Rl(t-T

...,xn ( t )'XI ( t )1,.

(i=1,2,.

.

.. , n )

( t - r ( t )1 I . .

,fin(t-T

(t)

I

0

I

(61)

I

f o r s i m p l i c i t y , w i t h o n e r e t a r d a t i o n Olr(t)'-r, ([132.10] I [ 7 7 . 2 ] ) .

As a l r e a d y shown, t h e meaning o f t h e d e f i n i t i o n s I - V of s t a b i l i t y , a s y m p t o t i c s t a b i l i t y , u n i form a s y m p t o t i c s t a b i l i t y , and so on remain unchanged. Only, i n c o n t r a s t t o e q u a t i o n s w i t h a r e t a r d e d argument, a l l of t h e s e d e f i n i t i o n s m u s t now b e f o r m u l a t e d i n terms o f t h e s p a c e C1. I n accordance w i t h t h i s , w e i n t r o d u c e t h e notation:

165

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

D e f i n i t i o n I.

The f u n c t i o n a l

( i n what f o l l o w s s a l w a y s v a r i e s w i t h i n t h e i n d i cated l i m i t s ) , is c a l l e d positive-definite i f there e x i s t s a c o n t i n u o u s f u n c t i o n cp ( r ) s u c h t h a t cp ( r ) > O f o r r # O and V[x(~),?(s),tI

$41I x ( s ) I

Ilr).

functional i s analogously

A negative-definite

defined

'

.

D e f i n i t -i.o n 11. The f u n c t i o r . a l V [ x ( s ) , 2 ( s ) , t ] h a s a n i n f i n i t e l y s m a l l u p p e r bound i f t h e r e e x i s t s > 0 , cpl(0) = 0 , s u c h a c o n t i n u o u s f u n c t i o n cp 1 ( r ) that V[X(S) ,jr(s)

,tl 2

cp10I x ( s ) I Ilr).

S t a b i l i t y Theorem. The t r i v i a l s o l u t i o n o f t h e system ( 6 1 ) i s s t a b l e i f t h e r e e x i s t s a continuous positive-definite functional

such t h a t along an i n t e g r a l curve x o ( t + s ) of t h e system ( 6 1 )

lim+

At+O

sup

AV a t5

0.

The p r o o f i s a n a l o g o u s t o t h e p r o o f of t h e s t a b i l i t y o f t h e t r i v i a l s o l u t i o n of t h e system of d i f f e r e n t i a l e q u a t i o n s w i t h a r e t a r d e d argument.

166

Ill. STABILITY THEORY

W e n o t e t h a t f o r systems of e q u a t i o n s of n e u t r a l t y p e , n a t u r a l l y a l l t h e o r e m s on Lyapunov's s e c o n d method a r e f o r m u l a t e d i n terms of d e r i v e d numbers and d e r i v a t i v e s , s i n c e t h e p e r t u r b e d s o l u t i o n s , g e n e r a l l y s p e a k i n g , d o n o t s a t i s f y t h e agreed condV d i t i o n s and, consequently, t h e d e r i v a t i v e - a l o n g dt an i n t e g r a l curve does n o t e x i s t a t each p o i n t .

Theorem on Asymptotic S t a b i l i t y I. The t r i v i a l s o l u t i o n of t h e s y s t e m ( 6 1 ) i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e i f t h e r e e x i s t s a continuous, positived e f i n i t e f u n c t i o n a l V [ x ( s ) , k ( s ) , t ] f o r t>t, and

I ] X ( S ) ] ] ~ ~ < HH>O, ,

Y

a d m i t t i n g an i n f i n i t e l y s m a l l

u p p e r bound and s u c h t h a t f o r t h e f u n c t i o n V ( t ) = AV

V[xo ( t + s ),Ao ( t + s ), t ], l i m At+O-

sup - i s negative-defiAt nite. Here x @ ( t + s ) is t h e s o l u t i o n o f t h e s y s t e m (611, d e f i n e d by t h e i n i t i a l v e c t o r - f u n c t i o n @(t) , where I 10 ( t o + s )I I <6, and 6 > 0 i s s u f f i c i e n t l y small. T h i s theorem i s proven analogously t o t h e c o r r e s p o n d i n g theorem f o r a s y s t e m o f e q u a t i o n s w i t h a r e t a r d e d argument (Theorem I o f N . N . K r a s o v s k i i ) . The t h e o r e m of a s y m p t o t i c s t a b i l i t y a d m i t s a c o n v e r s e - i n a n a l o g y t o t h e c o n v e r s e theorem ment i o n e d e a r l i e r (see [77.21)

.

F o r t h e s y s t e m o f e q u a t i o n s of n e u t r a l t y p e (61), it i s p o s s i b l e t o r e f o r m u l a t e a l l t h e theorems p r o v e n i n t h e p r e c e d i n g p a r a g r a p h s f o r a s y s t e m of e q u a t i o n s w i t h a r e t a r d e d argument ( 3 8 1 , [ 7 7 . 2 ] . For t h i s , t h e i d e a of t h e p r o o f o f t h e c o r r e s p o n d i n g t h e o r e m r e m a i n s unchanged. W e have

Theorem. I f t h e t r i v i a l s o l u t i o n o f t h e s y s t em ( 6 1 ) i s u n i f o r m l y a s y m p t o t i c a l l y s t a b l e and t h e functions fi s a t i s f y Lipschitz conditions i n a l l

167

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

arguments beginning w i t h t h e second, t h e n t h i s s o l u t i o n i s s t a b l e under c o n s t a n t l y a c t i n g d i s t u r b a n c e s . For a system o f e q u a t i o n s o f n e u t r a l t y p e , t h e r e h o l d t h e o r e m s a n a l o g o u s t o t h e o r e m s I and I1 a b o u t s t a b i l i t y i n t h e f i r s t approximation b u t c o n d i t i o n 1) o f t h e o r e m I i s r e p l a c e d by t h e r e q u i r e m e n t : a l l R e k . < - y < 0 , where y i s a c o n s t a n t . In the

7 -

work [77.11, t h e r e i s a l s o s t u d i e d some c r i t i c a l cases. I n [77.2] i s p r o v e n an a n a l o g o f t h e ChetayevShimanov t h e o r e m o f i n s t a b i l i t y . Thus, t h e problem o f g e n e r a l i z i n g t h e whole c i r c l e o f t h e o r e m s c o n n e c t e d w i t h Lyapunov's second method t o a s y s t e m o f d i f f e r e n t i a l e q u a t i o n s w i t h d e v i a t i n g a r g u m e n t s o f r e t a r d e d and n e u t r a l t y p e s i s , i n p r i n c i p l e , complete. Nevertheless, t h e p r a c t i c a l a p p l i c a t i o n of t h e s e theorems i s complic a t e d by t h e a b s e n c e o f any g e n e r a l methods f o r t h e c o n s t r u c t i o n of Lyapunov-Krasovskii f u n c t i o n a l s having t h e necessary properties. I n t h i s r e s p e c t , s y s t e m s o f e q u a t i o n s of neut r a l t y p e f u r n i s h many d i f f i c u l t i e s s i n c e t h e f u n c t i o n a l s depend n o t o n l y o n x ( s ) , b u t a l s o upon k ( s ) . Connected w i t h t h i s i s t h e s e a r c h f o r d i f f e r e n t m o d i f i c a t i o n s o f Lyapunov's s e c o n d method a d m i t t i n g t h e u s e , i n s o m e s e n s e , of m o r e c o n v e n i e n t f u n c t i o n als. For a c o n c r e t e system of d i f f e r e n t i a l e q u a t i o n s o f n e u t r a l t y p e , it sometimes t u r n s o u t t o b e conv e n i e n t t o u s e t h e metric

i=l

168

Ill. STABILITY THEORY

Under t h e a s s u m p t i o n t h a t t h e r i g h t - s i d e o f ( 6 1 ) i s c o n t i n u o u s and s a t i s f i e s a L i p s c h i t z c o n d i t i o n i n a l l arguments beginning w i t h t h e second, it i s p o s s i b l e t o prove t h e A s y m p t o t i c S t a b i l i t y Theorem 11. I f t h e r e e x i s t s a functional V[x(s) , G ( s ) , t l s a t i s f y i n g the conditions

AV l i m + sup < A t t+O

where W1 ( r ) and W2 ( r ) a r e m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n s f o r r > O , where W1 (0)=W, ( 0 ) =O

,

W ( r ) and

$ ( r ) a r e c o n t i n u o u s and p o s i t i v e f c r r > O , t h e n t h e s o l u t i o n of t h e s y s t e m ( 6 1 ) i s a s y m p t o t i c a l l y s t a b l e . The p r o o f i s a n a l o g o u s t o t h e proof o f N.N. K r a s o v s k i i ' s Theorem 11. However, i n c o n t r a s t t o a s y s t e m o f e q u a t i o n s w i t h a r e t a r d e d argument, f o r a s y s t e m o f e q u a t i o n s o f n e u t r a l t y p e such a n approach t u r n s o u t , i n p r a c t i c e , t o b e n o t so e f f e c t i v e . The d i f f i c u l t i e s c o n n e c t e d w i t h t h i s are t h a t i n t h e c o n d i t i o n ( 6 3 ) , (and n o t 11x1 1 , a s i n t h e r e e n t e r s t h e norm 11x1 (44) a b o v e ) .

Il

For surmounting t h e s e d i f f i c u l t i e s , w e i n t r o duce t h e following a u x i l i a r y n o t i o n s . W e w i l l say t h a t t h e t r i v i a l s o l u t i o n of t h e s y s t e m ( 6 1 ) i s s t a b l e i n t h e m e t r i c C i f , f o r any 0

w e may f i n d a 6 ( ~ ) > 0 , s u c h t h a t t h e i n e q u a l i t y 1 I @ ( t O +I sllTtO. A s y m p t o t i c s t a b i l i t y i n t h e metric C o i s c o r r e s p o n d E>O,

ingly defined.

-

Here

\

1x1

I

169

i s t h e norm i n t h e s p a c e

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Usually, i f w e succeed i n e s t a b l i s h i n g e f f e c t i v e conditions €or the s t a b i l i t y of solutions of systems o f d i f f e r e n t i a l e q u a t i o n s of n e u t r a l type i n t h e metric C o , a n d t h e n e s t a b l i s h a c o n n e c t i o n between s t a b i l i t y i n t h e m e t r i c s C

0

and C1, t h e n t h i s

i t s e l f w i l l be a s o l u t i o n about s t a b i l i t y of t h e b a s i c i n i t i a l v a l u e problem i n t h e metric C 1' W e show t h a t u n d e r t h e c o n d i t i o n s o f t h e p r e v i o u s theorem, t h e r e o c c u r s t h e

5.

Theorem on A s y m p t o t i c S t a b i l i t y i n t h e Metric I f t h e r e e x i s t s a f u n c t i o n a l V [ x ( s ) ,?(,I , t ] ,

where W (r) and W2 ( r ) a r e m o n o t o n i c a l l y i n c r e a s i n g

1

f o r r > O and W (O)=W,(O)=O,

a n d +(r) are con-

W(r)

1 -

t i n u o u s and p o s i t i v e f o r r > O , t h e n t h e s o l u t i o n o f t h e s y s t e m ( 6 1 ) is a s y m p t o t i c a l l y s t a b l e i n t h e metric C o . Proof.

For given E > O , w e choose 6>0 such t h a t

W1(6)+W2(26&)

c

W(E).

Then by (62)

V[Q ( s ) I 6 ( s ) , t o ] < f o r 1 j e ( t O + s )1 t h e function

IlT

#

(62')

Since along a trajectory,

< 6.

V ( t ) = V[x ( t + s ),it (P

is non-increasing,

W(E)

(P

( t + s ), t l

from (62) it follows t h a t

V[X@(t+s)'9, (t+s), t l

170

<

W(E)

Ill. STABILITY THEORY

f o r all t>tO, and keeping in mind that I 1x1 1x1 11, by (64) we obtain 1 Ix,(t) I I < € . This alone proves the stability of the solution in the metric C 0' Asymptotic stability is also proven as in Theorem I of N.N. Krasovskii. On the other hand, we have the

...,

Theorem. For the system (611, let T(t)=T>O, and let the €unctions €.(t,xl, xn-1 ,y ,...,y , 1 11 gl, z n satisfy Lipschitz conditions in all arguments beginning with the second with constants Lis(XI, (i,s=1,2,. ,n) in the region < H, t
...,

..

I Ilr

If the trivial solution of the system (61) is stable (asymptotically stable) in the metric Co, then this solution is also stable (asymptotically stable) in the metric C1. Proof.

Under the conditions of the theorem

We denote n Koi = C (Lis +Lis(Y)), K~ = max K oi s= 1 i n Kli - C Lis( 2 ) s=l

.

171

'

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

L e t tk =

tO+kT

and 11x1

By ( 6 6 ) , f o r any k = 1 , 2

BY (65) I

,... on

and l i m

C L ~ < O ~ - ~

k+m

a

k

f o r tk,2ititk.

the interval

tk,lzt$,

= 0.

Now l e t t h e t r i v i a l s o l u t i o n of system ( 6 1 ) be s t a b l e i n t h e metric C Then f o r any E > O , it i s 0' p o s s i b l e t o f i n d 6>0, such t h a t f o r 1 I @ ( t o + s I) IlT<6 uniformly i n k, E ~ < Eand,by v i r t u e of ( 6 5 ) and ( 6 7 ) I on any i n t e r v a l [ tO'tkl

Hence, t h e r e f o l l o w s s t a b i l i t y of t h e t r i v i a l s o l u tion i n the m e t r i c C 1' I f t h e t r i v i a l s o l u t i o n of t h e system ( 6 1 ) i s a s y m p t o t i c a l l y s t a b l e i n t h e metric C o , t h e n $&g E~ = 0 , and keeping i n mind ( 6 5 ) , w e o b t a i n Hence, by ( 6 7 ) and (68) t h e r e f o l l o w s l i m Bk = 0 .

k+=

a s y m p t o t i c s t a b i l i t y i n C1. W e mention some examples.

Example 1. 2(t)

+

ax(t)

+ b(t)A(t--r)

= 0,

where a , T > O are c o n s t a n t s , b ( t ) i s a c o n t i n u o u s f u n c t i o n f o r t 0-< t < w . The i n i t i a l f u n c t i o n is an a r b i t r a r y continuously d i f f e r e n t i a b l e function.

172

(69)

Ill. STABILITY T H E O R Y

lo

We consider the functional V[X(S),i(S) ,tl = x 2 (t) +

a

Z2(t+s) ds.

For a>O, this functional satisfies the conditions: 2 v[x(s),~(s),tl 5 r2 + CP I

V[x(s) , g ( s ) ,tl

2 r

.

. W e calculate the derivative along an integral curve. * dV dt =

-ax2 (t)

-

1-b2(t) ?2 (t-.c). a

For negative-definiteness of this quadratic form in the variables x(t) and Z(t--T), it suffices to satisfy the conditions a>O, Ib(t)l

<

1-6

6>0.

(70)

Consequently, by the asymptotic stability theorem in the metric Co and the theorem of connection between stability in the metrics Co and C1, under condition ( 7 0 ) , the trivial solution of Eq. (69) is asymptotically stable in the metric C1. By the same method, it is possible to obtain sufficient conditions for asymptotic stability for the system

*

At points of the form t=tO+k , the derivative k(t), generally speaking, does not exist. Therefore, we will assume that at these points, we consider not the derivative but the right upper derived numAVdt I berA t+8+ li sup At' 173

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

2 . (t) = 1

n

m n a x. (t) + c c bijk(t)x. (t-'ck) j=1 ij 3 k = l j=1 3 C

m

+

n

For this, the functional V may be chosen in the following form:

v

n =

c

m

i,j=l

6 . .x.x 11 1

j

+ c

n

c

k = l j=l

akj

j

0

x j

(t+s)ds

+

' k m

n

0

1 E Y k = l j=1 kj

k j 2 (t+s)ds.

-Tk

Example 2. &(t)

+

ax(t) = 4 (x(t-'c),ir(t-.c),t),

(711

where a and T > O are constants, the nonlinear function 4 satisfies the condition

uniformly in t. It is possible to show that the functional 1 x2 (t) + V[x(s),k(s),t] = x2(t+s)ds 2a 2

'1

+

-'c

satisfies the conditions of the theorem on asymptotic stability in the metric C o , if

174

1 1 1 . STABILITY T H E O R Y

On the other hand, by (72) and (731, the asymptotic stability of the trivial solution of Eq. (71) in the metric C0 implies by itself asymptotic stability in the metric C1. 10.

Absolute Stability We again consider the equation G(t)

+ ax(t) + bx(t-r)

= 0,

(74)

where a r b , and T Z O are constants. Earlier, Eq. (74) was studied for stability by the method of D-partitions and the domain of asymptotic stability of the solution of this equation was constructed. It is bounded by the lines

and the line a+b=O (Fig. 16; and Fig. 11). Of basic interest is the part of this domain represented by the inequalities

I

a>O, bl
(75)

.

(the shaded part in Fig. 16). As shown earlier, under the conditions (75), the solution of Eq. (74) is asymptotically stable for any T >-O . Definition. The solution x @ (t) of the differential equation with one o r several deviating arguments ?(t) = f(t,x(t) ,x(t-Tl) ,...,x(t--rrnH

(76)

(or analogously an equation of neutral type) we will call absolutely stable (absolutely asymptotically stable), if it is stable (asymr,totically stable) for arbitrary, constant, non-negative values T

j’

175

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

Thus, the conditions (75) define the domain of absolute asymptotic stability in the plane (a,b) of parameters for Eq. (74). We consider the linear equation n-1 n m x ( ~ )(t) + c akx(kl (t) + c c b . x ( ~(t-T.)=O, ) (77) k=O k=O j=1 k3 3 with constant coefficients ak,bkj and retardations r . > O (if for some j, T . > O , b # 0, then Eq. (77) is 37 nj of neutral type; if there is no such j,then (77) is an equation with a retarded argument). To Eq. (77) corresponds the characteristic quasipolynomial -T .z m @ ( z ) = P(z) + C Q..(z)e 7 j=1 I where n-1

P(z) = zn

+

C

n akzk, Q . ( z ) = 3

k=O

Z

k=O

k bkjz

.

We have the following Theorem.

(L.A. Zhivotovskii [29.2]).

Let in

(77)I

Then, in order that the solution of Eq. (77) be absolutely asymptotically stable, it is necessary and sufficient that the following two conditions be satisfied. 1. The real parts of all roots of the polynomial P ( z ) be negative. 2.

For any y > O , m C lQj(iY)I j=1

<

176

IP(iy)l.

Ill. STABILITY T H E O R Y

I t i s e a s y t o v e r i f y t h a t f o r Eq. ( 7 4 ) , t h e c o n d i t i o n s o f t h e t h e o r e m are e q u i v a l e n t t o t h e cond i t i o n s (75).

The c o n d i t i o n s o f a b s o l u t e s t a b i l i t y a r e a l s o s t u d i e d i n t h e works [ 1 0 2 . 2 1 , [ 1 4 7 . 2 1 . Remark. I n applications, equations with varia b l e r e t a r d a t i o n s are o f t e n encountered. Therefore it i s e x p e d i e n t , a l o n g w i t h t h e i d e a o f a b s o l u t e s t a b i l i t y , t o i n t r o d u c e t h e n o t i o n o f s t r o n g absolute stability. The s o l u t i o n x ( t ) o f Eq. ( 7 6 ) ( o r $

analogously an equation of n e u t r a l t y p e ) w i l l be c a l l e d s t r o n g l y , a b s o l u t e l y s t a b l e , i f it i s s t a b l e f o r a r b i t r a r y , continuous, .non-negative d e v i a t i o n s I t i s n o t d i f f i c u l t t o show examples of absol u t e l y stable but not strongly absolutely stable solutions. I t i s a l s o n o t d i f f i c u l t t o show examples o f s t r o n g l y a b s o l u t e l y s t a b l e s o l u t i o n s . F o r example, t h e t r i v i a l s o l u t i o n X E O , yz0 o f t h e s y s t e m mentioned on p. 30 i s s t r o n g l y , a b s o l u t e l y stable.

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

i

-i F i g u r e 7.

4 I I

I

F i g u r e 8. 178

W

Ill. S T A B I L I T Y T H E O R Y

I

I

I

-

I

.

I

Figure 9.

179

\

I,

DIFFERENTIAL EQUATIONS W I T H D E V I A T I N G ARGUMENTS

IV F i g u r e 11.

Figure 12. 180

Ill. S T A B I L I T Y T H E O R Y

p=2

F i g u r e 13.

Figure 14. 181

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS

b

F i g u r e 15.

F i g u r e 16.

182