stability and oscillations of solutions of integro-differential equations of pendulum—like systems

Math. Nachr. 177 (1996), 157-181

Stability and Oscillations of Solutions of Integro-Differential Equations of Pendulum-like Systems

By G.A. LEONOV and V.B. SMIRNOVA of St. Petersburg

(Received May 3, 1995)

1.

It is well known that boundary problems for differential equations often can be reduced to integral equations. More than that, various boundary problems for various differential equations can be transformed to the same integral equations. That is why integral equations give the opportunity of analytical investigation of common properties of wide classes of engineering, mechanical and physical problems [25, 261.

In this paper we appeal to Volterra equations because we are interested in the qualitative investigation of initial value problems for certain classes of differential equations (both ordinary differential equations and differential equations with time delay as well as partial differential equations). Each of them may be regarded as a certain generalization of the second order equation

(1 .1) i i+aU$$h(a) = 0 ( a > O ) ,

with a A-periodic nonlinear function q5(u). Equation (1.1) describes in particular the dynamics of a mathematical pendulum.

It is clear that if u(t) is a solution of ( 1 . 1 ) , then any function a( t )+Aj , where j is an integer, also is a solution of (1 .1) . Since this property is characteristic for the equation of the mathematical pendulum, any systems which possess i t are called pendulum-like systems.

For the first time the problem of investigation of global behaviour of solutions of (1.1) was stated by F. TRICOMI [37] in 1933. It turned out to be rather hard. A series of works devoted to (1.1) [2, 5, 7, 4, 13, 321 appeared since 1933, and only in the middle of the 70th the problem was exhaustively investigated. Two different types of global behaviour were fixed here. The first one corresponds to the situation when each solution tends to an equilibrium point when t tends to 00. (It is clear that (1 .1) has

158 Math. Nachr. 177 (1996)

a denumerable set of equilibria if any). The second one corresponds to the existence of a solution with the property

(1.2) .(t) 2 € > 0 (t 2 i > 0 ) .

The latter is called a circular solution.

system [la] An natural generalization of equation (1.1) is the multidimensional pendulum-like

i = A z + b $ ( u ) , ( Z E W , u E R 1 ) ,

6 = c * z + p q q u ) ,

with an n x n-constant matrix A , n-constant vectors b and c and a A-periodic function q4(u). So the problem of global behaviour of solutions of (1.3) appeared. System (1.3) is a typical system of indirect automatic control. But although there exists a theory of stability of automatic control systems developed, the system (1.3) proves to be out of its framework. The point is that standard Lyapunov functions, namely “a quadratic form” and “a quadratic form plus an integral of the nonlinearity”, which are exploited in control theory are of no use for a system with a periodic nonlinear function. In the space of parameters of system (1.3) standard Lyapunov functions give an empty stability region. Thus it turned out that the case of a periodic nonlinear function required a new type of Lyapunov functions. This new type was constructed in the papers [l, 15, 161 by means of a new method originally developed for the qualitative investigation of differential equations. It is called the nonlocal reduction method. It is based on the comparison principle. Its main idea is as follows. Together with the given multidimensional system we consider a specific system of the same type but of a lower order (of second order, as a rule) and such that its properties already have been investigated. This latter system is called a reduction system. The information about the reduction system is used while investigating the properties of the solutions of the given system. In the case of (1.3) the characteristic trajectories of the reduction system are injected into the Lyapunov functions. The nonlocal reduction method proved to be rather fruitful for the stability investigation of (1.3).

The most general form of a pendulum-like system is a Volterra integro-differential equation

(1.4) k( t ) = ~ ( t ) +&(a(t - h ) ) - g( t - T ) ~ ( u ( T ) ) ~ T ( t > 0; h 2 0 ) , Lt with A-periodic function 4(u). (It is evident that system (1.3) with a given initial value z ( 0 ) can be reduced to (1.4) with h = 0.) In particular, this equation serves as a mathematical description of various types of synchronization systems, both engineering [24, 381 and biological [35] ones. They differ from each other by the linear part which is characterized by p and g ( t ) , and by the form of the nonlinear function 4(u).

In this paper we shall present an investigation of the global behaviour of solutions of (1.4). We shall make certain assumptions concerning the functions a, g and 4. The properties of 4(u) are predetermined by the qualities of the technical devices which it is describing. We consider continuous functions 4(u) with two zeros in a period. We suppose that 4(u) is continuously differentiable in a neighborhood of each of the zeros

Leonov/Smirnova, Integro-Differential Equations of Pendulum-like Systems 159

and $'((TO) # 0 if #(go ) = 0. We suppose also that The properties of a ( t ) and g ( t ) are also assumed to correspond to the characteristics

of typical synchronization systems. We suppose that a ( t ) is continuous, tends to zero as t tends to +00, and there exits a positive constant ~1 such that

$(u) du < 0.

(1.5) a( t ) enlt E L ~ [ o , + m) .

We assume that g ( t ) is measurable and that there exists a positive constant ~2 such that (1.6) g ( t ) enzt E L2[0, + O0) . These properties of u and g guarantee the existence of a differentiable solution of (1.4) for any initial condition

and its extendability to the semiaxis [0, + co). Because of the peculiarities of d(u) equation (1.4) has a denumerable set of equilib-

ria. Indeed, any function u(t) = (20, where uo is a zero of d(u), is a stationary solution of (1.4). To obtain this fact it is sufficient in (1.4) to pass to the limit when t tends to 00. That is why we may expect for (1.4) the same types of global behaviour as for

The successful application of the nonlocal reduction technique for the stability analysis of (1.3) motivated the use of it also in (1.4) [21, 221. In this case it was accompanied by the a priori integral estimates method which, beginning with the paper [27], has been habitually applied when studying asymptotics of integral and integro-differential equations [ll , 20, 301.

Since we use here the method of a priori integral estimates, all the results will be formulated in terms of the transfer function of the linear part of (1.4),

(1.1).

K ( p ) = - p e - h p + G ( p ) ( p E a:' , Rep > - ~ g )

with

G ( p ) = Lrn g ( t ) e-p t dt ,

i.e., in the form of frequency domain inequalities. This is just the form used for stability and instability theorems which have been proved for the system (1.3) [l, 15, 161. This form is habitual for automatic control systems as it is convenient for applications. For the system (1.3) the frequency domain character of stability theorems is the result of employing the widely known Yakubovich-Kalman Theorem [14, 391 which gives necessary and sufficient conditions for the existence of a Lyapunov function. For the equation (1.4) the frequency domain stability statements appear as a consequence of the method of a priori integral estimates.

In this paper we shall begin with the introduction of the nonlocal reduction technique by a number of stability theorems, then we shall go on to sufficient conditions for the existence of circular solutions, and finally we shall consider possibilities of appearance of periodic solutions.

160 Math. Nachr. 177 (1996)

2.

This section is devoted to the stability of equation (1.4). We regard two types of stability.

Definition 2.1. Equation (1.4) is said to be Lagrange stable if each solution of the initial value problem (1.4), (1.7), for any uo( t ) E C'([-h,O]), is bounded for t E [O,+m].

Definition 2.2. Equation (1.4) is said to be gradient-l ike if each solution of the initial value problem (1.4), (1.7), for any u"(t ) E C'([-h, O]), tends to a certain equilibrium point as t tends to + 00.

It turned out that the property of Lagrange stability is the leading one for pendulum- like systems. If Lagrange stability is established for (1.4), then its gradient-like behaviour may be easily proved with the help of an additional frequency domain condition.

A number of stability theorems regarding equation (1.4) have already been published [al l 18, 191. We present here a new stability theorem. The results published can be deduced from it as special cases.

We assume in this section that $(u) is continuously differentiable on [O,A). We denote its zeroes on this interval by 81 and 82. Let @(el) > 0 and &(&) < 0. Let us consider numbers kl and kz such that

It is clear that k l k 2 < 0. In the chain (2.1) any of the inequalities may be missing. If the left side of (2.1) is missing we shall assume that k;' = 0, and if the right side of (2.1) is missing we shall assume that k z l = 0.

Let us draw attention to several properties of the solutions u(t ) of (1.4) and of the functions v ( t ) E qh(u(t)):

i)

ii)

iii)

iv)

v)

It is evident that ~ ( t ) is uniformly continuous on [0, m).

I t is not difficult to prove that the function &(t ) is uniformly continuous on [0, m) provided that a ( t ) and g ( t ) possess the properties described above.

According to the well-known Barbalat Lemma [28] it follows from i) and ii) that if k ( t ) E L2[0, + m) then u( t ) --+ 0 as t ---f + 00, and if ~ ( t ) E Lz[O, + m) then ~ ( t ) +. 0 as t +. + m.

In [19] the following statement is proved: If K ( 0 ) # 0, then for an arbitrary solution of (1.4) the inclusion u( t ) E L2[0, + m) implies that ~ ( t ) + 0 as t -+ +m.

It is not difficult to prove that if ~ ( t ) + 0 as t + $00, then u(t ) + uo where $(go ) = 0.


Let us define the function

41(u) = d(1- q 4 ' ( u ) ) ( l - &l4'(4)

It is nonnegative, A-periodic and continuous. With the help of 4(u) and h ( u ) we construct another function

F ( u ; r1, .2) = 4(u) + rll4(u)l + r241(u)l4(.>1 >

where the positive numbers rl and rg are such that

In this section for the equation (1.4) we shall choose a reduction equation of the form (2.3) U+aU+.F(a;r1,rg) = 0 ( a > 0) .

The function F ( u ; r1, rg) is A-periodic and its zeros coincide with the zeros of the function ~ ( c T ) . It is continuously differentiable for all u except for its zeros. When the argument u passes through a zero of F ( u ; T I , rg) the latter changes its sign.

As it has already been mentioned, equation (2.3) is by now exhaustively investigated. All the results accumulated here were for the first time methodically stated in the monograph [4]. Although in [4] the nonlinear function is assumed to be continuously differentiable, all the conclusions of this book remain true in the case when the derivative has discontinuities at the zeros of the nonlinearity, provided that the latter changes its sign when the argument passes through its zero. The possibility of such weakening of the restrictions on the nonlinear function follows from the interpretation of the properties of the solutions of (2.3) given in [19].

Together with equation (2.3) we consider the equivalent system

and the first order differential equation

It is known [4, 191 that for a nonlinear function F ( u ; T I , rg) there exists a bifurca- tional value of the parameter a = acr(rl,rg) such that for a > acr any solution of the system (2.4) tends to an equilibrium point (& + Ak,O) (i = 1, 2; k E Z) and for a < aer the system (2.4) has a circular solution. (If st F(u;r1,rg)du = 0, then for any a > 0 any solution of (2.4) tends to an equilibrium point, i.e., acr(rl,rg) = 0 [4, 191.) It is also known that in the case of a > acr the equation (2.5) has a solution y = Oo(u; r1, rg) with the following properties:

a) O0(&; r l , r2 ) = 0 but Oo(O; rl , r2 ) # 0 for 0 # 02;

162 Math. Nachr. 177 (1996)

b) @02(8; r1, r2) + + 00 as B ---f 00.

Note that every function

@k(o; ri ~ 2 ) = @ o ( Q + A I c ; TI, '2) (Ic E Z) satisfies equation (2.5).

(1.4) to the investigation of the second-order equation (2 .3) . The following assertion reduces the investigation of the integro-differential equation

Let K = f min{Klltc2}.

Theorem 2.3. Suppose that there exist numbers a1 2 0 , a2 2 0, A E ( O , K ) , 6 2 0, T 2 0, E > 0 and v E [ O , l ) such that the following hypotheses are fulfilled:

1) a1+ a2 = 1;

3) 2 & x ( r z J > ac7( 2 ( Y 1 m , 2 a 2 m ) ;

2) 2 a l G + ( Y Z ~ m a x g E [ 0 , ~ ) 41(a) < 1 with TI = ~ ( 1 - a12) ;

4) f o r all w E IR' the inequality

- rIc,'Ic;'(w2 + A 2 ) - (c + T ) l K ( i W - X)12

+ Re { (1 + T(L,' + I c F ' ) ( A + i w ) ) K ( i w - A ) } 2 b (2.6)

holds. T h e n equation (1.4) as Lagrange stable.

P r o o f . Let a( t ) be a solution of a certain initial value problem (1.4)1 (1.7). Let us construct a set C = (82 + A I c , Ic E Z} and a set 0 = {T I T > 0 a(T) E C}. If the set 0 is bounded, then the conclusion of the theorem is evident.

Let us consider the case when 0 is not bounded and ~ ( 0 ) E C. Let us define functions a( t ) = 4(dt) ) l

and

It is evident that ( V T ) X E Lz[O, + m) for T > 0. Note also that if T E 0, then q ~ ( t ) is continuous and continuously differentiable for all t except for t = 0 and t = T . The inclusion (1.6) guarantees that (<T)A E L2[0, + m) for any T > 0. Let us construct a set of functionals for T E 0. Define

M

ATX = 1 { ( V T ) X ( t ) ( C T ) X ( t ) + (f + m C T ) X ( w + 6 [ ( V T ) X ( t ) l 2

+ T ~ ~ ' ~ ~ ' [ ( ~ ] T ) x ( ~ ) ] ~ - ~ ( 6 ; ' + k . , ' ) ( iT)X( t ) (CT)A( t )} dt .


We denote by F [ f ] ( i w ) the Fourier transform of the function f ( t ) . Note tha t the functions ( q ~ ) x , ( < ~ ) x and ( I j ~ ) x , with T E 0, have quadratically integrable Fourier transforms and the following equalities are true:

Wi th the help of the Parseval equality for T E 0 we obtain

In virtue of the frequency domain inequality (2.6), we obtain

(2.8) ATX 5 0 for all T E 0 .

Let us take into account tha t for t E [O,T]

(2.9) (b)x(t> = (<T)X(t) + ( a o ) x ( t ) 7

where

Then ATX can be written as

ATX = Agi + A$i + ( 6 + T ) [(;.(t)]' e Z X t dt (T E 0) , Lrn where

Let us consider each functional separately. Since the functions u , q and I j are bounded on [0, co), in virtue of the choice of X we have tha t

(2.10) [A$,[ < 40 , qo independent of T .

164 Math. Nachr. 177 (1996)

Let us now denote q1(t) = p l ( u ( t ) ) and transform the integrand of ATA. Then ( 1 )

T A:? = I eZAt { b(t) ?)(t) + cb2(t ) + 6 7 1 2 ( t ) + u2(t) v f ( t ) } dt =

- - 1’ e 2 A t { tr(t) F ( u ( t ) , 2 a 1 m 1 2 a 2 r n ) + E ( l - v ) .“t)

+ cvU2(t) - 2 a 1 m 1v(t)l U(t) - 2 a 2 m 1q(t)l7/l(t) .(t)

+ 6 v 2 ( t ) + ru2(t) vf(t)} e2Xt dt .

Let

It follows from (2.8) - (2.11) that

(2.12) BTX 5 QO (T E 0 ) .

Let us consider now the reduction equation

U + 2d-U + F ( a; 2 a l G , 2 a z m ) = 0

and the corresponding first order differential equation

(2.13) 1 ~ + 2 ~ - ; j y + F ( ~ ; 2 n ~ ~ , 2 a ~ ~ ) = 0

By virtue of hypothesis 3 ) of the theorem, (2.13) has a solution of the type

y = @o(u; 2a1&, 2 a 2 J 3 )

which possesses the properties a) and b). We shall treat it further as @o(u). We shall also use the solutions @ N ( Q ) = @o(u) + A N (N E I). Then by means of change of variables y = f ig we transform (2.13) into the differential equation

dy(u) 1 (2.14) y(.) da + , / m y ( . ) + F ( e , 2 a l G , 2 a 2 & J ) = 0 .

It has the solutions ~ N ( u ) = & @ N ( u ) . Consider now a set of functionals

ITA = BTX+~~’’ [ N ( U ( T ) ) ] ’ - [ q N ( f l ( o ) ) ] 2 ( T E O ) ,

Leonov/Smirnova, Integro-Differential Equations of Pendulum-like Systems

(the number N will be calculated later). We then have

165

In the

Then

Since m ~ ( 0 ) is a solution of (2.14), we have that

(2.15) ITX 2 0 ( T E e ) .

It follows from (2.12) and (2.14) that

eZhT [N(~(T)) 1 > - po + [ T N ( ~ ( o ) ) 1 Let us now choose N in such a way that [ m*,(a(O))] for all T E 0 and all TI > N we have

(T E 0 ) .

2 > qo + 1 for all n 2 N . Then

[q*,(a(~)) 3 >

Consequently, if T E 0 then a(T) # 6'2fA n (TI = N , N+1, . . .). Thus &-A (N+1) < 0 a( t ) < 02 + A ( N + 1) (t E R1), and the theorem is proved.

Theorem 2.3 establishes suffic,ient conditions of Lagrange stability for equation (1.4). It abounds with free parameters. The parameters A, al , a2, 6, 7, v as well as the coefficients k.1 and k.2 are free and the "success77 of the employment of the theorem depends more or less on a good choice of these parameters. The testing of the hypotheses of the theorem must, of course, be carried out with the help of a computer.

For concrete problems some of the parameters may be fixed. Putting additional restrictions on the free parameters, on one hand we narrow the regions of the system parameters, where the frequency domain criterion is fulfilled, but on the other hand we obtain richer asymptotic properties of the solutions.

Theorem 2.4. Let p = 0 and K ( 0 ) # 0 , let a ( t ) E C1[O, + co) and g ( t ) be absolutely continuous functions, and suppose that there i s a positive number 1c3 such that

u(t) eK3t E L2([0, +a)) and g( t ) en3* E Lz([O, + co)).

166 Math. Nachr. 177 (1996)

Suppose also that all the hypotheses of Theorern 2.3 are fulfilled with the additional restrictions X E ( 0 , min { n, $63)) and

(2.16) Tk, 'k , ' = 0 .

Then equation (1.4) is gradient-like, and f o r every solution a ( t ) of the initial value problerri (1.4), (1.7) the following relation

(2.17) U ( t ) - 0 as t --+m

holds.

Proof . Take K O = 1 + 2X7(kT1 + k,') and consider the set of functionals

00

CJTX = 1 { nO(1/T)X(t) (<T)X( t ) + (f + T) ( (<T)X( t ) )2

+ T(G' + k y l ) (c.l.),(t) ( v T ) A ( t ) + 6 ( ( v T ) A ( t ) ) ' } dt (T > 0) ,

where 7 1 ~ and <T are as defined in the proof of Theorem 2.3. By means of the Parseval equality and of the relation

[$( (cT)A( t ) ) ] ( i ' " ) = iwF[ (CT)X( t ) ] ( iW)

we obtain

where

D$d = / T { n o q ( t ) u ( t ) + t u 2 ( t ) + 6q2( t ) 0

+ T [ b'(t) + (k;' + k ; l ) i i ( t ) q ( t ) ] } eZXt dt ,

and D$l depends on 71, uo and u o , with its modulas being bounded by a constant q1

which does not depend on T . It follows from (2.18) that

(1) D,, < Q 1 .

Leonov/Smirnova, Integro-Differential Equations of Pendulum-like Systems

Consider now the functional D$J

167

and transform its integrand. According to the second mean value theorem we have

0;; = D$JA (T* E [O,T])

and consequently for all T > 0

(2.19) D$J q1

The functional D$J can be represented in the following way

(2.20) + T ( k , l + k, ')(U(T) q(T) - b(0) q ( 0 ) )

[(It;' + k ; ' ) U ( t ) i ( t ) - U 2 ( 2 ) ] dt

Since IT, u and 77 are bounded, the first and the fourth terms of the right-hand side of (2.20) are bounded by a c,onstant independent of T. The last term is nonnegative because of (2.1) and (2.16). Thus it follows from (2.19) that

(2.21)

where q 2 does not depend on T . It follows from (2.21) that

(2.22) U ( t ) E LZ[O, + In iii) it is pointed out that (2.22) implies (2.17). If 6 # 0, then ~ ( t ) E L ~ [ O , + c o ) . Thus it follows from iii) that

(2.23) v( t )+-O as t + + c o ,

which according to v) implies the gradient-like behaviour of (1.4). If 6 = 0, then according to item iv) the relation (2.23) follows directly from (2.22). The theorem is proved. 0

The stability theorems can be supplemented by a theorem about dichotomy of solutions.

168 Math. Nachr. 177 (1996)

Theorem 2.5. Suppose that there exist numbers € 0 > 0 , TO 2 0 and 60 > 0 such that for all w E IR' we have

- T0k;'k,1w2 - ( € 0 + To)lli'(iw)(2 + Re { [ 1 + i ~ o ( I c Z ~ + k ~ ' ) w ] K ( i w ) } >_ 60 .

Then a n y bounded solution of (1.4) tends 20 a certain equilibrium point, with the limit relation (2.17) being true.

This theorem is proved in [19]. The following assertion is a direct corollary of Theorems 2.3 and 2.5.

Theorem 2.6. Suppose that there exist numbers a1 >_ 0 , a2 2 0 (a1 + a g = I), S > 0, T > 0, t > 0 and v E ( 0 , l ) such that

1)

(2.24) - ~ 1 c ~ ~ k , ' w ~ - ( 6 + ~ ) I I i ' ( i w ) 1 ~

+ R e { [ l + T ( k Z 1 + 1 c l 1 ) i w ] K ( i w ) } 2 6 ;

if a2 # 0 , t h e n

Then equation (1.4) is gradient-like, with the limit relation (2.17) being true for each of its solutions.

Proof . Let us show that if the hypotheses of the theorem are fulfilled, then the hypotheses of Theorem 2.3 are fulfilled as well. Note that the inclusion (1.6) guarantees that Il<(iw - A) ] is uniformly bounded with respect to w E IR provided that X < 6 2 .

Thus if the inequality (2.24) is true, then the inequality (2.6) is also true for sufficiently small X > 0.

Note also that in virtue of the hypothesis 3) the function F ( u ; 2 a 1 m ; 2 a z m ) is such that

A 1 F ( a; ~ C Y I ~ , 2nzJri6) d a = 0 .


It this case a e r ( 2 n l ~ , 2 a Z m ) = 0, and the hypothesis 3) of Theorem 2.3 is fulfilled for any positive numbers X and E . All the other hypotheses of the two theorems coincide. Thus any solution of (1.4), (1.7) is bounded for t E [O,+CQ). More than that, the hypothesis 1) of the theorem coincides with the hypothesis of Theorem 2.5. Thus equation (1.4) is gradient-like, with the limit relation (2.17) being true for each

0 of its solutions. The theorem is proved.

Remark 2.7. The frequency domain theorems proved in [21] can be considered as special cases of Theorems 2.3 and 2.4 for a2 = 0 and v = 0. The theorem of [18] is a corollary of Theorem 2.6 for the cases a1 = 0, a2 = 1 , and a1 = 1, a2 = 0.

Remark 2.8. The results of this section can be extended to the case of piecewise continuous functions p(a) [34]. The solution of the initial value problem (1.4)1 (1.7) is defined in this case by means of the approach suggested in [36].

3.

This section is devoted to the instability of the integro-differential equation (1.4).

Definition 3.1. We say that the solution u(t) of the initial value problem (1.4)1 (1.7) is a circular one if there exist numbers 6 > 0 and t̂ > 0 such that

d - 2 6 U

for all t 2 i

We present here a frequency domain criterion for the existence of circular solution of (1.4). An analogous criterion for a finite-dimensional pendulum-like system has been proved in [17]. We denote

r = lim p G ( p ) . P-m

Theorem 3.2. [33]. Let h = 0 , p 5 0 and r > 0. Suppose that there exists X > 0

1) the system such that the following requirements are fulfilled:

(3.3) X P dF fi y = --y-$Y(r); Ir = y+-p(a)

has a circular solution;

2) a ( t ) = ao( t ) + bOe-xt + lot e P X t

g ( t ) = y(t) + a e - X t + Bt e-xt

(bo , IO E IR1)l

(a, p E IR’ , a + 0, p fi 01, The proof of this theorem give11 in [33] is too brief, so we repeat it here in a more detailed form.

170 Math. Nachr. 177 (1996)

where ao(t) E C 1 [ O 1 m), y(t) E C2[0, m), and there exists s > 2 X such that

(3.4) y(t) e s t , j ( t ) e s t , ?( t ) e s t l ao(t) e b t , cEo(t) est are in L ~ [ O , +m) ;

3) the relations

(3.5) ReG(iw - A) < 0 for all w ,

lim w 2 G ( i w - A ) < o w-a3

(3.6)

and

(3.7)

are true. Then for an arbitrary q ( t ) satisfying (3.4) there exist numbers bo, 10 and an initial value u(0) such that the corresponding solution of (1.4), (1.7) is a circular one.

Remark 3.3. Condition 1) of Theorem 3.2 can be replaced by any condition which guarantees tha t (3.3) has circular solutions. Sufficient conditions of this type have been obtained in [2, 7, 4, 13, 321.

P r o o f of Theorem 3.2. We again use the notation ( f ) , . ( t ) = f ( t ) e r t . Together with the function q ( t ) = cp(o(t)) we introduce here the function p ( t ) = b ( t ) - p q ( t ) . We shall split the text, of the proof into several parts.

lo . For an arbitrary T > 0 let US consider a set of functionals

and transform it as follows:

Let us consider both co-factors of the intergrand of separately:

where


Then

( P ) X ( t ) = (ao)x(t) - (.l)X(t) - 6 ( t ) - PYA(2).

Consider the function R(t) and note that, as soon as r = a + y(O), we have

Thus

Let us single out in JTX the terms which depend on ( a o ) ~ and (a0)L. (We can suppose without loss of generality that y(0) = 0. Otherwise we could use the representation

where yl( t ) = y ( t ) - y(0) and cyl = y(0) + a.) So we consider the representation

(0) ( 1 ) ( 3 4 J T X = JTX + JTX 1

where T

$2 = ; 1 { (.l)X(.l)’X + ( .FA (4 + P Y X (.l)’x

+ B(.l ) A Yi + aPtX Yi + P2YX Yi 1 dt 1

and J;? contains all the terms which have ( U O ) ~ or ( ~ 0 ) ’ ~ as a co-factor. It is not difficult to prove that the hypothesis 2) guarantees the estimate

(3.9) IJ$JI 431

( 1 ) with 43 not depending on T . Let us transform J T X . We obtain

Consider now the functional

172 Math. Nachr. 177 (1996)

Let us define the function

Changing in JgX) the function &, by the function EAT and applying the Parseval rela- tionship we obtain

- L T ( y ) ; ( t ) E X T ( t ) dt

where x(p) = som y(t) e - p t dt. From the inequalities (3.5) - (3.7) we obtain

w 2 x ( i w - A ) - B = w2G(iw - A) < - 6 < 0 for all w E JR.

So for any T > 0

(3.11)

From (3.8) - (3.11) we have

(3 .12) JTX 2 &(T)7

where

Thus

1 1 (3.13) J T ( d x ( t ) 0 ( M t ) dt 2 - ,[(ddT)l2 + ~~;[(P)X(O)I~ + &(TI .

' 0 2 . We shall now introduce a trajectory of the reduction system (3.3) into the Popov functional. It follows from hypothesis 1) of the theorem that the system (3.3) has a


solution { u o , y o } such that u o ( t ) 2 6 > 0 for t 2 t^ > 0. Notice that we can always choose t^ in such a way that

(3.14) 4(ao(i)) = 0 .

Let us consider the equation

(3.15)

which corresponds to the system (3.3). Let Fo(u) be the solution of (3.15) corresponding to the circular solution {ao,yo}. Let 601 = uo(i) # 00, 6 0 2 = $ 0 0 . Note that Fo(u) has the following properties:

i) for all u E [u01,u02]

P Fo(u) + - 4 ( u ) 2 6 ; dF (3.16)

ii) Fo(uo1) 2 0 (it follows from (3.14) and (3.16));

iii) for all u E [ ~ 0 1 , a 0 2 1

(3.17) Fi(a) [Fo(u) + ( P / f i ) 4 ( 4 3 + (Vfi)Fo(a) + 4(u) = 0 .

According to the investigations of [17] the property i) implies the inequality

(3.18) ~ ; ( c r ) ~ o ( u ) + 4(u) 5 0 for all u E [ ~ O I , ~ O ~ I This is the trajectory Fo(u) which we are going to inject into the Popov func,tional.

3'. Let us consider the set of functionals

and transform it as follows:

where Fo = Fo(u(t)) . With the help of LTX we shall prove that for a fixed ao(t) it is possible to choose bo and lo in such a way that for t > 0 the inequality

(3.19) a( t ) > (To1

(3.20)

is true and thus for all t > 0 the inequality

F ; ( d t ) ) F o ( 4 t ) ) + 4(u(t)) I 0

is also true. Simultaneously we shall show that for all t > 0

174 Math. Nachr. 177 (1996)

is also true. Let us choose o(0) and bo in such a way that

(3.22) 4 0 ) = uo1,

(3.23) b(0) -p4(a(O)) = a o ( O ) + b o > 0 ,

(3.24) b(0) = ao(0) +bo + P 4 ( U O l ) > 0

and

(3.25) 1 1

F02(.(0)) < - (ao(O) + r = r (b(0) - P4( . (o) ) )2

According to ii) Fo(aol) 2 0, and thus it follows from (3.23) that the inequality (3.21) is true for t = 0. As all the functions in (3.21) are continuous, the inequality (3.21) will be valid in a certain right half-neighbourhood of zero. From (3.22) and (3.24) it follows that inequality (3.19), and consequently inequality (3.20), will be fulfilled in a certain right half-neighbourhood of zero. Suppose that the inequalities (3.19) and (3.21) hold for t E [0, To) (TO > 0). We can prove that no one of them can be violated for t = To. For this, suppose to the contrary that the inequalities do not hold for t = TO, that is, either

(3.26) a(To) I (701

It follows from (3.17) that

(3.28) LToX 5 0 1

whence 1 To 1

2 ( p ) x ( t ) ( 7 1 ) X ( t ) + 5 F&(O)). I - h - Fi(o(T0)) e2XTo

Taking into account (3.13) we have

- 1 F,2(o(To)) eZXT0 - - 1 [&(To) - p4(a(To))] 2 e2XT~ 2 2 r 1 1 (3.29)

< - F$(u(o)) - - [ b(0) - p4(a(O))] - &(TO). - 2 2 r


Let us investigate the right-hand side of (3.29). For this we consider

Note that

where 60 > 0 and 61 > 0 are arbitrary numbers. We choose 60 and 61 in such a way that

1 I x ( 0 ) < 6, -+- 461 460

Then

1 8’ P &(TO) 2 - ((ao)~)~(O) - ~ ( y x ( 0 ) ) ’ + r (ao)x(O) yx(0) - M

where M is positive and bounded by a value independent of TO but linearly dependent on I I x (0 ) l So”“r)lx(t)I’ dt and So”[(ao)’x(W dt .

so

176 Math. Nachr. 177 (1996)

If y~(0 ) = 0, and if lbol is large enough, then the right-hand side of (3.29) is negative and (3.27) cannot be valid.

From (3.21) and (3.16) i t follows that for all t E [0, TO)

(3.30) U(t) 2 € > 0 ,

and thus inequality (3.26) also cannot be true. We have proved that the inequalities (3.19) - (3.21) cannot be violated for any t = TO > 0. So they are true for all t > 0.

0 Thus inequality (3.30) is also true for all t > 0, and Theorem 3.2 is proved.

4.

This section is devoted to a type of c,ircular solution of (1.4) which is of most interest because it describes the important behaviour of technical systems described by (1.4).

Definition 4.1. We say that the solution a( t ) of the initial value problem (1.4), (1.7) is a periodic solution of the second kind if there exist a positive number T and an integer j # 0 such that

a( t ) = p ( t ) + j A - t , T

where p ( t ) is a T-periodic function. The number w = periodic solution of the second kind.

is called the frequency of the

The purpose of our investigation is to combine the idea of harmonic analysis, which is habitually used when solving approximately the equations with periodic nonlinearities [29, 31, 241 and the strict methods developed in Sections 2 and 3.

We prove here a frequency domain theorem which guarantees that the equation (1.4) has no periodic solutions of the second kind with certain frequencies. An analogous theorem has been proved in the paper [23] for a finite-dimensional system (1.3) (see also [9]).

Let the restrictions on the nonlinear function 4(a) be the same as in Section 2. We need here two numbers which have already been used in Section 1 . Let

Theorem 4.2. Suppose that there exzst numbers w o > 0 , r 2 0, S > 0 and E > 0 such that the following hypotheses are true:

1.

K ( 0 ) - (€ + T ) l K ( O ) l ~ > 6 ;


2. The inequality

Re { K ( i w ) - 7[ K ( i w ) + k;’iw]* [ K ( i w ) + k T ’ i w ] } - cIK(iw)(’ > 6

(where the symbol * denotes the complex conjugation) holds f o r all w 2 wo; 3. One of the following two inequalities is true:

either a) 466 2 (vo)>” or b) 4r6 > (v‘)’. Then the equation (1.4) has no periodic solutions of the second kind with a frequency w 2 wo.

Proof . We shall use here the method suggested by GARBER [lo] and extend it to integro-differential equations. We shall combine GARBER’S method with the BAKAEV- GUZH technique [3] and with its modification offered by BROCKETT [8]. This technique has been already used by us in the process of the proof of Theorem 2.3. It gave us the opportunity to single out positive definite quadratic forms in the integrand of the functional A T X .

Suppose that u(t) is a certain periodic solution of the second kind of (1.4) and it has the frequency w 2 W O . Consider again the function ~ ( t ) = 4(u(t)) . It is a periodic function with period T = G . Indeed,

4(a(T)) = 4(P(T) + jA) = 4(P(T)) = 4(P(O)) = 4(0(0)).

Let us expand ~ ( t ) into a Fourier series. So

(the restrictions on $(u) guarantee that the expansion (4.1) exists). Let us substitute d(u(t)) by (4.1) in the equation (1.4). Then

+m

.(t) = a ( t ) + p C BI e-i lw(h-t) - lt g ( 7 ) c BI e-ilw(T-t) d7 ,::M I = - 00

Hence it follows that

(4.2) +03

.(t) = a ( t ) + b ( t ) - C B~~i’(i lw)e-i’’w(T-t) , I=- M

where

Y I = - 03 t

By virtue of (4.1) and (1.6) we claim that b(t) + 0 as t --f $00. It is clear that b(t) is T-periodic. So it follows from (4.2) that d t ) + b( t ) = 0 and

I = - 00

178 Math. Nachr. 177 (1996)

Let us construct the functional

where T is a "period" of the periodic solution of the second kind. Let us consider each term of I separately. By virtue of (4.1) and (4.3) we have

T T / u * q dt = / (- E Zi ' ( i lw)B~ &Iu') * ( E BI ei"") d t . I = - w I = - w 0 0

Since s,' e i ~ w t eimut dt = 0 for I # m and B k = B* (-k)'

In a similar way we get

m

cu

h*Ijdt = r j*ud t = 2T X R e ( K * ( i l w ) i 1 w ) . i 0 i 0 I=1

We have tha t

03

I = T [ 6 + (C + 6)lZi ' (0)(2 - Zi'(O)] lBolZ + 2 C IB112 1=1

1 - 1 2 2

{ x {Icy k, 1 w T + (T + ~ ) ~ Z i ' ( i ~ / ) ~ ~ + + 6 - ReZi'(iw1) - T ( k T 1 + ~ ; ' ) R e ( Z i ' * ( i l w ) i l w ) } 1

The hypotheses 2) and 3) of the theorem guarantee tha t

(4.4) I < 0 .


Suppose now tha t inequality a) is true. Let us introduce the auxiliary function

Fo(a) = 4(u) + .o14(.)I It is clear t ha t st Fo(a) da = 0. Then

Fo(a)do = 0 .

T

CT' JOT Fo(u(t)) U(t) dt = (4.5)

Let fo(t) = Fo(a(t)) . Because of (4.5) it follows so U f o dt = 0. So I = where

+ I ( 2 ) ,

It followsfrom (2.1) tha t I ( ' ) 2 0. On the other hand

r(1) = { - UvO(q( + S(7]l2 + e(k2 ( } dt ,

and in virtue of a) i t is true tha t I ( ' ) 2 0. So I 2 0 which contradicts the negativeness of I . Thus in the case (a) the theorem is proved.

Suppose tha t the inequality b) is true. Let us introduce the auxiliary function

F1(a) = v ( 1 ) 1 4 1 ( 4 4 ( d + 4(u) A where 41(a) was introduced in Section 2. I t is clear tha t so Fl (u )da = 0 and conse-

quently sl::'F1(a)da = 0. Let f l ( t ) = F l ( u ( t ) ) and 71(t) = $l (a( t ) ) . Then

or

By virtue of inequality b) I 2 0 which contradicts (4.4). Thus the theorem is proved. 0

5 .

The results of the Sections 2, 3, 4 were applied t o the stability investigation of phase synchronization systems [19, 231, in particular of phase locked loops with t ime delay [21, 201. We have tha t estimates of boundaries of stability and instability regions in the space of system parameters have been obtained. They give the opportunity t o revise the results of approximate stability investigations [6 , 241.

180 Math. Nachr. 177 (1996)

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s t . Petersburg university Department of Mathematics a n d Mechanics Bibliotechnaya sq. 2 198904 St. Petersburg Russia

stability and oscillations of solutions of integro-differential equations of pendulum—like systems

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