Critical case for periodic solutions of a class of neutral equations with a small parameter

Pub . Mat . UAB Nó 13, Juliol 1979 CRITICAL CASE FOR PERIODIC SOLUTIONS OF A CLASSOF NEUTRAL EQUATIONSWITH A SMALL PARAMETER PedroMartínez Amores Secciónde Matemáticas - Universidad de Granada Introduction In this palier we study a class of neutral functio nal differentialequationswhich arises from a coupled sys tem of differential-difference and ordinary diff erence equa tions that occur in various applications, as electrical cir cuitswithlossless transmision lines, [11 . In [101, [151 is studied the stabilityof such system and in [131 are given - conditions for the existence of periodic solutions . In this paper, we consider a nonlinear system with a small parame-- ter and we study the existente of periodic solutionswhen - the corresponding linear system can have periodic solutions (critical case) . This is done by applying the method of Hale [5,61 andits extension to delay diferential equations - (see [141) to the neutral equations obtained from the coupled system after taking into account the resultsof [101 . l . Notation and summary of knownresults Let E n be a complex n-dimensional linearvector space with norm 1 .1 andlet r be a fixed positive number . C = C  ( [ - r,  o 1, E n) is the spaceof continuous functions 1 : [-r, 0 1 --~ E n with norm I~¡= sup~  ~(9)1 :9r, o1 ~~ Suppose D, L are bounded linearoperatorsfrom C to E n ,  0 = H f(0) - f [dl+ (A)~~ (9) -r 0 L ( 1 ) = f -r [d 1 (9)]f (8) where  li  is an n x n matrix, det li =~ 0, rt , ~ are n xn matrix Functions of . boundedvariation on [- r, 0] with M nonatomic at zero . We assume N has no singularpart . If .  x  is a continuous function mapping [a'- r,  ) into E n , then for any tE[0,w)we define xt in C by -  - x t (9) = x (t + A) ,9E[-r,  0] . An autonomouslinearhomoge- - neousneutralfuntionaldifferential equation (N F D E ) is - def ined to be d dt D(x t ) =  L (x t ) A solution x= x (¢) of  (1 .] .) through a point JE C  at  t = 0 is a continuousfunction taking [-r, A),  - A> 0, into E n such that  xo = f, D(x t ) is continuously di-- fferentiable on [0, A) and equation (1 .1) is sati,sfied on - this interval . It is proved in [2,4] that there is a unique solution  x (~) through  and x (1)(t)  is continuous in  (t, f b If . the transformation T(t) : C -+C is defined by - T (t) ~ = xt (1), then it is shown in [11] that 1 T (t ), t>,0 } - is a stronglycontinuoussemigroup of linear operators with the inf initesimal generatorA : -9(A) -r  C,  A  ( 9) _  ( 9 ),  - andthe spectrum  0'(A) of A consist of those X which satisfy det 62 0  0 Of%) = 0,  a(ñ) -_ ñH _~J  e e d~(9) _ J  é6dj(9 _r  -r The fundamental matrixsolution of (1 .1) is defined to be the n x n matrix solution of the equation t D(X t ) = I+J  L ( X) ds, t> 0 0 s 0 , -rc9<0 Xo (0)  0 (1 .6)  xt -X o G(t) = T(t-r)[f -X oG(C)]+ t  t +JT T(t-s)X o F(s)dsJ e . (d sT(t-s)X o IG(s) for t>r, f E C, where it is always understood that the inte gralsin (1 .6) are actuallyan integrals in E n . Formula (1 .6) suggert the changeof variables x tXo G(t) = z t ,  f -X o G (T) = f rom C - P C .  If this is done,  equation (1 .6 ) becomes t  t (1 .7) z =T (t-T)t+ f  T (t-s)X F(s)dsf [d s T(t-s),XJG(s) . t  T  o Definition 1 .1 . The operatorD is said to be stable if there is a -4 > 0  such that all rootsof the equations det D (e x . I) = 0 stisfy ReÁ<-a . If D(+) = H j(0) - M«-r), then D is stable if the rootsof the polynomialequation det (H -PM) = 0 sátisfy  - An important propertyof equation (1 .1) when D is stableis the following (see [3]) : If D is stable, then - there is a constant  aD< 0 such that for any a > a D , there - are only a finitenimber of rootsof det (j(ñ) = 0 with - - Reñ) a . Let D be stable .  If A = t ~ : det á (X) = 0,  Re X> .0 }, then A is a finite set and it is follows from [11] that the space C can be descomposed as C=P 49 Q, where P, Q are - subspaces of C invariant under T(t), the space P is finite dimensional and corresponda to the initialvalues of all - those solutions of (1 .1) which are of the form p(t)e xt ,  - where  p(t) is a polynomial in t and ÁEA . If í  is a basis for P then for ever y +E P thereexista a vector aE E  such that +=í a .  En this case, we can define T(t)+= l e Bt a,  - 6 3 If  F, G : [0, ~)  --~ En are continuous, a nonhoimo- : geneous linear N FD E  is defined as (1 .3)  dt  ID (xt)-G(t»=  L(x t )+ F(t) . A solution through f at  t = T  of (1 .3) is defined as before and is knownto existon [o'-r, c o) . The variation of constantsformula for (1 .3) (see [9]) statesthat the solution of (1 .3) through (r, f) is given by t  t+ (1 .4) x(t) = T(t-( - ) 1(0) +f ue X(t-s)F(s)dsf .. [d s X(t-s)]G(s)-G(r), for t> l T, where X is the fundamental matrix solution given by (1 .2) . Equation (1 .4) can be written as (1 .5)  x(t) -X(o)G(t)=T(t-Q»)~(0)-X(t-T)G(T) t  t for t>,(' . Now, let PC be the space of functionstaking  - [- r,0] into E n which are uniformly continuouson E -r .0) and may be discontinuous at zero . With the matrixX as defined 0 before,  it is clearthat PC= C+ ( X0 ), where (X 0 ) is the - - span of X 0 ; that is any1,EP C  is given as ^+ = 1+ X 0 b,  1 E C, n bEE .  We make P Canormed vectorspace by defining the norm 1+1= max{/41, b) . Let us define xt (~ ) = T (t) ~ , where fiEP C  and  - x(^r)  is the solution of (1 .1) through + . The operator -  - T(t) : PC - a (functions on [-r, 0] ) is linear, but T(t) does 64 not take P C --j PC . It is an extensionof the original semi group T .(t) on C . If we use this notation, then the variation of constants formula (1 .5) can be written as where  B  is an n x n gtatrix defined by  Aí =f B .  The spectrum of B is A . If  C  is decomposed by n as  C = P G Q then equa-- tion (1 .6) is equivalent to t xtXp G(t) = T(t-T)C,~PXpG(Q')]+ f . . T(t-s)XpF(s)ds - -fr [ds T(t-s)XP]G(s) 0 where the superscrits P and Q designate the projections of - the correspondinefunctions onto the subspaces P and Q, respectively, and they can be determinedby meansof adjoint differential equation to (1 .1), see (111 . 2 . The li nea r problem t xtXQ G(t) = T(t-~)[¢QXQ G(c)J + fT T(t-s) XQ F(s)ds - t (ds T(t-s)XQG(S) 0 1 In this section, we consider the system a) z(t) = A1x(t)+A2y(t-r) b) y(t)-A 3 x(t)+A 4 y(t-r)=0 where x, y are n-vector andall matrices are constants . For any aEE n,--tEC, onecan define a solution of (2 .1) with  - initial value x (o) = a,  y  = ^~ .  If we def ine,  C = C ([-r, 0], E n ) 0 (2 .2) D 1 L 1 D, L :  Enx  C ----> E nx n'  D= [D  L - 0 2 7t D 1 (a, -t)= a D 2 (a,^f) ='f(0) - A 3 a - A4 L 1 (a,°%) = A l a+A 2 +(-r)  65 thenequation (2,1) is a specialcase of the NF D E (2 .3) d dt andone obtain the system (2,1) by requiring that is a strcngly continuous semigroupo The infinitesimal gene_estor e of T (+) í_0 0 D (x (t ),  y t )  =  L(x (t ), y t ) D 2 (a,^%)= 0 Equátion (2-3) definesa semigroup T(t) on Enx C , If we define (E n xC) 0 = « a,t)EE n xC : D2(a,^f)=0} then (E n xC) 0 can be considered as a Banach spacee Furthermose, forany -  - (a, -+)E(E n xC) 0 , the solution of  (2,3) through (a,-t)  will - be in (E nx C) 0  since it corresponds to the solution of  (2 .1) through(a,-t)o Consequently, T  (t)  def  T (t) (  ,  (E n. C)  -~  (En . C ) 0  0  0 n  1 (E x C) 0 nitesimal generator of T(t)¿One shows that T(A 0)=Ii1E4 :: Observe that = A l  where A is the infi ¡(E nx C) 0 ~ I-A 1 det~(ñ)=0},  o (ñ) =  -A 3 a  I 0 a 0 0 a D( a ,^~ . ~(O) -A 3 a - A4 Y(-r)  -A 3 I  ^Y(0 )  0  A4 ^¡'(-r ) def =  H ~(O) -M 4(-r ) a  A1 0 a ío A L(a,~) -- - Y(O ) -A3a-A4  ~,(-r)  0  0  .~((0) + 0  0 66 df  Ni (0) + P ~(-r)o -A 2 e  Á r I-A,e Xr duli lessthan 1, then D is stable . trix solution X(t) of (2 .3) as If  X = ( X 11  X 121 ~ _where  X i .,  i, j = 1 9 2 1 are nxn matri 21 22  j  _ ces, then X mustbe a solution of (2 .3) with the initial - data specified above . Therefore, the matrices X, . must sa-- 1sJ tisfy (2 .4) Notice that  (2 .4a)  impliesX 11 ,  X 21 are solutions of  (2 .1) . The functions X12, X22 do not satisfy (2 .1 b) . This implies that the variation of X(t) satisfies the system (2 .1) . UsingLaplace transform or the same type of argu-- ments as in Hale () pag . 303, onecan prove the following Lemma 2 .1 .  If the eigenvalues of A 4 have moduliless than 1 and all roots of det á (ñ) = 0 satisfy Reñ<-d<0, then - there are positiveconstants K, o( such that Ix 11 (t)1,  1x21(t)ls  I X ij (t)j_4 Ke C(t s a.e  t>,0,  i,j=1,2 . Now we consider thenonhomogeneous system (2 .5) Thus, if the eigenvalues of the matrix A 4 have moAl so, f rom (1 .2) . we can define the fundamental ma fI 0 X(9) = H 1 o  (A 3Ie = 0, x0 (e) = o,  -r i6 9 <0 . a) D2 (x 11 (t), x 21 .t ) = o , t> " o b) D2 (x 12 (t), x 22 .t ) =  , t>, o . X(t)=A 1 x(t)+A 2 y(t-r)+f(t) y(t) -A 3 x(t) -A 4 y(t-r) - g(t) =0 n where f, g are continuous functions from [0, -0) to E . With D, L defined as in (2 .2), system (2 .5) is a specialcase of the N F D E (2 .6)  ct {D (wt)-G(t)}- L(wt)+F(t) where  w  = col (x (t ),  y  ),  w  _ ¢  col (a, ^~ )E E nxC =  , - t  t o G = col (0,g)EEnxn$ F = col(f, O)EE nxn and one obtains the system (2 .5) from (2 .6) by requiring that D 2 (a,y)= g(0) . As in Section 1, if we extend the definition of - T(t) to E n x (C + ( X_ ) ) de Y,  then the generalsolution of .  - (2 .6) is givenby the variation of constants formula (2 .7) t w t-Xo G(t) = T(t«- X 0 G(0),+ f  T(t-s)X 0 F(s)ds - 0 t -f [d s T(t-s) X0 ]G(s) . o This last formula suggests the change of variables w tX0 G(t) = z t s  JX0 G(O) =5,  1 from'E nx C  to  Y . If this is done, formula (24) becomes t  L (2 .8)  zt T(t) +f T(t-s) X0 F(s)dsf [ds T(t-s)X 0 ]G(s~ t>,0 . Onecan give an explicit decomposition of (2 .8) by using the adjointequation to (2 .3) . For this, we write - - (2 .3) in the form d át ~H w (t) - Mw(t-r) )= Nw(t) +P w(t-r), w0 =JEE nx C which is equivalent to (2 .9) dt {w(t)-Mw (t-r)}= Ñw(t)+Pw(t-r) lince H1 M=M and where H 1 N=N, H 1 P=P . We define the adjoint equation to (2 .9) as (2 .10) d t jv(t)-v .(t+r) M}= -v(t)Ñ-v(t+r) P, v O =t<GE n XC~ In the lame way, we may write (2 .6) in the form 68 (2 .11)  dt {w(t)-Mw(t-r)-H 1 G(t)}= Nw(t)+Pw(t-r)+H 1 F(t) Using the samearguments as in [7,13), it is - -- easily shown that if the eigenvalues of A4 have moduli less than 1 and EnxC  is -decomposed by /t= I% : det,&(a) =0 ,  Rein> .0 as P eQ thenequation (2 .8) is equivalent to (2 .12) t  - a)  zt= T(t)  P+ f O  T(t-s) Xó H 1 F(s) ds  - - f t [d  T(t-s) X P ] H 1 G(s) 0 s  0 t b) zQ=T(t)3 Q +f T(t-s) XóH 1 F(s) ds t -f [d s T(t-s) Xo]H _1 G(s) 0 where  JS  = 4 -XoG (0),  4  = o  tt t if  zt = íu (t ),  where  is a basis for P and T(t) ~_  eBt,  - the spectrum of B is 11 ., then  u  satisf¡es the equation (2 .13)  ü(t)=Bu (t)+^1(0)H 1 F(t)+B i(0) H 1 G(t), -oo4tcoo where IP is a basis forthe initialvalues of those solu- - tions of (2 .10) of the form p(t)e - ~ t , p a polynomial,IEA . If 9is the Banachspaceof continuous and  - T-periodic functions with norm 1 f  =  sup 1 (f (t )j , tE[0, T]}, - then onecan state the theorem on the Fredholm alternative for periodic solutions as : Theorem 2 .1,03] If the eigenvalues of A 4 have moduliless than 1 and f, gE -9, thensystem (2 .5) has a solution in if and only if (2 .14)  f0 v(t)H . 1 F(s)dsf jdv (t)]H 1 G(s)=0 . If the function z * (a, £ ) in Lemma 3.1  is differentiable  - t withrespect to a, we can apply the implicit function theo rem to (3 .9) to have a = a(6 ) . g = £ g, where  f (t,  g(t,  ) ble in ~ and we define In particular, i£ f= F -F ., are continuoslydifferentia (3 . .10)  F 1 (a, E ) =f T X11 (-t) H 1 f (t9 wt (a, F .) )  dt + o TX 12 Fi 1 g (t,  w* (a, E ) ) dt =  0 0 F 2(a, E . )  = , jT X 21 (-t ) H  1 f (t, wt (a, E)  )  dt  + o T 1 + f0  X 22 H  g (t, wt ( a, b) dt = 0 1 then, we have the followingtheorem for the firstapproximation Theorem 3 .2 . Let f, g satisfy the aboyé conditions . If - there is an  ao , I í eBtao 1 4 T ,  such that Ó(F,F) l (3 .11)  F1(a0, 0)=0, F 2 (a o , 0)=0,  det  1  2 (a 0 , 0) J 9E 0 áa then there is an E .7 0 suchthatsystem'(3 .1) has a T-perio o  - dic solutionwt(a 0 , E ) , 01IE) < E 0 , continuous in 6 : and Bt wt (a 0 , 0 ) _  e  a 0 . Proof .  The hypothesis (3 .11) andthe implicitfunction theo rem implythere is an  fi o> 0, such that equations (3 .10) ha - ve a solution  a ' (E),1 a(6)1-<a,  0-IEI<¿ 0 . Theorem 3.1 .  - implies the essertions of the theorem . REFE RENCES Brayton, R . Small - signal stability criterion for elec-- tricalnatworkscontaining lossles transmision lines . I . B .M . J . Res . Develop . 12(1968), 431-440- [2] Cruz, M . and J . Hale . Existence, uniqueness and conti- - nuousdependence for hereditarysystems . Annaly Math . Pu ra Appl . 85 (1970), 63-82 . Cruz, 11 . and J . Hale . 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