DSpace at VNU: On uniform stability of the characteristic spectrum for sequences of linear differential equation system

VNU J O U R N A L OF SCIEN C E N a t S c i \ XV n‘^5 - 1999 ON U N IF O R M STA BILITY OF TH E C H A R A T E R IS T IC SP E C T R U M FO R SEQ U EN C ES OF L IN E A R D IF FE R E N T IA L EQ U ATIO N S Y S T E M ’ N guyen T he H oan Facility o f Mỉitheiiìatic^, Mechanics and InibriuHtics College of Natiiial Scicjices - V N Ư D a o T h i L ie n TeHchei \s Trying Coilege Thai N guy en U n i v e r s it y A b s t r a c t: / / ? t/u.s paper we gwe a covdition for which the charateristic spectrum of tÃc sequence of l i n t a r drjfereĩittaỉ equation systeTUS are stable This condition IS imposed 071 the coeffi.cemts of sys tems The obtained results are applied f o r studying un tfon n roìighĩiess I INTRODUCTION C o n s i d e r a s e q i i e i i c e o f s y s t f ' i n s f o n s i s t i i i g OÍ l i n e a r c l i f f e r e i i t i a l e q u a t i o n T e n '\ ^ = „ ( n T , n = (1) fit where An{f ) a X - m atrix contimious 01Ì [^O'Oo) and satisfies th e condition ^ sup < C X ), '// — 1, 2, • • • (2) t>fa denote by the chaiatoristic spectrum of th e system (1) Let MS as s o c ia t e w i t h (1) a s c q u e u r e of non-liiu'ar syyteiiis ^ = A „(/).r + /„ (/,.'■ ) (if p ertu rb ed by the function ỷ n ự - x ) satisfying the relation | / n ( ^ - r ) l | < e iiiiifonuiy ìỉ ppcỉ-stỉihỉe if for Huy g i v e n e > i) t h e r e e x i s t s Ò = ố{ f ) Sììch t h a t the H s sỉ uii ptỉ un (4) i m p l i e s for all // € A „ If the assiiniption (4) implies then the chaiati'iistic sp(‘ctruni of (1) is said to be uniformly lower-stahlo If both the inoqualitios (5)-(6)hold thoỉỉ tlif* charateristic spectrum of (1) is said to be uniformly stable Tho notion of uniforin staỉ)ility of a charateristic spectrum for the sequence of differential equation syst(niis is used in tho stu d y of uniform roughness of this sequence and, in turn, the Iinifonii loughiioss of the sequence of dirft'rential equation systems is used in estiination of Iiunxber of stable periodic solutions of the differential systems [1 II SPECIAL CASE /„(/,.r) - First of all wo roiisiiler tlii‘ special case when* is linear in x,that is: fn{t.-r) 'I'hen th(' system (3) is of tilt' fonn tỉ-i' ih'iiuti \>\ \ \ - , aÍ, 4,,(/)./■ + /? , ( / - , ^ < Aj // - || ữ „ (/) |ị < the ( li nt t 1i^it ii < (S V / > / o (7) 11liiii of (T) A]>plyiiig P fM o n ’s tiansfoiin ation ■r^U„{t)y, (8) where ư„{t ) is an orthogonal m atrix, tlu' system (1) is reduced to the triangular one § wliere Pn{t) = {f ) A„{t ) Un{t ) - = p»(')!/- (9) It is easy to verify th a t \\Pn{f)\\ < M l , n = by the traiisfoniiation (8), the system (7) becomes dy (if where Qn{t ) Pn{t) = P n [ f ) y + Qn{f ) y, (10) ^{f ) Bn{t ) UnỰ) - Denote by P u \ t ) , p \ \ f ),P 22 Ự) elements of the m atrix N g u y e n The Hoan, D a o T h i Lien 3() w\ Vi ^ ) ~ P2 Ì ^ ) = P n ^ ( ^ ) - P 2 ^ ( = < 'o s < ^ o ^ "> (f)Ịn ị"’ ( f ) - a ị ’ ( ì + ^ in ự > < " > ( f ) la ị'Ị ^ + o ị: ^ > ( f ) / V ) V , ”*(o = /íÌ2V)-PÍ"V) = cos2^(0-«iiV)l-sni2v^('')(0[4i^+«i”V ) Th(MefoK\ p'j;'(n-/:;>it) = ựíi^lVí') - + [ « y ; ' + , / , ' ; ' ( P ) ^ < ™ | i ' " ' ( / i + 4-,.{() in which A " 'ư ì-" 'ăm and - p\:ht) = V {!"n V ) W'UoiV ^ t , ự ) - ;;V)]^ + [ " ^ + «i;;’( in- C'OS[ ^2 ’ ^1 by g i v i n g a l o w e r V)Ouiul f o r For ^7-2 denote the spectra of th(' adjoint syisteiiis conosponding to (1) and (7) Then hv P m o i r s theorem and Lyapunov's iiioqiiality W ( ' liaví' A‘" ) + Í ” ^ = , Ã Í ''’ + f , " > > () Applying (20) to the bigger charateristic exponent it yields 7',” ’ < ^ or Thus, ã 1'” Sumiĩig up, we heve the following: > À - (21) On u n i f o r m s t a b i li t y o f the c h a r a t e r i s t ic s p e c t r u m f o r 33 Lem m a, For f sinali ciioĩigb and iit,{r)(lT < (' < oo, 7Ì — 1,2, , ■ft where !!» (') = + l