JASEM ISSN 9-8362 All rigs reserved Full-ex Available Olie a p:// wwwbiolieorgbr/ja J Appl Sci Eviro Mg 25 Vol 9 3) 3-36 Corollabiliy ad Null Corollabiliy of Liear Syses * DAVIES, I; 2 JACKREECE, P Depare of Maeaics ad Copuer Sciece, Rivers Sae Uiversiy of Sciece ad ecology, PMB 58, Por Harcour, Rivers Sae, Nigeria Eail: davsdoe@ yaooco 2 Depare of Maeaics, Rivers Sae College of Educaio, Por Harcour, Rivers Sae, Nigeria Eail: preboj@yaooco Mobile Poe: +234-83-673-979 ABSRAC: is paper esablises sufficie codiios for e corollabiliy ad ull corollabiliy of liear syses e ai is o use e variaio of cosa forula o deduce our corollabiliy graia, by exploiig e properies of e graia ad e asypoic sabiliy of e free syse, we acieved our resuls @JASEM Differeial equaios, i geeral, are a ipora ool for aressig io sigle syse ad aalyzig e ier-relaiosip bewee differe copoes wic oerwise ay coiue o reai idepede o eac oer I is kow i Sebaky ad Bayoui 973) a, i e sudy of ecooics, biology ad pysiological syses as well as elecroageic syses coposed of suc subsyses iercoeced by ydraulic, ecaical ad various oer likages, oe ecouers peoea wic cao be readily odeled uless relaios ivolvig ie delays are adied Models for suc syses ca be corolled A delayed corol o suc syses will affec e x ) = p i= A ) x ) + i & ) i= B ) ) *Correspodig auor: Eail: davsdoe@ yaooco p i i evoluio of e syse i a idirec aer Arsei ad adore, 982), were e decisios i e corol fucio are sifed, wised or cobied before affecig e evoluio Models for syses wi delay i e corol occur i e sudy of gaspressures bipropella rocke syses, i populaio odels ad i soe coplex ecooic syses ec e corollabiliy of syses wi delays i e corol as bee sudied by several auors see Balacadra, 987; Balacadra ad Dauer, 996; Cukwu, 979; Owuau, 989) I paricular Maiius ad Olbro 972) sudied e syse ad gave sufficie codiios for e relaive corollabiliy of ) Our ieres, is o iegrae e cocep of ull corollabiliy io a geeralized syse wi delay i sae ad corol give by x& = L x + C u 2) ), ) ) ) We sall give sufficie codiios for e ull corollabiliy wi cosrai of 2) we relaive corollabiliy is assued Our resuls coplee ad exed kow resuls BASIC NOAIONS AND PRELIMINARIES Le ad be posiive iegers, E e real lie, ) We deoe by E e space of real - upples wi e Euclidea or deoed by If J is ay ierval of E e usual Lebesgue space of square iegrable equivale class of) fucios fro J o E will be deoed by L 2 J, E ) L, ], E ) deoes e space of iegrable fucios fro, ] o E Le > be give, for fucios x :, ] E,, ], we use x o deoe e fucios o,] defied by x = x + for s,] Cosider e syse x& ) = L, x ) + ) ) 3) were L, φ ) = dη, φ 4) saisfied alos everywere o, ] e iegral is i e Lebesgue- Sieljes sese wi respec o s L, φ ) is coiuous i, liear i φ η, is a arix fucio easurable i ad of
Corollabiliy ad Null Corollabiliy 32 bouded variaio i φ o,] for eac, ] C ) is a arix assued o be bouded ad easurable o, ] e corol fucio ) E, is assued o be easurable ad bouded o every fiie ierval rougou e sequel, e corols of ieres are, B = L2, ], E ), U L2, ], E ) a closed ad bouded subse of B wi zero i e ierior relaive o B If X ady are liear spaces is a appig, we sall use e sybols D ), R ) ad N ) o deoe e doai, rage ad ull spaces of respecively Defiiio - e coplee sae of syse 3) a ie is give by z ) = { x ), x, u } Defiiio 2 - Syse 3) is relaively corollable o, ], if for every z ) ad every vecor x E, ere exis a corol u B, suc a e correspodig rajecory of syse 3) saisfies x ) = x If syse 3) is relaively corollable o eac ierval, ], >, we say, syse 3) is relaively corollable Defiiio 3 - Syse 3) is said o be ull corollable a =, if for ay iiial sae { x,, x u } o, ], ere exiss a adissible corol ) B defied o, ] suc a e respose is broug o e origi of E a =, usig e corol effor ), o, ] u ) =, o, ] see Sebaky ad Bayoui 973) I is ull corollable wi cosrais a =, if for ay iiial sae { x, x, u } o, ], ere exiss a adissible corol ) U, defied o, ] suc a e respose x ) of syse 3) saisfies x ) =, usig e corol effor ) U, o, ] u ) =, o, ] Defiiio 4 - e doai D of ull corollabiliy of syse 3) is e se of all iiial pois wic e soluio ) x of syse 3) wi x ) = x saisfies x ) = E a soe usig x E for u U Defiiio 5 - A operaor : X Y, were X ady are liear spaces, is said o be closed if for ay sequece u D ) suc a u u ad u v, u belogs o D ) ad u = v e variaio of paraeer of syse 3) for > + iply e exisece of a uique absoluely coiuous soluio x ) of syse 3), wi iiial coplee sae z ) of e for x ) = X, ) x ) + X, 5) were X, saisfies e equaio X, = L, X, ), > s alos everywere
Corollabiliy ad Null Corollabiliy 33, s < < s X, = I, = s I Ideiy arix), x ) = L, x ) 6) X is called e fudaeal arix soluio of e syse We ow obai a ore coveie for of e soluio 3) by expressig 5) as x ) = X, x + X s + C s + ) ), ) ) u + X, e soluio x ) of syse 3) a = Klaka, 98) becoes x = + + + ) X, ) x ) X, u + X, We ow defie e corollabiliy arix of syse 3), give by W, ) = X, ] X, ] 9) Were deoes raspose 7) 8) Defiiio 6 - e reacable se of syse 3) a ie usig L 2 corols is e subse of { } X, u L2 { X, : u U} P, ) = : ad e cosrai reacable se of syse 3) is give by R, ) = e cosrai reacable se wi uspecified ed ie is give by R ) = U R, ) Defiiio 7 - Syse 3) is said o be proper i c X, ] alos everywere >, E, ], ] c E iplies δ, we say syse 3) is proper a ie we say e syse is proper i E if NULL CONROLLABILIY WIH CONSRAINED CONROLS Lea - e followig are equivale i) W, ) is o sigular for eac ii) Syse 3) is proper i iii) Syse 3) is relaively corollable o eac ierval, ] Proof -, ) c E give by c = If syse 3) is proper o, + ] δ, ], > > for eac If syse 3) is proper o eac ierval E for eac ierval, ] W is o sigular iplies, ) X, ] = alos everywere o, ] iplies = W is posiive defiie, a is c erefore, i) iplies ii),
Corollabiliy ad Null Corollabiliy 34 o prove e equivalece of ii) ad iii) le c X, ] alos everywere, ] c X c E, ad assue a, = c X for eac, = for u L2 I follows fro is a c is orogooal o e se P, ) We assue a syse 3) is relaively corollable, e P, ) = E, so a c =, eaig a iii) iplies ii) Coversely, assue a syse 3) is o corollable, so a P, ) E, for > e, ere exiss c, c E, suc a c P, ) = I follows a for all adissible corol u L2 = c Hece X, ] = c X, ] X, ] c, alos everywere, ], o sow a i) iplies ii) K We defie e operaor ) If we assue a, ) Bu is ol, if ad oly if c L2, ], E E by K u) = X, : W is sigular e e syeric operaor KK = W, ) is posiive defiie rak W, ) = eore - Syse 3) is proper o, ] Proof - If R, ) is a closed ad covex subse of if ad oly if i R, ) R, ) iplies a, ere is a suppor plae Π of, ) R, ) c y were c is a ouward oral o Π If X, e E Klaka, 976), e a poi y o e boudary of R roug, c X, y, a is c y y) for eac u is e corol correspodig o y, we ave for eac u U Sice U is a ui spere, is las iequaliy ol for eac u U, if ad oly if c X, c X, ad u ) sg c X, s + ) ) u c X, u = as y is o e boudary Sice we always ave R, ) If were o i e ierior of R, ) e is o e boudary, ece, fro e = + + foregoig, is iplies c X, so a X, ] = c alos everywere, ] is by our defiiio iplies a e syse is o proper sice c is coplees e proof eore 2 - Syse 3) is relaively corollable if ad oly if i R, ) for eac > Proof - By lea, syse 3) is relaively corollable o, ], > if ad oly if, i is proper o, ] erefore, by eore, syse 3) is relaively corollable if ad oly if i R, )
Corollabiliy ad Null Corollabiliy 35 eore 3 - If syse 3) is relaively corollable o, ] for eac, e e doai ull corollabiliy of syse 3) coais zero i is ierior Proof - Assue a syse 3) is relaively corollable o, ], > e by eore 2, i R, ), for eac > Sice x = is a soluio of syse 3) wi u =, we ave D Hece, If i D, e ere exiss a sequece x E suc a x as ad o x is i D, a is x Fro 5), we ave x ) = X, ) x ) + X, for ay > ad ay u U Hece, for u =, z = x ) = X, ) x ), is o i R, ) for ay > erefore e sequece z E is suc a z R, ), bu z as erefore, R, ) a coradicio Hece, i D def > z, eore 4 - Assue i) Syse 3) is relaively corollable o, ] for eac > ii) e zero soluio of syse 6) is uiforly asypoically sable, so a e soluio of 6) saisfies a ) x ) k x e, a >, k > are cosas e syse 3) is ull corollable wi cosrais Proof - By i) ad eore 3, e doai D of ull corollabiliy of syse 3) coais zero i is ierior erefore ere exiss a ball B suc a B D By ii), every soluio of syse 3) wi u = ) saisfies x ) as Hece a soe < wi u = ) x ) B D, for > erefore, usig ad x = x ) a u = as iiial daa ere exiss a u U ad soe 2 > suc a x 2 ) = us provig e eore e soluio ) x of syse 3) saisfies Lea 2 Syse 3) is relaively corollable o, ], ) e proof is quie sadard ad siple, for siilar proof see Klaka 976 ) if ad oly if rak W = eore 5 - Syse 3) is ull corollable wi cosrais if i) RakW, ) =, for > ii) e zero soluio of 6) is uiforly asypoically sable Proof - Iediaely fro eore 4 ad lea eore 6 - e syse 6) is ull corollable wi cosrais if i) RakW, ) =, for > ii) If all e caracerisics roos ave egaive real par Proof : Iediaely Coclusio: I is paper we ave developed ad proved sufficie codiios for e corollabiliy ad ull corollabiliy of lier syses wi delay i e sae ad corol a is, if e ucorolled syse is asypoically sable ad e corolled syse is relaively corollable e, e syse is ull corollable wi cosrais
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