On stability of first order linear impulsive differential equations

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1 Ieraioal Joural of aisics ad Applied Mahemaics 218; 3(3): IN: Mahs 218; 3(3): as & Mahs Received: Acceped: IM Esuabaa Deparme of Mahemaics, Uiversiy of Calabar, P.M.B 1115, Calabar, Cross River ae, Nigeria UA Abasiewere Deparme of Mahemaics ad aisics, Uiversiy of Uyo, P.M.B. 117, Uyo, Awa Ibom ae, Nigeria O sabiliy of firs order liear impulsive differeial equaios IM Esuabaa ad UA Abasiewere Absrac I his paper, we focus o he sabiliy problems of firs order liear impulsive differeial equaios. We cosruc a ordiary differeial equaio represeaio of he impulsive sysem such ha i is suiable for he qualiaive aalysis of he laer. This process is achieved by a rasformaio ha biecively maps he soluios of he iiial value problems for impulsive differeial equaios o he soluios of he iiial value problems for ordiary differeial equaios. A relaioship bewee sabiliy properies of impulsive differeial equaios ad he correspodig ordiary differeial equaios was esablished. Keywords: abiliy, Impulsive differeial equaios, asympoic 1. Iroducio The heory of impulsive sysem was developed o log ago as a idepede area of mahemaical aalysis. The developme came ou of curiosiy o build a mahemaical framewor ha ruly describes physical ad biological process as hey occur i aure. Prior o his oble developme, scieiss had ofe made a uderlyig assumpio ha he behaviour of physical ad biological sysems described by ordiary differeial equaios is coiuous ad iegrable i some sese. The sae of a sysem is suscepible o chage. I some processes, hese chages are ofe characerized by shor-ime perurbaios (impulses) whose duraios are egligible whe compared wih he oal duraio of heir eire ime of evoluio [1, 4, 5, 6, 7, 8, 9, 1, 11]. Impulsive differeial equaios are adequae mahemaical models for he descripio of evoluio processes characerized by he combiaio of coiuous ad ump chages of heir sae. For he coiuous chage of such processes, ordiary differeial equaios are used, while he momes ad he magiude of he ump are give by codiios [1, 2, 3]. Impulsive sysems are sysems whose saes are characerized by small perurbaios (impulses) i he form of umps [12, 13, 14]. I his wor, we focus o he sabiliy problems of impulsive differeial equaios which is a emergig area of research presely a is ifacy compared o he sabiliy aalysis of ordiary differeial equaios. This is probably due o he aure of he impulsive processes which are momearily eposed o harsh impacs. The aim of his sudy is o esablish sufficie codiios for sabiliy of impulsive differeial equaios from he sabiliy codiios for he associaed ordiary differeial equaios. To help i our ivesigaio, we will defie some impora coceps ad esablish some facs. 1.1 Impulsive differeial ysem: Impulsive differeial equaios wih fied momes of impulsive effec are of he form Correspodece UA Abasiewere Deparme of Mahemaics ad aisics, Uiversiy of Uyo, P.M.B. 117, Uyo, Awa Ibom ae, Nigeria '( ) f (, ( )), T \ ( ) f (, ( )),, Where ad he real umerical sequece (, ) R R ~231~ 1 (1.1) is icreasig ad has o fiie accumulaio poi. I he case of ufied momes of impulsive effecs he impulsive pois may be ime ad sae depede; ha is, : (, ( )).

2 Ieraioal Joural of aisics ad Applied Mahemaics Whe he fucio depeds o he sae of he sysem (1.1) i is said o have impulses a variable imes. This is refleced i he fac ha differe soluios will ed o udergo impulses a differe imes. However, if he fucios are all cosas he sysems is said o have impulses a fied imes i which case all soluios udergo impulses acios a he same ime, he quesio of eisece of soluio of he sysem (1.1) is o-rivial whe impulses occur a variable imes. Eve he precise oio of wha a soluio is mus be carefully saed. I is fairly clear ha soluios should be piecewise coiuous ad i fac piecewise coiuously differeiable (or piecewise absoluely differeiable whe cosiderig geeralized ypes of soluios). A soluio will udergo simple ump discoiuiy whe i iersecs impulse hyper-surfaces. Eve afer focusig o a paricular class of relaios ( s, ( s )) give by impulse hyper-surfaces, impulsive differeial equaios sill ehibi some uusual behaviour (Balliger, 1999). We shall cocerae oly o hose wih fied momes. Be ha as i may, o obai or discuss he soluio of a impulsive differeial equaio, we mus ae io cogizace cerai peculiariies of he model. We assume ha for, he soluio of equaio (1.1) is deermied by he ordiary differeial equaio ' f,. For, a chage by ump of he soluio occurs so ha ad f,. Afer he ump a he mome he sysem (2.1) coicides wih he soluio y of he iiial value problem (Baiov ad imeoov, 1995): T \, he soluio of y ' f,, y,. 1 This simply meas ha, afer he ump a differeial equaio is give by (1.2), a ew fucio y() aes over corol from (). The corollig impulsive '( ) f (, ( )), T \ ( ) g (, ( )),, ( ), T \, (, ) (1.3) 1.2 Cosrucio of a associaed ordiary differeial equaio from a impulsive differeial equaio Le he impulsive differeial equaio be defied by equaio (1.3). We will follow he cosrucio seps of he cosrucio of he absolue coiuous raecory of he Caraheodory ype absolue coiuous equaio wih a mappig coecig he wo raecories. As a preparaory sep, we will esablish a relaioship bewee a se : R,, 1 Of ime poi (impulse pois) ad sequeces of iervals c : ˆ, ˆ ˆ ˆ, R, 1, ; i :, ˆ 1 ˆ ˆ ˆ, 1 R, 1,. Noaio 1.1: Le us deoe by ˆ ˆ :, : ˆ, U ad by : ˆ, O The se of images of impulse pois ad uios of he iervals i Defiiio 1.1: Le ˆ : Ad :, U 1 be defied as follows: : ad is ierior respecively. c ~232~

3 Ieraioal Joural of aisics ad Applied Mahemaics ˆ 1 :, ˆ, ˆ. Moreover le _ : ( ). 1 Defiiio 1.2: We defie a ODE wih righ side measurable i, coiuous or Lipschizia i (,y) for each fied, o he impulsive differeial equaio (1.3) as follows: f,,,,,,, y :,, y,, y,, y,ˆ R R,,g, ˆ, 1 1 1, :. Lemma 1.1. The ordiary differeial equaio ˆ 1, ', y ',, y,,g,, 1, y, y,,, y, ˆ R R, (1.4) 1 Has a uique soluio. Theorem 1.1. The soluio of he iiial value problem of he differeial equaio wih righ side,,, y,,, y, R R ', y ',, y, s, y s, y, s,, y R (1.5) Eiss ad is uique provided ha he soluio of he iiial value problem s of he impulsive differeiable equaio (1.3) eiss ad is uique. Moreover, he raecory of soluio (, y ) of iiial value problems s, s,, saisfies 1 he codiio ha, s, is a soluio of he impulsive differeial equaio (1.3). Proof: The saeme of he heorem follows from he cosrucio, relaio (1.5) ad Defiiio 1.2. Now, le us cosider he composie sysem discussed i heorem 4.1 usig he dyamical sysems ad he associaed ordiary differeial equaio. Codiio 1.1 C.1 1 1, f,,,, R,, ; C.2, g,,, R ; C.3 ( f (, ), ) ( f (, ), ), f, a d f 1 1 f (, ),, (, ), R,, ; ~233~

4 Ieraioal Joural of aisics ad Applied Mahemaics C.4 (g (, ), ) (g (, ), ) g,, a d g g (, ),, (, ) R ; We will ow prove some basic iequaliies useful i our sabiliy aalysis. Lemma 1.2 Le he righ side of he oe dimesioal ordiary differeial equaio defied i (1.5) fulfil codiio 1.1, C.1 he he soluio of he iiial value problem i (1.5) saisfies he iequaliy: e,,, 1 R,, 1. If i fulfils Codiio 1.1, C.3 he (1.6) e,,, 1 R,, 1. (1.7) Proof: Iegraig boh sides of he codiio C.1, wih : : we ge 1 : f s, s d s s d s 1. 1 By iducio, we assume ha wih q p q Holds. p!, 1 q p : f s, s d s ad he iequaliy The usig he codiio C.1 we ge 1 : f s, s d s s d s p 1 p ( ) ( ) ds p! p! p p. ( ) By he successive approimaio for he soluio,, he upper esimae eds o e, ad hece proves iequaliy (1.6). The proof of iequaliy (1.7) is esseially he same, ecep ha he basic esimae is as saed i codiio C.3: f, holds i. The same seps are repeaed as i he firs case ad we ge he saeme of he lemma. 2. aeme of he Problem This wor is aimed a ivesigaig he codiios uder which a soluio will remai close o aoher soluio whe a he iiial ime poi hey are sufficiely close o each oher. This issue is eve more emphasized for impulsive differeial equaios where he sysem is eposed o sigificaly harsh impacs a he impulse pois. Oe was o ow which sysem ca remai close o he behaviour of he origial dyamical sysem. As earlier meioed, hese aspecs of he ivesigaios are a heir ifacy whereas heir pracical imporace are highly raed. Here, we aalyze sabiliy issues of liear firs order impulsive ordiary differeial equaios wih iiial codiios. 3. Mai Resuls We are ow sufficiely equipped o formulae heorems abou he composie sysem. Theorem 4.3: Assume ha he compoes of he impulsive differeial equaio (1.3) or equivalely of he associaed ordiary differeial equaio (1.5) fulfil Codiio 1.1, C.1 ad C.4 or C.3 ad C.2 or C.4 i = 1 dimesio ad i addiio ~234~

5 Ieraioal Joural of aisics ad Applied Mahemaics i) ˆ ˆ 1 ˆ ˆ 1 m i e, e 1, ; he he sequece ˆ is sable. ii) ˆ ˆ 1 ˆ ˆ 1 m i e, e 1, ; he he sequece ˆ is asympoically sable. iii) If Codiio 1.1, C.1 holds ad C.3 does o hold he M ˆ such ha e M,. If codiios (i) ad (ii) are rue wih C.3, or codiio (iii) is rue he he soluio he sabiliy of he sequece ˆ respecively. : ˆ, is sable or asympoically sable by Proof: i) By Lemma 1.2, usig Codiio 1.1, C.1 ad codiio (iii) of he heorem, ˆ ˆ ˆ ˆ e M,, Ad by codiio C.2 ˆ ˆ e,, 1. ii) By he secod saeme of he same lemma, usig Codiio 1.1, C.3, C.4, ˆ ˆ ˆ e,, ad e, ˆ, 1 iii) From pois (i) ad (ii) we have ha ˆ ˆ 1 ˆ ˆ m a, M,,,,. iv) By codiio (i) of his heorem, he se ˆ N is bouded hece he same holds for he raecory. v) By codiio (ii) of his heorem, ˆ sable. if Theorem 4.4: Le equaio (1.3) fulfil he followig codiios ad le = 1: hece by codiio (iii) of his heorem, he sysem is asympoically i) f,,, Ad g,,. hece or oherwise, ( ) is a soluio of he iiial value problem i [, ). ii) Le f,,,,. iii) Le g,, 1,, R ; If 1, ad ~235~

6 Ieraioal Joural of aisics ad Applied Mahemaics 1 1, 1 e, (3.1) The he ideically zero soluio is asympoically sable. Proof: Codiios 2. Ad 3. Ad iequaliy (3.1) gra he codiios of heorem 4.3 case C.3 codiio 2. hece he sysem is sable or asympoically sable subec o he way of fulfilme of iequaliy (3.1). 4. Coclusio I his case he sabiliy of he combied sysem comes from he sabiliy of he ordiary differeial equaio compoe of he impulsive differeial equaio. The sabiliy properies of impulsive sysems are iheried by he associaed ordiary differeial equaios ad vice versa. 5. Refereces 1. Baiov D, imeoov P. Impulsive differeial equaios: Asympoic properies of he soluios, World scieific pub. Coy. Pe. Ld, Baiov DD, imeoov P. ysem wih impulse effec: sabiliy heory ad applicaios. Ellis Horwood, Chicheser, Baiov DD, imeoov P. Impulsive Differeial equaios: periodic soluios ad applicaios. Logma scieific ad Techical, Harlow, Balliger G, Liu X. Eisece, uiqueess resuls for impulsive delay differeial equaios. DCDI. 1999; 5: Lashmiaham V, Baiov DD, imeoov P. Theory of impulsive differeial equaios. Word cieific, igapore, Oyelami BO. O miliary model for impulsive reiforceme fucios usig eclusio ad margializaio echiques. Noliear Aalysis. 1999; 35: amaoileo AM, Peresyu K. Impulsive differeial equaios. Word cieific, New Yor, Abasiewere UA. Oscillaory Properies ad Asympoic Behaviour of he oluios of Neural Delay Impulsive Differeial Equaios, Uiversiy of Calabar, Cross River ae, Nigeria, Upublished Ph.D. hesis, Esuabaa IM. abiliy Aalysis of Impulsive Differeial Equaios From Ordiary Differeial Equaios Usig Dyamical ysem Techiques, Uiversiy of Calabar, Cross River ae, Nigeria, Upublished Ph.D. hesis, Abasiewere UA, Esuabaa IM. Oscillaio Theorem for ecod Order Neural Delay Differeial Equaios wih Impulses, Ieraioal Joural of Mahemaics Treds ad Techology (IJMTT). 217; 52(5): Abasiewere UA, Esuabaa IM, Isaac IO, Lipscey Z. Eisece Theorem For Liear Neural Impulsive Differeial Equaios Of The ecod Order, Commuicaios i Applied Aalysis (UA). 218; 22(2): Ario O, Piu M. More o liear differeial sysems wih small delays. Joural of differeial equaios. 21; 17: Basiec J, Dibli J, marda Z. Covergece es for oe scalar differeial equaio wih vaishig delay. Arch. Mah. (Bro) CDDE. 2; 36: Oyelami BO, Ale O. B-rasform ad is applicaio o fish -hyacih model. Ieraioal Joural of Mahemaics, Educaio, ciece ad Tecology. 22; 33(4): ~236~