Integral Φ0-Stability of Impulsive Differential Equations

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1 Ope Joural of Appled Sceces, 5, 5, Publsed Ole Ocober 5 ScRes p://wwwscrporg/joural/ojapps p://ddoorg/46/ojapps5564 Iegral Φ-Sably of Impulsve Dffereal Equaos Aju Sood, Sajay K Srvasava Appled Sceces Deparme (Researc Scolar-, Pujab Teccal Uversy, Kapurala, Ida Appled Sceces Deparme (Maemacs, Bea ollege of Egeerg ad Tecology, Gurdaspur, Ida Emal: ajusood6@yaoocom Receved 4 Sepember 5; acceped 7 Ocober 5; publsed Ocober 5 opyrg 5 by auors ad Scefc Researc Publsg Ic Ts wor s lcesed uder e reave ommos Arbuo Ieraoal Lcese ( BY p://creavecommosorg/lceses/by/4/ Absrac I s paper, e oos of egral φ -sably of ordary mpulsve dffereal equaos are roduced Te defo of egral φ -sably depeds sgfcaly o e fed me mpulses Suffce codos for egral φ -sably are obaed by usg comparso prcple ad pecewse couous coe valued Lyapuov fucos A ew comparso lemma, coecg e soluos of gve mpulsve dffereal sysem o e soluo of a vecor valued mpulsve dffereal sysem s also esablsed Keywords Iegral φ -Sably, oe Valued Lyapuov Fucos, Impulsve Dffereal Equaos, Fed Tme Impulses Iroduco Impulsve dffereal equaos ave bee developed modelg mpulsve problems pyscs, populao dyamcs, ecology, bologcal sysems, dusral robocs, opmal corol, bo-ecology ad so for I vew of e vas applcaos, e fudameal ad qualave properes e sably, boudedess ec of suc equaos are suded eesvely pas decades Several ypes of sably ave bee defed ad esablsed leraure by academcas for mpulsve ordary dffereal equaos Varous ecques suc as scalar valued pecewse couous Lyapuov fucos, vecor valued pecewse couous Lyapuov fucos, Rajum meod, comparso prcple ec ave bee employed o esabls sably resuls To e bes of our owledge, e cocep of egral sably ad φ -sably were roduced for ordary dffereal equaos by Lasmaam 969 [] ad by Apa 99 [] respecvely Laer, ese sa- How o ce s paper: Sood, A ad Srvasava, SK (5 Iegral Φ -Sably of Impulsve Dffereal Equaos Ope Joural of Appled Sceces, 5, p://ddoorg/46/ojapps5564

2 A Sood, S K Srvasava bles were developed [] ad [4] by Apa, Solma ad Abdalla bu for ordary dffereal equaos I, Iegral sably was esablsed for mpulsve fucoal dffereal equaos by Hrsova Movaed by ese wors, s paper, we roduce ad esabls egral φ -sably for mpulsve ordary dffereal equaos: (,,, f I ( were, N, R, I( ( ( R, f : R R R, < < < < ad I : R R are a sequece of saaeous mpulse operaors ad ave bee used o depc abrup cages suc as socs, arvesg, aural dsasers ec ad K s a coe defed Seco Te paper s orgazed as follows: I Seco, some prelmares oes ad defos are gve I Seco, a ew comparso lemma, coecg e soluos of gve mpulsve ordary dffereal sysem o e soluo of a vecor valued mpulsve dffereal sysem s wored ou Ts lemma plays a mpora role esablsg e ma resuls of e paper Suffce codos for egral φ -sably are obaed by employg comparso prcple ad pecewse couous coe valued Lyapuov fucos Prelmares Le R deoe e -dmesoal Eucldea space w ay covee orm ad e scalar produc ( y, y, R [,, J [,, (, For ay (,,,, y ( y, y,, y R, Le ( ;, R we wll wre y ff y for all,,, be e soluo of sysem (, avg dscoues of e frs ype (lef couous a e momes we ey mee e yper plaes Togeer w sysem (, le us cosder, s perurbed IDS: (, (,,, f f I I were, f (, : R R R, I ( : R R Le f (, f (,, I I ( N so a e rval soluo of ( ad ( ess Le us defe e followg: Defo A proper subse K of R s called a coe f ( λκ Κ, λ ( ΚΚ Κ ( ΚΚ (v Κ (v Κ { Κ } { }, were Κ ad Κ are eror ad closure of Κ respecvely Κ deoes e boudary of Κ Defo Te se Κ { φ R :( φ, Κ } s called e adjo coe f sasfes e properes (-(v of defo, φ Κ, Κ Κ Te se Κ ff ( φ for some { } Defo A fuco g: D R, D R R s sad o be quas moooe relave o e coe Κ f for eac R, uv, D ad v u Κ mply a ere ess ( g ( v g ( u φ,,, osder e followg ses: φ Κ suc a ( v u {, a R R : a(, a( r s s rcly creasg r} K ( φ, ad 65

3 {, b R R R : b (, ad for ay f ed [, }, { : : (,,,,,, } K K P R R R f R R R f R R { Κ < > } { R Κ > } { ρφ } ( R ( S ρ, φ, : φ, ρ, ρ S ρ, φ, : φ, ρ, ρ ( T,, ρ R :(, S(,, s [, T] Ω Defo 4 A fuco V : R R Κ s sad o belog o class L f:, G,, u S ρ, φ ; V( s a couous fuco ( V(, For eac N, lm V (, V (, V (, ad lm V(, s Lpscz couous relave o coe K, s secod argume; es Ad for :,,, we defe dervave of e fuco (, ( by DV( (, lmsup V(, f(, V(, A Sood, S K Srvasava V alog e rajecory of e sysem Now referrg [5], le us defe e followg: Defo 5 Le φ Κ Te fuco V(, L s sad o be φ -wealy decresce, f ere ess a δ > ad a fuco a K suc a e equaly ( φ, < δ mples a ( φ (, V, < a, φ, Defo 6 Le φ Κ Te fuco V(, L s sad o be φ -srogly decresce, f ere ess a δ > ad a fuco a K suc a e equaly ( φ, < δ mples a ( φ (, V, < a φ, Trougou e paper was assumed a φ Le us cosder e followg comparso mpulsve dffereal sysems (referrg [] for Ordary dffereal sysems ad alog w s perurbed sysem ( u g u, ξ u u u( u ( w g w, η w w w w ( γ w g w p, η w w (5 w w were g P R K R s quas moooe o decreasg s secod argume ad ξ :K K s quas moooe o decreasg sasfyg g (,, ξ g P R K R, η : K K, p: R R, γ : R K, g (, η, p( ad γ are o be cose laer suc a q γ Defo 7 Te zero soluo of ( s sad o be φ -sable, f for every α > ad for ay J ere ess a posve fuco β β (, α K, wc s couous for eac α suc a e equaly ( φ, < β mples a ( φ, r ( r s e mamal soluo of ( relave o e coe K < α, were φ Κ ad ( (4 65

4 A Sood, S K Srvasava Defo 8 Te zero soluo of ( s sad o be egrally sable, f for every α ad for ay J ere ess a posve fuco β β (, α K, wc s couous for eac α suc a for ay soluo ( ;, of perurbed sysem (, e equaly < β olds provded a α ad for every, I,,,, of RHS of ( sasfy T >, e perurbaos f ( ad T sup f ( s, ds sup I : : < β < T : ( < β α Defo 9 Te rval soluo of ( s sad o be egrally φ -sable, f for every α ad for ay J ere ess a posve fuco β β (, α K, wc s couous for eac α suc a for ay soluo ( ;, of perurbed sysem ( ad for φ Κ, e equaly ( φ, < β olds provded a ad, for every Ma Resuls ( φ, α T >, e perurbaos f (, ad (6 I,,,, of RHS of ( sasfy T sup f ( s, ds sup I Ω ( T,, β : < T : φ, ( < β α (7 ( Lemma : osder e comparso sysem ( ad assume a ( g P R K R were g s quas moooe o decreasg s secod argume; ( V L suc a ( ρ, φ ( V P S K ad sasfes ( φ φ (, D V,, g, V, : :,, ξ K suc a ( φ, V (, I ( ξ (( φ, V (, for :,,, r :, u be e mamal soluo of ( esg o J Te for ay soluo ( :, ( φ, V(, ( φ, r ( :, u provded a ( φ, V (, ( φ, u :, be e soluo of ( esg for suc a ( φ, V (, ( φ, u m ( φ, V (, for suc a m ( ( φ, V(, u Le of ( esg o J, we ave Proof: Le Defe Te for small >, we ave m ( m φ, V, φ, V, φ, V, V, ( ( ( ( ( ( ( φ ( ( (, V, V (, f, V (, f, V (, ( φ, V (, ( V (, f (, ( φ, V (, f (, V (, V(, f(, V(, ( φ V (, ( V (, f (, φ, (, (, (, V f V Mφ ( f(, φ, were M s e Lpscz cosa ( ],, 654

5 A Sood, S K Srvasava Terefore we ave (,, (, ( f (, ( m m V f V φ M φ, (, (, (, m ( m ( V f V φ M f(, φ, D m( ( φ, D V (, ( ( φ, g(, V (, Also m ( ( φ, u ad ( ( φ,, ( φ, (, ( φ, ξ (, ( m V V I V Te by eorem (4 [6], we observe e desred equaly ( φ, V(, ( φ, r ( :, u for all Teorem : Le us assume e followg: Le f P R R R ad I R R :,,, Tere es V(, L, V(, suc a ( V s φ -wealy decresce ( For :,,, e equaly ( φ, D( V (, φ, g (, V (, olds for all S ( ρ φ were g moooe o decreasg s secod argume,,, ( ( φ, V(, I ( ( φ, ξ ( V(, for all (, ( ρ, φ, :,, s moooe o decreasg, sasfyg ξ ( ( µ For ay umber µ > ere ess V (, L, V(, suc a ( (v b(,, V µ φ φ (, a( φ, for (, S( ρφ, S ( µφ, were ab K, (v For :,,, e equaly olds for ay were g P R K R (v ( ( ( φ, D V, D V µ (, φ, g (, V (, V µ (, ( ( (, S( ρ, φ S ( µ, φ S were ξ s moooe o decreasg s secod argume ( ( µ ( µ ( φ, V (, (,, (, (, I V I φ η V V for S( ρ φ S ( µ φ,,,,,,, were : K η 4 Te sysem ( ad (4 ave soluos, for ay al po 5 For ay al po (, R R, e sysem ( as soluo Le e zero soluo of ( be φ -sable, ad scalar IDE (4 s egrally φ -sable, e e sysem ( wll be egrally φ -sable Proof: Sce V (, L s φ -wealy decresce, erefore ere ess a ρ > ( ρ < ρ ad a fuco φ, < ρ mples a were η, ψ K suc a e equaly ( ( φ, V(, ψ, ( φ, φ Κ < (8 655

6 A Sood, S K Srvasava Le be a fed me oose a umber α > suc a α < ρ re- As V(,, V( µ (, specvely Le L, ere es Lpscz cosas M ad M M φ α α M of V, ad V ( µ (, As e zero soluo of ( s φ -sable, erefore for every α > ad for ay J ere ess a pos- δ δ, α for eac α suc a e equaly ( φ,u < δ mples a α ( φ, r( :, u <, (9 ve fuco were r( :, u s e mamal soluo of ( As ψ K, ere ess δ δ( δ > ad ece δ δ(, α ( φ, u δ ψ, ( φ, u δ suc a < < ( Aga vew of e fac a e perurbaos (5, deped oly o ad sysem (4 s φ -egrally sable, β β, α K, couous for eac α (ae parcular α b( α ere ess a fuco suc a for every soluo w ( ;, w of perurbed sysem (5, e equaly ( φ, w ( ;, w β olds provded a ( φ, w α ad for every T >, e perurbao erms Sce b K, lm b( s s < ( p ad γ sasfy T p( s ds γ ( α ( : < T le us coose β β( β ψ K s a fuco sasfyg ψ( α < ρ Selec δ δ ( α β, α δ m { δ, ρ }, Le ( ;, > suc a b( β β ad β ψ ( α < < suc a e equales a α δ < ad > were ψ δ < β old ( be e soluo of ( Now we wll prove a f e equales (6 ad (7 are sasfed e ( φ, < β, (4 If possble le s be false Terefore ere ess a po ase : Le for ay,,, φ, β ( I s case frs we oe a ( φ, ( For f ( φ (, δ (5 > suc a ( φ ( (, β ad φ, β,, < (5 Te e soluo ( ;, > δ, e by e coce of s couous a δ we ge (, ( Terefore ψ φ < β wc s a coradco o Now le us cosder e erval (, Subcase : Le ere ess (,, :,,, suc a δ ( ( φ, (, ( S( β, φ S ρ, φ :, If r( :, u s e mamal soluo of ( w u ( ( V, ad ( of eorem, usg lemma, we oba ( ( ad, e vew of e assumpos ( φ, V, :, φ, r :, u :, (6 656

7 A Sood, S K Srvasava were ( :, s a soluo of (, sarg a As δ ( ( φ, s cose erefore we ave ( φ (, δ < δ ad ( φ ( (, u φ, V, by usg (8 ad e (, we ge (, u (, V ( ( (,,, ( φ, V(, :, ( φ, r( :, u :, α ( ( ( Now φ φ ψ φ < δ by vrue of (9 gves: φ, V, :, φ, r :, u < for, (7 Now from equaly ( ad codo (v of eorem, we ge Le us defe e fuco V : R R Now, for codo o V ad, ad ( α (, (,, V α, we ave ( α φ V a φ < a δ < (8 Κ, V L by ( α V, V, V,, (, S( β, φ S ( α, φ, vew of (v of eorem ad lpscz ( α ( φ (, D V, φ, D V, D ( V (, φ,lmsup { V(, f(, f (, V(, } ( α ( α { V ( f( f ( V ( } lmsup,,,, { ( ( φ,lmsup V, f, f, V, f, ( α (, (, (, } φ,lmsup V (, f(, f (, ( α ( α ( α ( (,,,, (, },lmsup { V(, f(, f (, V(, f(, } V f V V f V f V φ ( α ( α,lmsup { V (, f(, f (, V (, f(, } φ ( { ( { } { ( } ( α ( α α φ,lmsup V, f, V, φ,lmsup V, f, V, M sup f, M sup f,, D V,, D V, φ φ φ φ Ω(, T, β Ω(, T, β ( α ( ( φ, g V,, V (, φ ( M M sup f (, Ω(, T, β ( φ, g( V, (, φ ( M M sup f (,, were T Ω(, T, β ( ( Aga for (, suc a (, ( β, φ ( α, φ Lpscz codos o V ad V α, we ge (9 S S, by usg codo (v of eorem ad 657

8 A Sood, S K Srvasava ( φ, V (, I I φ ( (, V, I { (, (, V I I V I } ( φ, V (, I (, (, (, φ V I I V I ( φ, V (, I ( α ( α φ, V (, I (, (, (, I V I I V I V I ( φ, V (, I, ( φ V(, I (, I V I ( α ( α ( φ, V (, I (, I V I ( φ, η ( V (, ( M φ I ( M φ I ( ( φ, η ( V (, ( ( M M φ I ( φ, η ( V (, ( M M φ sup I ( : < T : ( φ, < β For e mpulsve dffereal sysem (5 wc s e perurbed sysem of (4, se e perurbaos o RHS of (5 as ad ( φ γ ( φ Ω(, T, β p M M sup f, ad M M sup I Terefore (9 ad ( ca be wre as : ( φ, < β ( φ, (, φ, (, (, D V g V p ( (, (,, (, φ V I I φ η V γ If we cosder e comparso sysem (5 w mamal soluo r ( :, w, roug e po (, w were w ( ( V,, usg (9, ( ad lemma, we ge ( ( φ, V, ;, φ, r ;, w, H, were H s e erval of esece of mamal soluo ( ;, ( r w ( Now by usg e equaly (7 for T e erval, ad from e coce of α, p s ds : < ( sup (, d φ ( sup (,, : : (, Ω T β < φ ( < β γ φ M M f s s M M I φ ( M M sup f ( s, ds sup I ( < αφ ( M M < α ( T,, : : (, Ω β < φ < β Le us coose a po T > suc a p( s ds ( T p( < α ( 658

9 A Sood, S K Srvasava Now le us defe a couous fuco :, ad e sequece of umbers p R gve by p :, p ( ( p T :, T T : T ( :, γ γ : > We see a f (7 olds e from (, for every T > T p ( s ds γ < α ( : < T le r ( :, w be e mamal soluo of (5, roug e po (, defed by p ( s ad γ Noe a ere we ave r ( : (, w r :, w ;, From equales (7 ad (8 we see, ( ( ( α, V,, V, V (, ( ad ece from (, we ge w were e perurbaos erms are ( φ φ < α e ( φ, w ( φ, r ( :, w α (4 < β for (5 Now from e coce of β, equales (, (5 ad codo (v of saeme of eorem, we ge wc yelds b( β b( β ( β β > ( φ, ( :, ( φ, ( :, ( φ, (, ( ;, ( α ( α ( φ, V(, ( ;, w V (, ( ;, w ( φ, V (, ( ;, w b( ( φ, ( ;, w b( β b r w r w V w >, a coradco ad erefore e equaly (4 s vald for for some,,, δ φ ad Subcase : Le ere es a po (, (, ( S( β, φ S ρ, φ :, oose δ sasfyg δ < δ < β w δ ( ( ( φ, ;, :, suc a, Now f we ae δ place of δ ad repea e proof of subcase we arrve a coradco a assures e valdy of (4 ase : If for some N e from (5, Le us selec β β( β (, φ ( β ad φ (, < β, [, ( φ ( ( (, ;, φ, I I ( > suc a b( β sup φ, η ( ( ;, r w Now by adopg e procedure as case, we ge e equales ( ad (5 Te by usg ese equales alog w e codos (v ad (v of e saeme of eorem, we ave 659

10 A Sood, S K Srvasava sup (, ( ;, (, ( ( ;, (, ( (, ( ( φ, V (, ( ( φ, V α (, ( b( φ, ( b( β b β φ η r w > φ η r w φ η V ad a aga s a coradco Terefore equaly (4 s vald Tus all e cases, valdy of (4 proves a sysem ( s egrally φ -sable 4 ocluso Resuls [] [4] [7] ave bee eploed ad eeded o esabls e ew ype of sably e egral φ -sably for e mpulsve dffereal sysems Suffce codos are obaed by employg comparso prcple ad pecewse couous coe valued Lyapuov fucos Refereces [] Lasmaam, V ad Leela, S (969 Dffereal ad Iegral Iequales Teory ad Applcaos Academc Press, New Yor, -9 [] Apa, EP ad Ayele, O (99 O e φ -Sably of Nolear Sysems of omparso Dffereal Sysems Joural of Maemacal Aalyss ad Applcaos, 64, 7-4 p://ddoorg/6/-47x(996-u [] Apa, EP (99 O e φ -Sably of Perurbed Nolear Dffereal Sysems Ieraoal ere for Teorecal Pyscs, - [4] Solma, AA ad Abdalla, MH ( Iegral Sably rera of Nolear Dffereal Sysems Maemacal ad ompuer Modellg, 48, p://ddoorg/6/jmcm7 [5] Hrsova, SG ad Russov, I (9 Sably Terms of Two Measures for Ial Tme Dffereces for Dffereal Equaos by Perurbg Lyapuov Fucos Ieraoal Joural of Pure ad Appled Maemacs, 5, 9- [6] Lasmaam, V, Baov, D ad Smeov, PS (989 Teory of Impulsve Dffereal Equaos World Scefc Publsg o Pv Ld, Sgapore, USA, Eglad [7] Hrsova, SG ( Iegral Sably Terms of Two Measures for Impulsve Dffereal Equaos Maemacal ad ompuer Modellg, 5, -8 p://ddoorg/6/jmcm99 66