Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations

Transcription

1 Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie ISSN: & Prit ISSN: Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios By Dilbaj Sigh & Sajay Kumar Srivastava Lovely Professioal Uiversity, Pujab, Idia Abstract - Strict stability is the id of stability that ca give us some iformatio about the rate of decay of the solutio There are some results about strict stability of fuctioal differetial equatios O other had, i the study of stability, a iterestig set of problems deal with brigig sets close to a certai state, rather tha the equilibrium state The desired state of a system may be mathematically ustable ad yet the system may oscillate sufficietly ear this state that its performace is acceptable May problems fall ito this category Such cosideratios led to the otio of practical stability which is either weaer or stroger tha stability I this paper, strict practical stability of Impulsive fuctioal differetial equatios i which the state variables o the impulses are related to time delay is cosidered By usig Lyapuov fuctios ad Razumihi techique, some criteria for strict practical stability for fuctioal differetial equatios, i which the state variables o the impulses are related to the time delay, are provided Keywords : impulsive fuctioal differetial equatio, strict stability, lyapuov fuctio, razumihi techique, time delay GJSFR-F Classificatio : MSC 00: H0, 34A09 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios Strictly as per the compliace ad regulatios of : 03 Dilbaj Sigh & Sajay Kumar Srivastava This is a research/review paper, distributed uder the terms of the Creative Commos Attributio-Nocommercial 30 Uported Licese permittig all o commercial use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited

2 Ref 6 She J H, "Razumihi techiques i impulsive fuctioal differetial equatios", Noliear Aalysis, vol 36, pp 9-30, 999 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios Dilbaj Sigh α & Sajay Kumar Srivastava σ Abstract - Strict stability is the id of stability that ca give us some iformatio about the rate of decay of the solutio There are some results about strict stability of fuctioal differetial equatios O other had, i the study of stability, a iterestig set of problems deal with brigig sets close to a certai state, rather tha the equilibrium state The desired state of a system may be mathematically ustable ad yet the system may oscillate sufficietly ear this state that its performace is acceptable May problems fall ito this category Such cosideratios led to the otio of practical stability which is either weaer or stroger tha stability I this paper, strict practical stability of Impulsive fuctioal differetial equatios i which the state variables o the impulses are related to time delay is cosidered By usig Lyapuov fuctios ad Razumihi techique, some criteria for strict practical stability for fuctioal differetial equatios, i which the state variables o the impulses are related to the time delay, are provided Keywords : impulsive fuctioal differetial equatio, strict stability, lyapuov fuctio, razumihi techique, time delay I Itroductio The impulsive differetial equatios represet a more atural framewor for mathematical modelig of may real world pheomea tha ordiary differetial equatios I recet years, sigificat progress has bee made i the theory of impulsive differetial Impulses ca mae ustable systems stable, so it has bee widely used i may fields such as physics, chemistry, biology, populatio dyamics, idustrial robotics ad so o The impulsive differetial equatios represet a more atural framewor for mathematical modelig of may real world pheomea tha ordiary differetial equatios I recet years, sigificat progress has bee made i the theory of impulsive differetial equatios [3-4] I additio to that, fuctioal differetial equatios have a wide applicatio i our society So it is importat to study them There are some results o impulsive fuctioal differetial equatios [5,6,7,8,,3,4] We ca easily see that i the previous wors about impulsive fuctioal differetial equatios the authors always suppose that the state variables o the impulses are oly related to the preset state But i most cases, it is more applicable that the state variables o the impulses are also related to the former state But there are rare results about impulsive fuctioal differetial equatios i which state variable o the impulses are related to the time delay I additio to that strict stability is aalogous to 03 Joural of Sciece Frotier Research Volume XIII Issue I V ersio I Global Year F Author α : Departmet of Mathematics, Lovely Professioal Uiversity, Pujab, Idia dilbajsigh@lpucoi Author σ : Beat College of Egieerig ad Techology, Gurdaspur, Pujab, Idia ss64_bect@yahoocom 03 Global Jourals Ic (US

3 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios 03 Global Joural of Sciece Frotier Research F Volume XIII Issue I V ersio I Year Lyapuov s uiform asymptotic stability It gives us some iformatio about the rate of decay of the solutios I[], the authors have explored further the defiitios of strict stability of differetial equatios ad have gotte some results I [3] authors have gotte some results about the strict stability of impulsive fuctioal differetial equatios i which the state variables o the impulses are ot related to the time delay Moreover, i the study of Lyapuov stability, a iterestig set of problems deal with brigig sets close to a certai state, rather tha the equilibrium state The desired state of a system may be mathematically ustable ad yet the system may oscillate sufficietly ear this state that its performace is acceptable May problems fall ito this category icludig the travel of a space vehicle betwee two poits, a aircraft or a missile which may oscillate aroud a mathematically ustable course yet its performace may be acceptable, the problem i a chemical process of eepig the temperature withi certai bouds, etc Such cosideratios led to the otio of practical stability which is either weaer or stroger tha Lyapuov stability I [] authors have gotte some results about the practical stability of impulsive fuctioal differetial equatios i Equatios i terms of two measures I [] authors have gotte some results about strict practical stability of delay differetial equatios I this paper, strict practical stability of impulsive fuctioal differetial equatios i which the state variables o the impulses are related to the time delay is cosidered This paper is orgaized as follows I Sectio II, we itroduce some basic defiitios ad otatios I Sectio III, some criteria i the form of theorem for strict practical stability of impulsive fuctioal differetial equatios is obtaied i which state variables o the impulses are related to the time delay is ivestigated Fially, cocludig remars are give i Sectio IV II PRELIMINARIES Cosider the followig Impulsive fuctioal differetial equatio i which the state variables o the impulses are related to time delay xt ( = ftx (,, t t t τ t 0 x( τ I ( x( τ = J ( x( τ τ, =,,3,, ( Where x R, f CR [ DR, ], I, J CR [, R], D is a ope set i PC([ τ,0], R, where τ = costat > 0, PC([ τ,0], R = { φ:[ τ,0] R, φ( t is cotiuous everywhere except a fiite umber of poits ˆt at which φ ( tˆ ad φ ( tˆ exist ad φ ( tˆ = φ ( tˆ }, f( t,0 = 0, for all t RI, (0 0, J 0,0 = τ < τ < τ < τ < < τ <, τ, for ad 0 3 = = s t s t xt ( lim xs (, xt ( lim xs ( For each t t0, xt D is defied by xt ( s = xt ( s, τ s 0 For φ PC([ τ,0], R φ is defied by φ = sup τ s 0 φ, φ is defied by φ = if τ s 0 φ, where deotes the orm of a vector R We ca see that xt ( 0is a solutio of ( which we call the zero solutio A fuctio xt ( is called a solutio of ( with the iitial coditio x σ = ϕ Where σ ad ϕ PC([ τ,0], R, the iitial value problems of equatio ( is t 0 Ref Lashmiatham V ad Zhag Y, "Strict practical stability of delay differetial equatio", Applied Mathematics ad Computatio, vol 8, pp 75-85, Global Jourals Ic (US

4 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios xt ( = ftx (,, t t t τ t 0 x( τ I ( x( τ = J ( x( τ τ, =,,3,, x σ = ϕ ( Throughout this paper we let the followig hypothesis hold (H For t [ στσ, ], the solutio xtσϕ (;, coicides with the fuctio ϕ( t σ (H For each fuctio xs ( :[ σ τ, ] R, which is cotiuous everywhere except at the poit { } τ at which x( τ,( τ exist ad x( τ = x( τ, ftx (, is cotiuous for t almost all t [ σ, ad at the discotiuous poits f is right cotiuous (H 3 f(, t φ is Lipschitzia i φ i each compact set i PC([ τ, 0], R (H 4 The fuctios I, J, =,,, are such that if x D, I 0 ad J 0, the I( x J ( xt ( τ D Uder the hypothesis (H (H, there is a uique solutio of problems( through-out ( σϕ, We are usig the followig otatios: S( ρ = { x R : x < ρ}, K= { a CR [, R ]: at ( is mootoe strictly icreasig ad a(0 = 0}, We have followig defiitios 03 Joural of Sciece Frotier Research Volume XIII Issue I V ersio I 3Global Year Defiitio : The trivial solutio of ( is said to be Strictly practical stable, if for ay σ t0 there exist (, A, A, we have ϕ implies xt ( A for some t 0 R ; ad for every there exist A, such that ϕ > implies xt ( > A for all t σ Uiformly strictly practical stable if ( holds for all t 0 R Defiitio : The fuctio V( x :[ t0, ] S( ρ R belogs to class V 0 if: The fuctio V is cotiuous o each of the sets [ τ, τ S( ρ ad for all t t0, Vt (,0 0 ; Vtx (, is locally Lipschitzia i x S( ρ ; 3 For each =,,, there exist fiite limits lim Vty (, V( τ, x ty = (, ( τ F With V( τ, = x Vtx (, satisfied Defiitio 3 : Let V V0, for ( t, x [ τ, τ S( ρ, DVis defied as DV( t, xt ( = lim x δ sup { V( t δ, xt ( δ V( txt, ( } δ 03 Global Jourals Ic (US

5 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios III MAIN RESULT Now we cosider the uiformly strict practical stability of the impulsive fuctioal differetial equatio i which the state variables o the impulses are related to the time delay We have the followig theorem about the uiform practical stability of the system ( Theorem 3 Assume that (i There exist 0< A, a, b, V ( txt, ( v Such that 0 b( xt ( V ( txt, ( a( xt ( 03 Global Joural of Sciece Frotier Research F Volume XIII Issue I V ersio I Year 4 (ii For ay solutio of (, ( t, xt ( 0 (iii For all Z, x S( ρ V ( t sxt, ( s V ( txt, ( for s [ τ,0] ad c V ( τ, ( ( ( ( [ (, ( (, ( ] I x τ J x τ τ V τ x τ V τ τ x τ τ Where c 0 (iv For ay 0 <, there exist a, b ad c < =, V ( txt, ( v Such that 0 b( xt ( V ( txt, ( a( xt ( (v For ay solutio of (, V ( t sxt, ( s V ( txt, ( for s [ τ,0] ad (vi For all Z, x S( ρ ( t, xt ( 0 V ( τ, I( x( τ J( x( τ τ d [ V (, ( (, ( ] x V τ τ τ τ x τ τ Where 0 d < ad d < The the trivial solutio of ( is uiformly strictly practical stable Proof : Sice It follows that = = = = b <, c < c b = M ad = N = Obviously M,0< N Let 0 A < ρ ad σ t0 be give ad σ [ τ, τ] for some Z Choose 0 < A such that Ma( < b( A 03 Global Jourals Ic (US

6 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios We claim that ϕ implies xt ( < A for t σ Obviously for ay t [ σ τσ, ], there exist a θ [ τ,0], Such that V ( txt, ( = V ( σ θ, x( σ θ The we claim that a ( x( σ θ = a ( x ( θ = a ( ϕθ ( a ( σ V ( txt, ( a( for σ t < τ (3 I fact, if iequality (3 does ot hold, the there exist a tˆ ( στ, such that V ( tˆ, xt ( ˆ > a( V ( σ, x( σ which implies that there is a t ( σ, tˆ such that ad ( t, xt ( 0 > (4 V ( t sxt, ( s V ( t, xt ( for s [ τ,0] By coditio (ii, which implies ( t, xt ( 0 This cotradict iequality (4 so iequality (3 holds I view of iequality (3 ad coditio (iii, we have 03 Joural of Sciece Frotier Research Volume XIII Issue I V ersio I 5Global Year V ( τ, x( τ = V ( τ, I ( x( τ J ( x( τ τ c [ V ( τ, ( (, ( ] ( x τ V τ τ x τ τ c a ( Next we prove that V ( txt, ( ( c a( for τ t τ (5 If iequality (5 does ot hold, the there is a sˆ ( τ, τ such that F V ( sxs ˆ, ( ˆ > ( c a( V ( τ, x( τ Which implies that there is a s ( τ, sˆ such that ( s, xs ( 0 > (6 ad V ( s sxs, ( s V ( s, xs ( for s [ τ,0] By coditio (ii which implies that ( s, xs ( 0 This cotradicts iequality (6, so iequality (5 holds I view of iequality (6 ad the coditio (iii, we have = V ( τ, x( τ V ( τ, I ( x( τ J ( x( τ τ c [ ( τ, ( τ ( τ τ, ( τ τ] V x V x ( c ( c a ( 03 Global Jourals Ic (US

7 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios 03 Global Joural of Sciece Frotier Research F Volume XIII Issue I V ersio I Year 6 This together with iequality (3 yields V ( t, x( t Ma ( From this ad coditio (i we have b( xt ( V ( txt, ( Ma( b( A for t σ Thus, we have xt ( < A for t σ Now, let 0 <, ad Choose 0 < A such that a( A < Nb( Next we claim that ϕ > implies xt ( > A for t σ If this holds the A< xt ( < A for t σ Obviously for ay t [ σ τσ, ], there exist a θ [ τ,0], Such that V ( t, x( t = V ( σ θ, x( σ θ b ( x( σ θ = b ( x ( θ = b ( ϕ( θ b ( σ The we claim that V ( txt, ( b( for σ t < τ (7 I fact, if iequality (7 does ot hold, the there exist a t ( στ, such that V ( t, xt ( < b( V ( σ, x( σ (8 which implies that there is a t ( σ, t such that DV ( t, xt ( < 0 (9 ad V ( t sxt, ( s V ( t, xt ( for s [ τ,0] By coditio (v, which implies DV ( t, xt ( 0 This cotradict iequality (9 so iequality (8 holds I view of iequality (7 ad coditio (vi, we have d V ( τ, ( (, ( ( ( ( [ (, ( (, ( ] ( ( x τ = V τ I x τ J x τ τ V τ x τ V τ τ x τ τ d b Next we prove that V ( txt, ( ( d b( for τ t τ (0 If iequality (0 does ot hold, the there is a rˆ ( τ, τ such that Which implies that there is a Ad V ( rxr ˆ, ( ˆ < ( d b( V ( τ, x( τ r ( τ, rˆ such that ( r, xr ( 0 < ( V ( s sxs, ( s V ( s, xs ( for s [ τ,0] By coditio (v which implies that ( r, xr ( 0 This cotradicts iequality (, so iequality (0 holds I view of iequality ( ad the coditio (vi, we have 03 Global Jourals Ic (US

8 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios Which implies that there is a r ( τ, rˆ such that ( r, xr ( 0 < ( Ad V ( s sxs, ( s V ( s, xs ( for s [ τ,0] By coditio (v which implies that ( r, xr ( 0 This cotradicts iequality (, so iequality (0 holds I view of iequality ( ad the coditio (vi, we have V ( τ, x( τ = V ( τ, I ( x( τ J ( x( τ τ 03 d [ ( τ, ( τ ( τ τ, ( τ τ] V x V x ( d ( d b ( By simple iductio, we ca prove that i geeral that for =0,,, V ( txt, ( ( d ( d b( for τ m t < τ m m This together with iequality (7 yields From this ad coditio (iv we have V ( t, x( t Nb ( a( xt ( V ( txt, ( Nb( > a( A for t σ Thus, we have xt ( > A for t σ Thus, the trivial solutio of ( is uiformly strictly practical stable Joural of Sciece Frotier Research Volume XIII Issue I V ersio I 7Global Year F IV CONCLUSION I this paper, the strict practical stability of impulsive fuctioal differetial equatios i which the state variables o the impulses are related to the time delay is cosidered By usig Lyapuov fuctios ad Razumihi techique, we have obtaied some results for the strict practical stability Strict practical stability theorem for impulsive fuctioal differetial equatio have bee exteded to impulsive fuctioal differetial equatios i which the state variables o the impulses are related to the time delay Refereces Référeces Referecias Lashmiatham V ad Mohapatra RN, "Strict stability of differetial equatios", Noliear Aalysis, vol 46, pp 95-9, 00 Lashmiatham V ad Zhag Y, "Strict practical stability of delay differetial equatio", Applied Mathematics ad Computatio, vol 8, pp 75-85, 00 3 Lashmiatham V, Baiov D D, ad Simeoov P S, "Theory of Impulsive Differetial Equatios", Vol6 of Series i Moder Applied Mathematics, World Scietific, Teaec, NJ, USA, Global Jourals Ic (US

9 Uiform Strict Practical Stability Criteria for Impulsive Fuctioal Differetial Equatios 03 Joural of Sciece Frotier Research Volume XIII Issue I V ersio I 8Global Year F 4 Lashmiatham V ad Vasudhara Devi J, "Strict stability criteria for impulsive differetial systems" Noliear Aalysis: Theory, Methods ad Applicatio, Vol, o 0, pp , Kaie L ad Guowei Y, Strict stability criteria for impulsive fuctioal differetial equatios, Joural of Iequalities ad Applicatio, Vol 008, Article ID 43863, 8 pages 6 She J H, "Razumihi techiques i impulsive fuctioal differetial equatios", Noliear Aalysis, vol 36, pp 9-30, Sigh Dilbaj ad Srivastava SK, "Strict Stability Criteria for Impulsive Differetial Systems", Advace i Differetial equatio ad Cotrol Processes, Vol 0, pp 7-8, 0 8 Sigh Dilbaj ad Srivastava SK, "Strict Stability Criteria for Impulsive Fuctioal Differetial Equatios", Lecture i Egieerig ad Computer Sciece, Vol 97, pp 69-7, 0 9 Su J T ad Zhag Y P, "Stability aalysis of impulsive cotrol systems", IEE Proceedig-Cotrol Theory ad Applicatio, vol 50(4, pp , Su J T, "Stability criteria of impulsive differetial system,", Applied Mathematics ad Computatio, vol 56(, pp 85-9, 004 Zhag Yu ad Su J T, "Practical Stability of Impulsive fuctioal Differetial Equatios i terms of two measures", Comp Math Appl, Vol 48, pp , 004 Zhag Yu ad Su J T, "Strict Stability if Impulsive Differetial Equatios", Acta Mathematica Siica, Eglish Series, Vol, No 3, pp 83-88, Zhag Yu ad Su Jitao, "Stability of impulsive fuctioal differetial equatios", Noliear Aalysis, vol 68, pp , Zhag Yu ad Su Jitao, "Strict stability of impulsive fuctioal differetial equatios", Mathematical Aalysis ad Applicatio, vol 30, pp 37-48, Global Jourals Ic (US