Asympoic relionship beween rjecories of nominl nd uncerin nonliner sysems on ime scles Fim Zohr Tousser 1,2, Michel Defoor 1, Boudekhil Chfi 2 nd Mohmed Djemï 1 Absrc This pper sudies he relionship beween rjecories of nominl nd uncerin nonliner sysems evolving on non-uniform ime domins. The heory of dynmic equions on ime scle is used o nlyze he sbiliy of perurbed nonliner sysems. Firs, i will be shown h he error beween he uncerin nd he nominl rjecories remins bounded for priculr clss of sysems. Then, using he Lypunov heory, some condiions re derived o gurnee h he rjecory of he perurbed sysem exponenilly converges o he rjecory of he corresponding nominl sysem. These resuls re useful o sudy he robusness properies of uncerin nonliner sysems evolving on non-uniform ime domins. Index Terms Time scles, Lypunov funcion, Exponenil sbiliy. I. INTRODUCTION The heory of dynmic equions on n rbirry ime scle T ws inroduced in S. Hilger s PhD hesis [1 in 1988. This heory ws found promising becuse i demonsres he inerply beween he heories of coninuous-ime nd discree-ime sysems [2, [3, [4. As expeced, once resul hs been esblished for dynmic equions on n rbirry ime scle, his resul holds for sndrd coninuous differenil equions (T = R nd sndrd difference equions (T = hz, h is rel number. Sbiliy for liner sysems on ime scles hs been sudied in [5, [6, [7. In [5, sufficien condiions were derived o ensure exponenil sbiliy using he concep of ime scle generlized exponenil funcion. A specrl chrcerizion of sympoic sbiliy on ime scle ws given in [6. In [7, uniform exponenil sbiliy for posiive liner imeinvrin sysems ws sudied using he sbiliy rdius. Exension o liner sysems wih srucured perurbions hs been invesiged in [8. In [9, uniform exponenil sbiliy hs been invesiged for clss of uncerin liner sysems using some properies of he ime scle generlized exponenil funcion. Recenly, he sbiliy of swiched liner sysems hs been sudied using common qudric Lypunov funcion [10, [11, [12 or specrl chrcerizion [13. This work ws suppored by he Inernionl Cmpus on Sfey nd Inermodliy in Trnsporion, he Europen Communiy, he Regionl Delegion of Reserch nd Technology, he Minisry of Higher Educion nd Reserch nd he Nord/Ps-de-Clis Region, he Nionl Cener of Scienific Reserch. 2 LAMIH, CNRS UMR 8201, UVHC, F-59313 Vlenciennes, Frnce; ousser222000@yhoo.fr, michel.defoor@univ-vlenciennes.fr, mohmed.djemi@univ-vlenciennes.fr 2 Lborory of Mhemics, Universiy of Sidi Bel Abbes, 22000, Algeri. chfi b@yhoo.com Sbiliy of nonliner sysems on ime scles ws sudied in [14, [15. Using Lypunov funcions, some condiions hve been derived o gurnee he sympoic sbiliy [14 nd exponenil sbiliy [15 for dynmic equions on ime scles. One should noify h mos of he exising works nlyze he sbiliy on ime scle for nonliner sysems wihou perurbion. The objecive of his pper is o deermine which sbiliy properies of nonliner sysem re preserved under bounded perurbions. Insed of direcly sudying he behvior of he perurbed nonliner sysem [16, we compre he uncerin nd he nominl sysem rjecories. Firs, i will be shown h he error beween he uncerin nd he nominl rjecories remins bounded for priculr clss of nonliner sysems. Then, using he Lypunov heory on ime scles, some condiions re derived o gurnee h he rjecory of he perurbed sysem exponenilly converges o he rjecory of he corresponding nominl sysem. The ouline of he pper is s follows. Secion II reclls some useful conceps on ime scle heory. The problem is sed in Secion III. Secion IV derives sufficien condiions o gurnee h he error beween he uncerin nd he nominl rjecories remins bounded. Oher condiions bsed on Lypunov heory on ime scles re proposed o ensure he exponenil sbiliy of he error rjecory. Secion V shows n exmple o illusre he proposed scheme. II. PRELIMINARIES ON TIME SCALE THEORY In his secion, we recll he bsic noions, min definiions nd properies regrding ime scle clculus [2, [3, [4. Then, he sudied clss of nonliner sysems is described. A. Descripion of he ime scle A ime scle T is n rbirry nonempy closed subse of R. We ssume hroughou he pper h T is unbounded bove. The mos populr exmples of ime scles re he rel numbers T = R, he inegers T = hz = hz ; z Z} where h is rel consn, T = P k ;b k } formed by union of disjoin inervls wih vrible lengh k nd vrible gp b k nd T = q N = q ; n N}, q > 1, for qunum nlysis. For ny T, he forwrd nd bckwrd jump operors re defined by nd σ( := infs T : s > } (1 ρ( := sups T : s < } (2
The operors σ : T T nd ρ : T T llow he following clssificion of poins in T. A poin T is clled righdense, lef-dense, righ-scered, or lef-scered if σ( =, ρ( =, σ( >, or ρ( <, respecively. I is ssumed h inf /0 = supt (i.e. σ( = if T hs mximum nd sup /0 = inft (i.e. ρ( = if T hs minimum where /0 denoes he empy se. A se T k is defined s T k = T\m} if T hs lef-scered mximum m} nd T k = T oherwise. The disnce from n rbirry smpling ime T o he nex smpling ime is clled he grininess funcion of he ime scle T nd is deermined by µ( := σ(, T (3 If T = R, hen σ( = = ρ( nd µ( = 0. If T = hz, hen σ( = + h, ρ( = h nd µ( = h. These necessry definiions re required o define he differenil operor for funcions wih ime scle domins. B. Differeniion on ime scle Now, le us consider funcion f : T R nd le us inroduce he so-clled -derivive poin T k. The funcion f is sid o be -differenible poin T k if here exiss γ R such h for ny ε > 0, here exiss W-neighborhood of T k sisfying he following inequliy ( f (σ( f (s γ(σ( s ε σ( s for ll s W. We denoe f ( = γ. If f is -differenible for ny T k, hen f : T R is -differenible on T k. Some useful relionships concerning he -derivive of f re reclled in he nex heorem. Theorem 1: [2, [3 Assume h f : T R nd T k. The following semens hold 1 If f is -differenible, hen i is coninuous. 2 If f is coninuous nd is righ-scered, hen f is -differenible wih f ( = f (σ( f ( µ( ; (4 3 If is righ-dense, hen f is -differenible if nd only if he limi f ( f (s lim s s exiss s finie number, nd f ( = lim s f ( f (s s. 4 If f is -differenible T k, hen f (σ( = f (+ µ( f (. Theorem 1 generlizes boh he clssicl differenil operor nd he forwrd difference operor. Indeed, if T = R, hen f ( = ḟ (. If T = Z, hen f ( = f ( + 1 f (. C. Inegrion on ime scle A funcion f : T R is clled rd-coninuous if i is coninuous righ-dense poins in T nd is lef-sided limi exiss lef-dense poins in T. The se of ll rd-coninuous funcions f : T R is denoed by C rd = C rd (T = C rd (T,R Considering funcion F : T R wih -derivive F ( = f (. Then, F is clled -niderivive of f ( nd for ny,b T we cn define he following inegrl b f ( = F(b F( (5 I cn be shown h every rd-coninuous funcion hs - niderivive [17 nd he following semens holds: If f ( 0 on [,b nd s, T, s b, hen f ( = f (s + s f (ττ f (s, i.e. funcion f ( is incresing in T. If supt =, hen he improper inegrl is defined by f ( = lim (F(b F( b for ll T. The nex heorem describe he chin rule for he - differeniion of ( f g wih g : T R nd f : R R. Theorem 2: [2, [3 If f : R R is coninuously differenible nd g : T R is -differenible, hen ( f g : T R is -differenible nd he following formul holds ( 1 ( f g ( = ḟ (g( + hµ(g (dh g ( (6 D. Time scle exponenil funcion 0 A funcion f : T R is clled regressive if 1+ µ( f ( 0 for ll T k, nd posiively regressive if 1 + µ( f ( > 0 for ll T k. The se of ll rd-coninuous nd regressive funcion f : T R is denoed by R = R(T = R(T,R If n ddiion is defined by T (p q( = p( + q( + µ(p(q( hen, pir (R, is n Abelin group. The inverse elemen is p( p( = 1 + µ(p( nd he circle subsrcion is defined by p q = p ( q One should noify h if p,q R, hen p, q, p q, p q,q p R. The se of ll posiively regressive funcions of R is R + = R + (T = R + (T,R
To define he generlized ime scle exponenil funcion, le us inroduce he following conceps [2, [18. For some h > 0, we consider he ses Z h = z C : π h < Im(z π } h nd C h = z C : z 1 } h For h = 0, le Z 0 = C 0 = C be he se of complex numbers. For h 0, he cylinder rnsformion ξ h : C h Z h is defined by 1h Log(1 + zh, if h > 0 ξ h (z = z, if h = 0 where Log denoes he principl logrihm funcion. For funcion p R, he generlized ime scle exponenil funcion e p (,s is defined by ( e p (,s = exp ξ µ(τ (p(ττ, (,s T T s where ξ µ( (p( is he cylinder rnsformion of p(. Herefer, some properies of he ime scle exponenil funcion re given. 1 e p (,s = e p (,s e p (,s = 1 e p (s, = e p (s, e p (,se p (s,r = e p (,r e p (,s e q (,s = e p q (,s Theorem 3: [2, [3 Suppose h x, f C rd nd p R. Le T nd x 0 R. Then, he unique soluion of he iniil-vlue problem is given by x ( = p(x( + f (, x( = x 0 (7 x( = e p (, x 0 + e p (,σ(τ f (ττ (8 Using Theorem 3, he Gronwll s inequliy on T cn be expressed s follows. Lemm 4: [19 Le y, f C rd nd p R +. Then implies y( y( e p (, + C rd y ( p(y( + f (, for ll T e p (,σ(s f (ss, for ll, T. Theorem 5 (Gronwll s inequliy: [19 Le y, f, p nd p 0. Then implies y( f ( + y( f ( + y(sp(ss, e p (,σ(s f (sp(ss, for ll T for ll T III. PROBLEM STATEMENT Le T be ime scle wih bounded grininess funcion. I is ssumed h supt =. Le us consider he dynmic sysem x ( = f (,x(,, T x( = x 0, T, x 0 R n (9 where x( R n is he se nd f : T R n R n. I is ssumed h condiions for he exisence of n unique soluion of (9 re sisfied. The objecive of his pper is o deermine which sbiliy properies of sysem (9 re preserved under bounded perurbion. Insed of direcly sudying he behvior of he perurbed nonliner sysem, we compre he nominl sysem (9 nd he following uncerin sysem y = f (,y( + g(,y(,, T y( = y 0, T, y 0 R n (10 where y( R n is he se nd g : T R n R n is he perurbion. If he ddiionl erm g(, y is smll in some sense, i is resonble o expec h he rjecory of (10 will be similr o he rjecory of (9, provided h he iniil vlues for he sysems re sufficienly close. I is ssumed h condiions for he exisence of n unique soluion of (10 re sisfied. The conribuion of his pper is o sudy he sbiliy properies of (10 wih respec o (9. Herefer, we give some definiions nd lemm which will be useful o derive he min resuls. Definiion 6: The soluion of he perurbed dynmic equion (10 is sid o be sble wih respec o he corresponding nominl equion (9, if ε > 0, δ > 0 such h T, x 0 R n, y 0 x 0 < δ y( x( < ε nd i is sid o be sympoiclly sble wih respec o (9 if i is sble wih respec o (9 nd δ 0 > 0, such h x 0 R n, y 0 x 0 < δ 0 lim y( x( = 0. + Definiion 7: The soluion of he perurbed dynmic equion (10 is sid o be exponenilly sble wih respec o he corresponding nominl equion (9, if α < 0 wih α R +, β 1, δ 0 > 0 nd p > 0 such h T, x 0 R n, y 0 x 0 < δ 0 y( x( β y 0 x 0 e p α(, IV. MAIN RESULTS The following resul presens sufficien condiions o ensure boundedness of he error beween he rjecory of he perurbed dynmic equion (10 nd he nominl one (9.
Theorem 8: Assuming h funcion f sisfies he generlized Lipschiz condiion, i.e. for ll T, ξ 1 R n, ξ 2 R n f (,ξ 1 f (,ξ 2 L( ξ 1 ξ 2 (11 where L C rd. I is lso ssumed h C > 0 such h e L (, < C < (12 The perurbion g is bounded s follows, ξ R n, g(,ξ h( (13 where h : T R n is inegrble on T nd D > 0 such h + h(s s D < (14 If condiions (11-(14 re fulfilled, hen he error beween he rjecory of he uncerin dynmic equion (10 nd he nominl one (9 remins bounded. Proof: From sysems (9, (10, one ges x( = x 0 + f (s,x(ss (15 Le us pply he Gronwll s inequliy, i.e. Theorem 5 o he bove relion. One cn obin y( x( A + e L (,σ(sal(s s A 1 + e L (,σ(sl(s s A 1 + e L (, e L (,σ(sl(s s A 1 + e L (, e L (σ(s, L(s s A 1 + e L (, (1 + µ(s( L(se L (s, L(ss L(s A 1 + e L (, 1 + µ(sl(s e L(s, s [ A 1 e L (,( L(se L (s, s A 1 ( L(se L (s, s A 1 e L(s, s A (1 [e L (s, 0 A(1 e L (, + e L (, A(1 1 + e L (, Ae L (, ( y 0 x 0 + + Using (12 nd (14, one cn derive h(s s e L (, y( x( ( y 0 x 0 + DC = y 0 x 0 C +CD nd y( = y 0 + f (s,y(ss + g(s,y(ss (16 Seing δ 0 = D, one hs x 0 R n, T, y 0 x 0 < δ 0 y( x( 2CD I concludes he proof. Equions (15-(16 yield y( x( y 0 x 0 + f (s,y(s f (s,x(s s + g(s,y(s s y 0 x 0 + L(s y(s x(s s + h(s s y 0 x 0 + L(s y(s x(s s + + h(s s A + L(s y(s x(s s wih A = y 0 x 0 + + h( s. The nex resul presens sufficien condiions in erms of Lypunov funcion o ensure exponenil sbiliy for perurbed dynmic equion (10 ccording o Definiion 7. Theorem 9: Suppose h here exiss Lypunov funcion V : T R n R + (,ξ V (,ξ (17 -differenible in T nd coninuously differenible in ξ R n such h i For ll (,ξ T R n, λ 1 ξ p V (,ξ λ 2 ξ p (18 ii The -derivive of V sisfies he following inequliy V (,y( x( λ 3 y( x( p (19
wih consns λ 1 > 0, λ 3 > 0, λ 2 > µ(λ 3 T (i.e λ 3 λ R + nd p > 0. 2 Then he soluion of he perurbed dynmic equion (10 is exponenilly sble wih respec o he corresponding nominl equion (9. Proof: Equions (18 nd (19 yield V (,y( x( ( λ 3 y( x( p λ3 λ V (,y( x( 2 (20 Inegring boh sides of (20 nd using Theorem 3, one cn obin V (,y( x( V (,y 0 x 0 e λ4 (, (21 wih λ 4 = λ 3 λ 2. From (18, i follows h V (,y( x( λ 1 y( x( p Hence, ( 1 y( x( p Using (18, one ges λ 1 V (,y 0 x 0 e λ4 (, V (,y 0 x 0 λ 2 y 0 x 0 p Therefore, he following inequliy holds ( 1 1 p 1 1 p y( x( y0 x 0 λ2 e p λ (, 4 0 λ 1 ( λ2 λ 1 This complees he proof. 1 p 1 y0 x 0 e p (, λ4 0 V. NUMERICAL EXAMPLES In his secion, he proposed scheme is illusred hrough numericl exmples. A. Exmple 1 Le us consider he uncerin nonliner sysem y ( = 1 (+1 2 y + g(,y(, 0, T y(0 = y 0, y 0 R (22 where y R is he se nd = 1+µ( ( is consn nd µ is he grininess funcion of T. The perurbion is s follows g(,y( = e 1 (,0sin(y( (23 The corresponding nominl sysem is x ( = 1 (+1 2 x, 0, T x(0 = x 0, x 0 R (24 where x R is he se. The objecive is o sudy he sbiliy properies of (22 wih respec o (24. One should noify h (23 sisfies he generlized Lipschiz condiion wih L( = 1 (+1 2. The perurbion g is bounded, i.e. e 1 (,0sin(y e 1 (,0 sin(y e 1 (,0 (25 To sudy he boundedness of he error beween he rjecory of he perurbed dynmic equion (22 nd he nominl one (24, le us consider he following ime scles: For T = R, 0, one ges nd 0 0 1 e L (,0 = e (s+1 2 ds = e 1 +1 +1 < e 1 < e 1 (s,0 s = e s ds = e + 1 1 < 0 Therefore, from Theorem 8, one cn conclude h he error beween he rjecory of he perurbed dynmic equion (22 nd he nominl one (24 remins bounded. For he nonhomogeneous ime scle T = P σk, k+1 } = k=0 [ σ k, k+1 wih k < σk < k+1, k R (k N nd σ0 = = 0 (one cn refer o [13 for furher deils bou his ime scle, he forwrd jump operor sisfies σ( k = σk nd he grininess funcion is such h µ( k = σ( k k = σk k which is bounded by µ mx. On his ime scle, one ges, [ σk, k+1, e L (,0 = e 0 log(1+µ(sl(s s µ(s m( 1 e m(+1 + k i=0 log(1+µ( il( i m( 1 e m(+1 + k i=0 µ( il( i m( 1 m(+1 e +µ mx k i=0 1 ( i +1 2 wih m( = k i=0 µ( i. Since k 1 i=0 is bounded, ( i +1 2 here exiss C > 0 such h e L (,0 < C < Furhermore, on [ σk, k+1, he equliy yields e 1 (,0 = e m(+ k i=0 log(1 µ( i e 1 (s,0 s So, here exis D > 0 such h e m(s s e 1 (s,0 < D < Therefore, from Theorem 8, one cn conclude h he error beween he rjecory of he perurbed dynmic equion (22 nd he nominl one (24 remins bounded.
B. Exmple 2 Le us consider he nonliner dynmic equion x = x + e 1 (,0sin(x, 0, T x(0 = x 0, x 0 R (26 where x R is he se nd = 1+µ( ( is consn nd µ is he grininess funcion of T. The objecive is o sudy he sbiliy properies of y = y + e 1 (,0sin(y, 0, T (27 y(0 = y 0, y 0 R wih respec o (24. One should noify h sysem (27 is only perurbed ime = 0 (i.e. y 0 x 0. Le us consider he cndide Lypunov funcion V : T R n R + (,ξ V (,ξ = ξ 2 (28 Condiion (18 is sisfied wih λ 1 = λ 2 = 1 nd p = 2. The rcking error e = y x fulfills he following dynmics e = e + e 1 (,0(sin(y sin(x e(0 = y 0 x 0 (29 The -derivive of V long he rjecories of (29 on n rbirry ime scle T is given by V (e = e e + e(σ(e = e e + (e + µ(e e = 2e e + µ((e 2 = 2( e 2 + 2e 1 (,0(sin(y sin(xe +µ([ e + e 1 (,0(sin(y sin(x 2 = 2( e 2 + 2e 1 (,0(sin(y sin(xe +µ([( e 2 + µ([e 1 (,0(sin(y sin(x 2 +2µ([ e 1 (,0(sin(y sin(xe = [2( + µ(( 2 e 2 +2e 1 (,0[1 + 2µ(( (sin(y sin(xe +µ(e 2 1 (,0(sin(y sin(x2 [2( + µ(( 2 + µ(e 2 1 (,0e2 + 2e 1 (,0(1 + 2µ(( e 2 Since on n rbirry ime scle, e 1 (,0 M (M is posiive consn, one ges V (e λ 3 e 2 wih 2 λ 3 = 1 + µ( µ(2 (1 + µ( 2 2M 1 µ( 1 + µ( µ(m2 Therefore, he condiions of Theorem 18 re sisfied if λ 3 > 0 1 > µ(λ 3 (30 If inequliies (30 hold, hen he soluion of (27 is exponenilly sble wih respec o he soluion (26. Le us consider he following ime scles: For T = R, 0, µ( = 0 nd M = 1. Inequliies (30 men > 1. For T = hz wih h = 1 2, µ( = 2 1 nd M = 1. Inequliies (30 men 12.9282 > > 1.0381. VI. CONCLUSION In his pper, he sympoic equivlence of nominl nd uncerin nonliner sysems on ime scles hs been sudied. Two heorems hve been proposed. Firs, i hs been shown h he error beween he uncerin nd he nominl rjecories remins bounded for priculr clss of sysems. Then, using he Lypunov heory, some condiions hve been derived o gurnee h he rjecory of he perurbed sysem exponenilly converges o he rjecory of he corresponding nominl sysem. REFERENCES [1 S. Hilger, Ein Mβkeenklkäul mi Anvendung uf Zenrumsmnnigfligkeien, PhD hesis, Universiä Würzburg, 1988. [2 M. Bohner nd A. Peerson, Dynmic Equions on Time Scles, An Inroducion wih Applicions, Birkhuser, Boson 2001. [3 M. Bohner, A. Peerson, Advnces in Dynmic Equions on Time Scles, Birkhuser Boson, Inc. Boson, MA, 2003. [4 J. M. Dvis, I. A. Grvgne, B. Jckson nd R. J. Mrks, Conrollbiliy, Observbiliy, Relisbiliy nd Sbiliy of Dynmic Liner Sysems, Elecronic Journl of Differenil Equions, 37, pp. 1 32, 2009. [5 J. DCuhn, Sbiliy for Time Vrying Liner Dynmic Sysems on Time Scles, J. Compu.Appl. Mh., 176(2, pp. 381-410, 2005. [6 C. Pozsche, S. Siegmund nd F. Wirh, A specrl chrcerizion of exponenil sbiliy for liner ime-invrin sysems on ime scles, Discree Conin. Dyn. Sys., 9, pp. 1223-1241, 2003. [7 T. S. Don, A. Kluch, S. Siegmund nd F. R. Wirh, Sbiliy rdii for posiive liner ime-invrin sysems on ime scles, Sysems nd Conrol Leers, 59, 2010. [8 N. H. Du, D. D. Thun nd N. C. Liem, Sbiliy rdius of implici dynmic equions wih consn coefficiens on ime scles, Sysems nd Conrol Leers, 60(8, pp. 596 603, 2011. [9 Z. Brosiewicz nd E. Piorowsk, On sbilisbiliy of nonliner sysems on ime scles, In. J. of Conrol, 86(1, pp. 139 145, 2013. [10 J. M. Dvis, I. A. Grvgne nd A. A. Rmos, Sbiliy of Swiched Liner Sysems on non-uniform Time Domins, IEEE Souhesern Symposium on Sysems Theory, Texs, 2010. [11 F. Z. Tousser nd M. Djemi, Sbiliy of Swiched Liner Sysems on Time Scle, In. Conf. on Sysems nd Conrol, Algeri, 2013. [12 F. Z. Tousser, M. Defoor nd M. Djemi, Sbiliy nlysis of clss of uncerin swiched sysems on ime scle using Lypunov funcions, Nonliner Anlysis: Hybrid Sysems, 2015. [13 F. Z. Tousser, M. Defoor nd M. Djemi, Sbiliy nlysis of clss of swiched liner sysems on non-uniform ime domins, Sysems nd Conrol Leers, 74, pp. 24 31, 2014. [14 Z. Brosiewicz nd E. Piorowsk, Lypunov funcions in sbiliy of nonliner sysems on ime scles, Journl of Difference Equions nd Applicions, 17(03, pp. 309 325, 2011. [15 A. C. Peerson nd Y. N. Rffoul, Exponenil Sbiliy of Dynmic Equions on Time Scles, Advnces in Difference Equions, 2, pp. 133 144, 2005. [16 B. B. Nsser nd M. A. Hmmmi, On Prcicl Sbiliy of Time Scle Perurbed Sysems. Journl of Dynmicl Sysems nd Geomeric Theories, 12(1, pp. 51 67, 2014. [17 G. Guseinov, Inegrion on Time Scles, J. of Mhemicl Anlysis nd Applicions, 285, pp. 107 127, 2003. [18 S. Hilger, Anlysis on mesure chins : unified pproch o coninuous nd discree clculus, Resuls Mh., 18, pp. 18 56, 1990. [19 E. Akin-Bohner, M. Bohner nd nd F. Akin, Pchpe inequliies on ime scles, Journl of Inequliies in Pure nd Applied Mhemics, pp. 1 23, 2005.