LYAPUNOV-KRASOVSKII STABILITY THEOREM FOR FRACTIONAL SYSTEMS WITH DELAY

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1 LYAPUNOV-KRASOVSKII STABILITY THEOREM FOR FRACTIONAL SYSTEMS WITH DELAY D. BALEANU 12 A. RANJBAR N. 3 S.J. SADATI R. 3 H. DELAVARI 3 T. ABDELJAWAD 1 (MARAABA) V. GEJJİ 4 1 Deparme of Mahemais ad Compuer Siee Fauly of Ars ad Siees Cakaya Uiversiy-653 Akara Turkey 2 Isiue of Spae Siees P.O.Box MG-23 R Magurele-Buhares Romaia dumiru@akaya.edu.r 3 Deparme of Elerial ad Compuer Egieerig Fauly of Elerial Egieerig Babol Uiversiy of Tehology Babol Ira 4 Deparme of Mahemais Uiversiy of Pue Pue 4117 Idia Reeived Jauary Fraioal alulus ehiues ad mehods sared o be applied durig he las deades i several fields of siee ad egieerig. I his paper we sudied he sabiliy of fraioal order oliear ime-delay sysems for Capuo s derivaive ad we exeded Lyapuov-Krasovskii heorem for he fraioal oliear sysems. Key words: fraioal oliear sysems sabiliy Lyapuov-Krasovskii heorem ime-delay sysems. 1. INTRODUCTION Fraioal alulus is a emergig field wih various appliaios i siee ad egieerig. Fraioal alulus is a good adidae o solve he dyamis of omplex sysems. Durig he las years fraioal alulus was subjeed o a iese debae [1-4]. Fraioal differeial euaios sared o play a impora role i modelig aomalous diffusio proesses havig log rage depedee ad so o. Several ope problems remai usolved or here were parially solved wih his ype of alulus. Amog hose kids of problems we meio he uesio of sabiliy whih is of mai ieres i orol heory. Also he problem of imedelay sysem has bee disussed over may years. Time delay is very ofe eouered i differe ehial sysems e.g. eleri peumai ad hydrauli eworks hemial proesses ad log rasmissio lies. The exisee of pure ime delay regardless of is presee i a orol ad/or sae may ause udesirable Rom. Jour. Phys. Vol. 56 Nos. 5 6 P Buhares 211

2 2 Lyapuov-Krasovskii sabiliy heorem for fraioal sysems wih delay 637 sysem rasie respose or geerally eve a isabiliy. Numerous repors have bee published o his maer wih pariular emphasis o he appliaio of Lyapuov s seod mehod [5 6]. I ree years osiderable aeio has bee paid o orol sysems whose proesses ad/or orollers are of fraioal order. This is maily due o he fa ha may real-world physial sysems are well haraerized by fraioalorder differeial euaios i.e. euaios ivolvig oieger-order derivaives. I pariular i has bee show ha visoelasi maerials havig memory ad herediary effes [7] ad dyamial proesses suh as semi-ifiie lossy RC rasmissio [8] mass diffusio ad hea oduio [9] a be more adeuaely modeled by fraioal-order models ha ieger-order models. Moreover wih he suess i he syhesis of real oieger differeiaor ad he emergee of ew elerial irui eleme alled fraae [1] fraioal-order orollers [11 12] iludig fraioal-order PID orollers [13] have bee proposed o ehae he robusess ad performae of orol sysems. Some lieraures published abou sabiliy of fraioal order liear ime delay sysems [14 15]. I he base of Lyapuov s seod mehod some work has bee doe i he field of sabiliy of fraioal order oliear sysems wihou delay [16-18]. Bu i seems ha a few aeios have bee paid o he sabiliy of fraioal order oliear ime-delay sysems. The purpose of his paper is o develop he Lyapuov-Krasovskii heorem for fraioal order oliear ime-delay sysems. The mausrip is orgaized as follows: I Seio 2 some basi defiiios of fraioal alulus are meioed. Seio 3 is devoed o fraioal oliear ime-delay sysems. Seio 4 preses he geeralizaio of he fraioal Lyapuov-Krasovskii heorem whe boh fraioal derivaives ad delay are preseed. Fially he Colusios are show i Seio PRELIMINARIES AND DEFINITIONS I he fraioal alulus he Riema-Liouville ad Capuo fraioal derivaives are defied respeively [15 16] x( s) D x = d s ( 1 < ) (1) 1 d () + 1 Γ( ) d s ( x ) ( s) 1 1 () Γ( ) s + D x = d s 1 < (2)

3 638 D. Baleau e al. 3 where x () is a arbirary differeiable fuio ad D ad D are he Riema-Liouville ad Capuo fraioal derivaives of order o[ ] respeively ad Γ( ) For < 1 we have ad deoes he Gamma fuio. x( s) D x = d s ( < 1) (3) 1 d () Γ( 1 ) d s x ( s) 1 () Γ( 1 ) s 1 D x = d s < 1. (4) Some properies of Riema-Liouville ad Capuo derivaives are realled below [15 16]: Propery 1. Whe < < 1 we have () () I pariular if x( ) = we have Propery 2. For ad v > 1 we have I pariular if 1 Propery 3. x( ) 1 D x = D x Γ D x D x. (5) =. (6) Γ ( 1 + v) 1 v v v D = Γ + < < ad x () ( ) v = he from Propery 1 we have Γ ( 1 + v) 1 v v v D = Γ + ( ) D ax + by = a D x + b D y (9) where a ad b are arbirary osas... (7) (8)

4 4 Lyapuov-Krasovskii sabiliy heorem for fraioal sysems wih delay 639 Propery 4. From he defiiio of Capuo s derivaive (4) whe < 1 we have () I D () ( x = x x ) (1) where 1 f()d s s 1 Γ( ) A ( s) ( I f) = > ( ) >. 3. FRACTIONAL NONLINEAR TIME-DELAY SYSTEM Le C [ a b] be he se of oiuous fuios mappig he ierval [ ab ] o. I may siuaios oe may wish o ideify a maximum ime delay r of a sysem. I his ase we are ofe ieresed i he se of oiuous fuio C = C r. mappig [ r] o For ay A > ( ) for whih we simplify he oaio o [ ] ad ay oiuous fuio of ime C [ r A] ψ + + A le ψ C be a segme of fuio ψ defied as ψ ( θ ) =ψ ( +θ ) r θ. Cosider Capuo fraioal oliear ime-delay sysem D x f ( x ) = (11) x < < 1 ad f : C. As suh o deermie he fuure where evoluio of he sae. I is eessary o speify he iiial sae variables x ( ) i a ime ierval of legh r say from where r o i.e. x = φ (12) φ C is give. I oher words x( + θ ) =φ( θ) r θ. φ C( a b ) defie he oiuous orm by for a fuio [ ] φ = max φ θ. (13) a θ b Defiiio. For he sysem desribed by (11) he rivial soluio x = is said o be sable if for ay ad ay ε > here exiss a δ =δ( ε ) > suh ha x <δ implies. I is said o be asympoially sable if i is x <ε for

5 64 D. Baleau e al. 5 sable ad for ay ad ay x <δ implies x ε > here exiss a ( ) δ =δ ε > suh ha lim =. I is said o be uiformly sable if i is sable ad x δ=δ ε > a be hose idepedely of. I is uiformly asympoially sable if i is uiformly sable ad here exiss a δ > ad fuios δ( ε ) T ( ε ) suh ha x < δ ad + T ( ε ) implies x( ) < ε. I is globally (uiformly) asympoially sable if i is (uiformly) asympoially sable ad δ a be a arbirary large fiie umber [21]. 4. FRACTIONAL LYAPUNOV-KRASOVSKII THEOREM As i he sudy of sysems wihou delay a effeive mehod for deermiig he sabiliy of a ime-delay sysem is Lyapuov mehod. Sie i a ime-delay r i.e. sysem he sae a ime reuired he value of x i he ierval [ ] x i is aural o expe ha for a ime-delay sysem orrespodig Lyapuov fuio be a fuioal V ( x ) depedig o x whih also should measure he deviaio of x from he rivial soluio. Le ( ) x τ φ be he soluio of (11) a ime wih iiial odiio x τ =φ. The we alulae he Capuo derivaive of V ( x ) wih respe o ad evaluae i a = τas follows V φ be differeiable ad le DV where < < 1. Theorem: Suppose f : bouded ses i ( τφ ) = DV( x( τφ )) 1 V ( s xs ) = Γ ( 1 ) ( s) = τ x =φ d s = τ x =φ C i (6) maps ( bouded ses i C ) (14) io ad α1 α2 α3 : + + are oiuous odereasig α s α s are posiive for s > ad fuios where addiioally 1 2 α =α =. If here exiss a oiuously differeiable fuioal 1 2 V : S ρ where Sρ = { φ C: φ <ρ} C suh ha ( ) V ( ) α1 φ φ α2 φ (15)

6 6 Lyapuov-Krasovskii sabiliy heorem for fraioal sysems wih delay 641 ad DV φ α φ < 1. (16) 3 The he rivial soluio of (11) is uiformly sable. If ( s) is uiformly asympoially sable. If i addiio lim α ( s) s α 3 > for s > he i = he i is globally uiformly asympoially sable. The ieger order derivaive versio of his heorem a be foud i [21 22]. Proof. For ay ε> sie α 2 is oiuous ad α () 2 = we a fid a suffiiely small δ=δ( ε ) > suh ha α2 ( δ ) <α1( ε ). Hee for ay iiial ime ad ay iiial odiio x = φ wih φ <δ we have ( ) DV x ad herefore by propery 4 V ( x ) V ( φ ) for ay. This implies ha 1 ( x() ) V ( x ) V ( ) α φ α φ α δ <α ε (17) whih implies ha x( ) <ε for To prove uiform asympoi sabiliy le < ε<ρ ad δ =δ( ε ) > ε ρ ad desigae by uiform sabiliy. Choose a fixed. Le us ow hoose δ( ε) orrespods o uiform sabiliy. Suppose ha x δ ε for all Therefore. This proves he uiform sabiliy. orrespod o δ =δ ε > where ε is α2 δ x δ ad T( ε ) = Γ ( 1 + ) α3 ( δ( ε) ) ( x() ) ( ) where x δ ad we would have α α δ ε (18) ( ) ( ) DV x α δ ε for (19) 3 ad hee by properies 2 ad 3 we olude D V x + α δ ε. The by usig he propery 4 we have ( ) 3 Γ ( 1 + ) (2)

7 642 D. Baleau e al. 7 As a resul we obai whih for T ( ) Γ ( 1 + ) 3 V x + α δ ε V φ. ( ) V φ α3 δ ε Γ ( 1 + ) ( ) α2 φ α 3 δ ε Γ ( 1 + ) ( ) α2 δ α3 δ ε Γ ( 1 + ) V x = + ε redues o ( δ( ε) ) ( 1) α3 <α1( δ( ε) ) V ( + T x ) + T α2 δ T =. Γ + This oradiio proves ha here exiss a 1 + T( ε) x( 1 ) <δ( ε ). Thus i ay ase we have x( ) < ε + T( ε) wheever x of (11). (21) (22) (23) suh ha < δ provig he uiform asympoi sabiliy of he rivial soluio Fially if lim α 1 ( s) = he s δ is give o saisfy δ above may be arbirary large ad ε a be hose afer α2 δ <α1 ε ad herefore global uiform asympoi sabiliy a be oluded. We observe from he above proof ha α1 α2 α3 ad V(.) eed oly o be defied i a eighborhood of zero exep for he ase of global sabiliy. We also oie ha he lower boud of V eed oly o be a posiive fuio of φ (). 5. CONCLUSIONS The ombiaio of he fraioal alulus ad delay ehiues seems o desribe beer he dyamis of he omplex sysems amely beause boh heories ake io aou he memory effes. I his paper we geeralized he fraioal Lyapuov-Krasovskii heorem i he presee of Capuo fraioal derivaives ad delay. The obaied heorem oais as pariular ases he fraioal alulus

8 8 Lyapuov-Krasovskii sabiliy heorem for fraioal sysems wih delay 643 versio as well as he ime-delay oe. The use of he Capuo fraioal derivaive was ruial for provig he obaied resuls. REFERENCES 1. R. Hilfer Appliaios of Fraioal Calulus i Physis World Sieifi Publishig Compay Sigapore G.M. Zaslavsky Hamiloia Chaos ad Fraioal Dyamis Oxford Uiversiy Press Oxford R.L. Magi Fraioal Calulus i Bioegieerig Begell House Publisher I. Coeiu J.A.T. Mahado A.M. Galhao ad A.M. Oliveira Opimal approximaio of fraioal derivaives hrough disree-ime fraios usig geei algorihms Commu. Noli. Si. Num. Simul. 15(3) (21) J. Che D. Xu ad B. Shafai O suffiie odiios for sabiliy idepede of delay IEEE Tras.Auoma. Corol AC 4 (9) (1995) T.N. Lee ad S. Dia Sabiliy of ime delay sysems IEEE Tras. Auoma. Corol AC31(3) (1981) R.L. Bagley ad O. Torvik O he appearae of he fraioal derivaive i he behavior of real maerials J. Appl. Meh. 51 (1984) E. Weber Liear Trasie Aalysis (Vol. II) New York Wiley V.G. Jeso ad G.V. Jeffreys Mahemaial Mehods i Chemial Egieerig (2d ed.) New York Aademi Press M. Nakagava ad K. Sorimahi Basi haraerisis of a fraae devie IEICE Trasaios Fudameals E75-A(12) (1992) P. Lausse A. Ousaloup ad B. Mahieu Third geeraio CRONE orol Proeedigs of ieraioal oferee o sysems ma ad ybereis 2 (1993) I. Podluby Fraioal-order sysems ad PI D λ µ-orollers IEEE Trasaios o Auomai Corol 44(1) (1999) H.F. Rayaud ad A. Zergaioh Sae-spae represeaio for fraioal order orollers Auomaia 36(7) (2) M.P. Lazarevi Fiie ime sabiliy aalysis of PD α fraioal orol of roboi ime-delay sysems Meh. Res. Comm. 33 (26) X. Zhag Some resuls of liear fraioal order ime-delay sysem Appl. Mah. Comp. 197 (28) S. Momai ad S. Hadid Lyapuov sabiliy soluios of fraioal iegro-differeial euaios I. J. Mah. Mah. Si. 47 (24) J. Sabaier O sabiliy of fraioal order sysems I Pleary Leure VIII o 3 rd IFAC Workshop o Fraioal Differeiaio ad is Appliaios AkaraTurkey Y. Li Y.Q. Che ad I. Podluby Sabiliy of fraioal-order oliear dyami sysems: Lyapuov dire mehod ad geeralized Miag-Leffler sabiliy Comp. Mah. Appl. (i press) A.A. Kilbas H.M. Sirvasava ad J.J. Trujillo Theory ad Appliaios of Fraioal Differeial Euaios Elsevier B.V I. Podluby Fraioal Differeial Euaios Aademi Press Sa Diego G. Keui V.L. Kharioov ad J. Che Sabiliy of Time-Delay Sysems Birkhauser A. Halaay Differeial euaios: Sabiliy Osillaios Time Lags Mahemais i Siee ad Egieerig series vol