-HYBRID LAPLACE TRANSFORM AND APPLICATIONS TO MULTIDIMENSIONAL HYBRID SYSTEMS. PART II: DETERMINING THE ORIGINAL

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1 UPB Sc B See A Vo 72 I 3 2 ISSN MUTIPE -HYBRID APACE TRANSORM AND APPICATIONS TO MUTIDIMENSIONA HYBRID SYSTEMS PART II: DETERMININ THE ORIINA Ve PREPEIŢĂ Te VASIACHE 2 Ace co copeeă oă - pce he ] c eoeee e ee ş ă o ee covoţe po oeo voe ţă ş voe ă S peee eoe e eee o Se oţe o oă e vee e p Me-oe ş e ă o eoeă e evoe eeă Th ppe copee he y o he -Hy pce To o ] wh he heoe e eo o he o eo o he e covoo poc o o ve Soe eho o o ee he o o ve - Hy pce To e pove A eee Me-oe ype veo o ehe wo ohe o e eve epecvey he Ree Theoe ve e ee expo A eee Expo Theoe o ve Keywo: Hy pce To Me-oe ype veo o expo heoe Ioco I P I ] he pe hy pce- o w ee popee wee pove c ey hoohey wo e-ey heoe o eeo eece o he o eeo o he e Sch oo ecey o he y o he coo-cee eo ye 4] 5] 6] 9] ] whch ppe oe y poe o ce he y o e epeve pocee 2] 3] 2] o he eve e coo yhe 7] Th ppe copee he y o he -Hy pce To o ] wh he heoe ve Seco 2 e eo o Po Mhec Ioc Depe I Uvey POITEHNICA o Bche Ro e-: 2 ece Mhec Ioc Depe I Uvey POITEHNICA o Bche Ro

2 8 Ve Pepeţă Te Vche he o eo o he e covoo poc o o ve Seco 3 pove oe eho o o ee he o o ve -hy pce o A eee Me-oe ype veo o ehe wo ohe o e eve epecvey he Ree Theoe ve e ee expo A eee Expo Theoe o ve 2 Mpe -hy pce o We eoe y he e { 2 } Deo 2 A co : R Z C o e coocee o co o py o h he oow popee: < o < o oe o pecewe ooh o R o y Z M > σ R > ch h M expσ R 2 > Deo 22 o y o he co e e ce he -hy pce o -HT o he e o Soee we h e he oo 22 Deo 23 o β { } he β - o he o co he co

3 Mpe -hy pce o ] P II: Dee he o 9 β o < β S β ohewe β β β whee β β wh β 23 o α { p} { } R wh > α we eoe y D α he ce poc Dα ] D α α he p * * pe e p whee α α Theoe 24 Ieo o he o o y α { p} β { } S D β α α β 24 Poo e eoe y he co D α Sβ By Deo 23 o oe β o o oe α The β -eece o Sβ ee Deo 29] Δ β Sβ hece Δβ Dα By ev wh p epec p we o Δ β By ppy he p opeo y Theoe 22] h ey ecoe hece α β

4 2 Ve Pepeţă Te Vche ] β α Theoe 25 Ieo o he e I he oow pe pope e covee he 25 Poo e eoe y he co ee y he pe pope e 25 y o By ev oe o By ppy Th 222] wh p γ γ we e ] Theeoe 25 e o ] Theoe 26 Covoo o y o co ] 26 Poo Sce he o e oe o he e o eve he he eo o he -hy covoo Deo 23] c e we y epc y The

5 Mpe -hy pce o ] P II: Dee he o 2 exp ] By ch he oe o eo o o he y he che o ve o eo he che o he ce o o we e exp ] o whch ecoe ce e o eve e: ] ] exp exp ] Theoe 27 Poc o o o y o co σ < < σ Re R R < < we hve 2 ] 27 whee we e he oow oo:

6 22 Ve Pepeţă Te Vche Poo e eoe he pe pce o ee Deo 24] exp ] ] φ The 22 c e ewe 28 By Theoe 26] 28 oe o exp 29 whee ] ] By Me-oe o 2 25] 8 Ch III 72 ] eeo o pe pce o we e o σ > σ σ > exp 2 2 exp 2 exp We h e he oo o he pe e epecvey e exp The he pco o 2 y he eee Me-oe o ve

7 Mpe -hy pce o ] P II: Dee he o e e e A y he eee Me-oe o we o 2 ] 2 By epoy 28 2 oe o ce e e e 2 The 2 pe 2 2 ] ] We h e he oo o oo o

8 24 Ve Pepeţă Te Vche e E e he y o he vo e } { o Theoe 28 I ve o C wh 2 2 A E ] 22 Poo By Deo 24 Theoe 27] E Z 23 By Re 29] he e h ee o 23 e o We c we H ] Z whee H H e yc co o he o R > heeoe ] Z Sy 22 e y he 23 Theoe 29 ve I he oow ex he 24 Poo By Theoe 28 o 25 wh β γ ] we e ] ] Δ Z 25 whee Z e he -eo pce oo o he e h ee o 25 c e we ee

9 Mpe -hy pce o ] P II: Dee he o 25 whoe c e expee he o he eece o p y ec oe e we o Δ e 2 2 ] By he ve heoe o D pce -oo we hve ] Z ] By he 25 y ec hee e oe o 24 3 Meho o ee he o We coe he oow poe: ve co whch yc o o D 5 24] eee o co ch h ] y we h eh veo o o he -hy pce o Theoe 3 I he o o he 2 3 exp whee > σ > R Poo We h ee y co he oow D pce o Z-o :

10 26 Ve Pepeţă Te Vche exp ] 32 exp ] 33 2 ] ˆ Z 34 ˆ ] ˆ ˆ Z 35 Ovoy ˆ ] ˆ Z 36 By he o c 2 Γ o he coece o e ee c y he Me-oe o ppe o he pce o y o he coe coo Γ he cce we o o he e ee exp 2 37

11 Mpe -hy pce o ] P II: Dee he o 27 By epc 37 he ece Z -o oe o he veo o 3 Re 32 Ue ce coo poe y Jo' e he pe copex e 3 c e cce Ree Theoe Theoe 33 The o o ve y exp!! 2 38 whee ce h he e cce Poo I epce y 36 ecoe ˆ hece ˆ c e coee he e o he coece o he Tyo ee expo o o he o o he Tyo ee coece o co ey! c ve! ˆ S o c e oe y epc y o y we e!! c e oe y epc ve y 39 37

12 28 Ve Pepeţă Te Vche Now e e h he o epe wh epec o h whee ech he coo o Jo' e o e o h he po N ch h Re < By ppy he Ree Theoe o he e 38 we e: Cooy 34 I epe wh epec o he o!! e exp e eoe y δ he cee pe co δ y δ δ δ Z he co Aohe eho o ee he o ve y he oow expo heoe: Theoe 35 I he e h he e ee expo o y α β αβ α β he o h he Tyo ee expo αβ α δ β α β α! α! whee α α α N β β β N αβ α α β β C α α α β β β α α α α e α β e β Poo By Deo 26 2]

13 Mpe -hy pce o ] P II: Dee he o 29 α δ α! By ey oe o δ β ] β α α α! e δ α α ] α β β β α β e α α β αβ α β αβ α δ β] α! α! 4 Coco I h ppe ] copee heoy o pe -Hy pce oo h ee eveope I ee ppe ppco w e pove c oo o ee-eece e eo we he eecy-o epeeo o eo hy coo ye R E E R E N C E S ] B Dve Ie To The Appco Spe Ve 978 2] M Dyov h K ov E Roe DH Owe Expoe y o cee e epeve pocee I J Coo ] K ov E Roe DH Owe New 2D oe o x o cee e epeve pocee I J Coo ] K ov Se-pce Reo o e 2-D Sye wh Exeo o he ee D > 2 Ce ece Noe Coo Ioo Scece 263 Spe Ve oo 2 5] T Kcoe Cooy eey coo o 2D coo-cee e ye App Mh Cop Sc ] T Kcoe S wo-eo coo-cee e ye Dyc o Coo Dcee Ipve Sye ] J Ke MB Ze Ieve e coo yhe o 2D ye heoy IEEE T A Coo AC

14 3 Ve Pepeţă Te Vche 8] V O V Pepeţă Teo ţo cţ copexe ş pcţ E Şţcă ş Eccopecă 986 9] V Pepeţă Ce o echy o 2D coo-cee epe ye Rev Roe Mh Pe App ] V Pepeţă 2D Coo-Dcee pce Too Appco o 2D Sye Rev Ro Mh Pe App ] V Pepeţă Mpe -Hy pce Too Appco o Meo Hy Sye P I o ppe 2] E Roe DH Owe Sy Ay o e Repeve Pocee ece Noe Coo Ioo Scece 75 E Tho H Wye W Spe Ve Be 999

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