REACHABILITY OF FRACTIONAL CONTINUOUS-TIME LINEAR SYSTEMS USING THE CAPUTO-FABRIZIO DERIVATIVE

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1 ECHLY OF FCONL CONNUOUS-ME LNE SYSEMS USNG HE CPUO-FZO DEVVE usz Kczor iłyso Uivrsiy o chology Fculy o Elcricl Egirig Wijs 45D, 5-5 iłyso E-il: czor@isppwupl KEYWODS Frciol, coiuous-i, lir, sys, Cpuo- Fbrizio iiio, rchbiliy SC h Cpuo-Fbrizio iiio o h rciol riviv is ppli o lysis o h posiiviy rchbiliy o coiuous-i lir syss Ncssry suici coiios or h rchbiliy o sr posiiv rciol coiuous-i lir syss r sblish NODUCON yicl sys is cll posiiv i is rjcory srig ro y ogiv iiil coiio s ris orvr i h posiiv orh or ll ogiv ipus ovrviw o s o h r i posiiv sys hory is giv i h oogrphs Fri ili ; Kczor i h pprs Kczor 997, 998, b, 4, 4b, 5b Mols hvig posiiv bhvior c b ou i girig, cooics, socil scics, biology ici, c h posiiv sr scripor syss hir sbiliy hv b lyz i Kczor 997, 998,, b, 4b, 5b h posiiv lir syss wih ir rciol orrs hv b rss i Kczor b, h scripor iscr-i lir syss i Kczor 998 Dscripor posiiv iscr-i coiuous-i olir syss hv b lyz i Kczor 4 h posiiviy lirizio o olir iscr-i syss by s-bcs i Kczor 4b Nw sbiliy ss o posiiv sr rciol lir syss hv b ivsig i Kczor h sbiliy robus sbilizio o iscr-i swich syss hv b lyz i Zhg, Xi, Zhg Wg 4; Zhg, H, Wu Hug 4 Miiu rgy corol o D syss i Hilbr spcs hs b lyz i Kl 98 Corollbiliy o yicl syss hs b ivsig i Kl 96; Kl 99, 997, 998 cly w iiio o h rciol riviv wihou sigulr rl hs b propos i Cpuo Fbrizio 5; Los Nio 5 his ppr h Cpuo-Fbrizio iiio o h rciol riviv will b ppli o lysis o h rchbiliy o h sr posiiv lir syss h ppr is orgiz s ollows scio cssry suici coiios or h rchbiliy o rciol sr coiuous-i lir syss r sblish Ncssry suici coiios or h posiiviy o h rciol syss suici coiios or h rchbiliy o h posiiv syss r propos i scio Cocluig rrs r giv i scio 4 h ollowig oio will b us: - h s o rl ubrs, - h s o rl rics, - h s o rics wih ogiv ris, M - h s o - h iiy ri Mzlr rics, ECHLY OF SNDD FCONL SYSEMS h Cpuo-Fbrizio iiio o rciol riviv o orr o h ucio or < < hs h or Cpuo Fbrizio 5; Los Nio 5 D p, & &, Cosir h rciol iril s quios D u, < <, y C Du, b whr, u, ipu oupu vcors p y r h s,,, C, D hor h soluio o h quio or giv iiil coiio ipu u hs h or Procigs h Europ Corc o Mollig Siulio ECMS hors Clus, Fr Hrr, Michl Miz, Olivr os Eiors SN: / SN: CD

2 u whr [ [ [ u u&, u u, [ βu u&,, β, -, L { s }, b Proo h proo is giv i Kczor 5 Diiio s o h sr sys is cll rchbl i i, i hr iss ipu [ u or, which srs h s [ o h sys ro zro iiil coiio o h il s vry s is rchbl i i, h h sys is cll [ rchbl i i, h sys is cll rchbl i or vry ipu [ hr iss u or, which srs h s o [ h sys ro o hor h sr rciol sys is rchbl i i, i oly i h ri [ 4 is ivribl h ipu which srs h s o h sys ro o is giv by u β u u Proo Subsiuig, [, 5 h soluio o h iril quio 6 or u u hs h or u β u o show h h ipu u, [, 8 9 srs h s ro o i i [, w subsiu 9 io 7 w obi Subsiuig 9 io 8 w obi 5 Fro hor is proo ollows h corollry Corollry h rciol sys is rchbl i i, i oly i h rciol sys [ u is rchbl i i, [ h ipu u srs h s ro o i i, o h sys i oly i h [ ipu 8 srs h s ro o i i, o h sys [ Epl Cosir h rciol sys scrib by h quio wih 5, zro iiil coiio, u,, - prr Copu h ipu u which srs h sys ro o [ os rspos i i [, Usig b w obi u β u u& 6 io or, u w obi u 7 [ ,

3 [ b ig io ccou h h igvlus o h ri r, usig h Sylvsr orul w obi 4 Usig 4 or 4, b w obi h ri 5 is osigulr sic or by hor h rciol sys wih is rchbl i i [, or h ipu srig h sys ro o [ i i [, is giv by [ 6 u β 6 For pl or w obi u 7

4 Ncssiy L u, i i-h colu o h iiy ri h rjcory os o lv h orh oly i D wh i iplis ij or i j, i, j,, M h D u sic u his iplis is rbirry Fro b or u, w hv y C C sic is rbirry ssuig ro b w hv y Du D sic u is rbirry L,,, r h igvlus o h ri h h igvlus o h ri [ r giv by [ Figur : pu sigl s vcor or [s ECHLY OF POSVE FCONL SYSEMS Cosir h rciol sys Diiio h rciol sys is cll irlly posiiv i h s vcor h oupu vcor ll ipus p y u Diiio rl ri, or ll iiil coiios, u ij &, [ is cll Mzlr ri i is o-igol ris r ogiv, i or i j ; i, j,, L L or ij M < < h 8 Proo h proo is siilr o h o giv i Kczor hor h rciol sys is posiiv i oly i M,, C, D 9 Proo Suicicy M w hv h ro, sic by L, u, u &, Proo is wll-ow Gchr 959 h i is wll-i o h spcru,,, o h ri h h igvlus o h ri r giv by,,, his cs [ L h ri or < < is sypoiclly sbl i oly i h ri is sypoiclly sbl Proo h igvlus,,, o h ri r rl wih h igvlus,,, o h ri by,,, sic h chrcrisic polyoils o h rics r rl by h quliy [ [ [ hror, ro i ollows h <,,, i oly i <,,, L 4 h ri [ M is sypoiclly sbl i oly i h igvlus jβ,,, o h ri sisy h coiio

5 [ β > Proo Fro or jβ jβ,,, w hv j β [ [ jβ j β [ [ β jβ [ β [ [ β [ β β j [ [ β [ β [ [ β,,, jβ 4 5 Fro 5 i ollows h >,,, i oly i >,,, L 5 h rics [ [ i M M, is sypoiclly sbl Proo h ri ri M 6 [ i h is sypoiclly sbl Kczor hror, by L or < < 5 hols i M is sypoiclly sbl Fro L 4 hor w hv h ollowig hor 4 h rciol sys is posiiv i M is sypoiclly sbl C, D Diiio 4 s, o h posiiv sys is cll rchbl i i, i hr iss ipu u [ or, which srs h s [ o h sys ro zro iiil coiio o h il s vry s is rchbl i i, h h sys is cll rchbl i [ i, h posiiv sys is cll [ rchbl i or vry ipu u hr iss or, which srs h s o [ h sys ro o Diiio 5 ri is cll ooil i i ch row i ch colu oly o ry is posiiv h riig ris r zro hor 5 h posiiv rciol sys is rchbl i i, i h ri [ 7 is ooil h ipu which srs h s o h sys ro o is giv by β u 8 Proo is wll-ow Kczor h i oly i h ri is ooil siilr wy s i proo o hor i c b show h h ipu 8 srs h s o posiiv sys ro o i i [, Fro 8 i ollows h u sic β > or β >, < <, Epl Coiuio o Epl No h h ri giv by 5 is ooil oly or 6 hror, w co sy yhig bou h rchbiliy o h posiiv sys wih i i [, or CONCLUDNG EMKS h Cpuo-Fbrizio iiio o h rciol riviv hs b ppli o lysis o h posiiviy rchbiliy o coiuous-i lir syss Ncssry suici coiios or h rchbiliy o sr coiuous-i lir syss hv b sblish hor Ncssry suici coiios or h posiiviy o h rciol lir syss hv b giv hors 4 Suici coiios or h rchbiliy o h rciol posiiv lir syss hv b lso sblish hor 5 h cosirios r illusr by uricl pls o sr posiiv rciol lir syss

6 h cosirios c b o coiuousiscr lir syss CKNOWLEDGEMEN his wor ws suppor by Niol Scic Cr i Pol ur wor No 4///S7/467 EFEENCES Cpuo M Fbrizio M 5 Nw Diiio o Frciol Driviv wihou Sigulr Krl Progr Frc Dir ppl, Vol, No, - Fri L ili S Posiiv Lir Syss J Wily, Nw Yor Gchr F 959 h hory o Mrics Chls Pub Cop, Loo Kczor 997 Posiiv sigulr iscr i lir syss ull Pol c ch Sci, Vol45, No4, 69-6 Kczor 998 Posiiv scripor iscr-i lir syss Probls o Nolir lysis i Egirig Syss, Vol, No7, 8-54 Kczor Posiiv D D Syss Sprigr- Vrlg, Loo Kczor Nw sbiliy ss o posiiv sr rciol lir syss Circuis Syss, Vol, No4, 6-68 Kczor b Posiiv lir syss cosisig o subsyss wih ir rciol orrs EEE rs Circuis Syss, Vol58, No6, - Kczor Slc Probls o Frciol Syss hory Sprigr-Vrlg, rli Kczor 4 Dscripor posiiv iscr-i coiuous-i olir syss Proc o SPE, Vol99, oi:7/74558 Kczor 4b Posiiviy lirizio o clss o olir iscr-i syss by s bcs Logisy, Vol6, Kczor 5 lysis o posiiv sbl rciol coiuous-i lir syss by h us o Cpuo- Fbrizio riviv Subi o EEE rs Circuis Syss Kczor 5b Posiiviy sbiliy o iscr-i olir syss Proc o CYCONF Kl E 96 O h grl hory o corol syss Proc o h irs r Cogrss o uoic Corol, Loo, Kl J 98 Miiu rgy corol o D syss i Hilbr spcs Syss Scic, Vol9, No-, -4 Kl J 99 Corollbiliy o Dyicl Syss Kluwr cic Prss, Dorrch Kl J 997 Corollbiliy o -D syss: survy J ppl Mh Copu Sci, Vol7, No4, - Kl J 998 Cosri corollbiliy o posiiv -D syss ull Pol c ch Sci, Vol6, No, 95-4 Los J Nio J 5 Propris o w rciol riviv wihou sigulr rl Progr Frc Dir ppl, Vol, No, 87-9 Zhg H; Xi D; Zhg H; Wg G 4 Sbiliy lysis or iscr-i swich syss wih usbl subsyss by o-p vrg wll i pproch S rscios, Vol5, 8-86 Zhg H; H Z; Wu H; Hug J 4 obus sbilizio o iscr-i posiiv swich syss wih ucriis vrg wll i swichig Circuis Sys Sigl Procss, Vol, 7-95 UHO OGPHES DEUSZ KCZOEK rciv h MSc, PhD DSc grs i lcricl girig ro h Wrsw Uivrsiy o chology i 956, , rspcivly h yrs h ws h o h Elcricl Egirig Fculy, i h prio o 97 7 h ws puy rcor o h Wrsw Uivrsiy o chology 97 h bc prossor i 974 ull prossor h s uivrsiy Sic h hs b prossor h iłyso Uivrsiy o chology 986 h ws lc corrspoig br i 996 ull br o h Polish cy o Scics h yrs h ws h ircor o h srch Cr o h Polish cy o Scics i o 4 h ws lc hoorry br o h Hugri cy o Scics H ws gr hoorry ocors by uivrsiis His rsrch irss covr syss hory, spcilly sigulr uliisiol syss, posiiv uliisiol syss, sigulr posiiv D D syss, s wll s posiiv rciol D D syss H iii rsrch i h il o sigulr D, posiiv D posiiv rciol lir syss H publish 8 boos 8 i Eglish ovr sciiic pprs H lso suprvis 69 PhD hss H is h ior-i-chi o h ulli o h Polish cy o Scics: chicl Scics br o ioril bors o iriol jourls