AN UNCERTAIN CAUCHY PROBLEM OF A NEW CLASS OF FUZZY DIFFERENTIAL EQUATIONS. Alexei Bychkov, Eugene Ivanov, Olha Suprun

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1 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, AN UNCERAIN CAUCHY PROBLEM OF A NEW CLASS OF FUZZY DIFFERENIAL EQUAIONS Alexei Bychkov, Eugee Ivaov, Olha Supru Absrac: he cocep of fuzzy percepio roamig process is eered A ew iegral is buil by he fuzzy percepio roamig process Properies of his iegral are sudied he ew class of differeial equalizaios is acquired A heorem of exisece ad uiqueess of a soluio for he ew class of fuzzy percepio differeial equaios are proved Keywords: possibiliy-heoreical approach, dyamic processes, sof modelig, fuzzy percepio processes, Cauchy problem ACM Classificaio Keywords: G17 Ordiary Differeial Equaios Iroducio I is impora o aswer a quesio of how exac he resul of desig is a o-crisp supervisio whe we deal wih mahemaical modelig of complex sysems A sochasic approach is he mos widespread o describe a o-crisp a he complex sysem desig However, such descripio i probabiliy erms is uaural for he uique pheomea I appears ha he mos suiable for his are he ideas of possibiliy heory Firs, he heory of possibiliy was mos fully preseed by Dubois D Developme of ideas [Dubois, 1988] is offered i his aricle here are differe approaches o formalize o-crisp dyamics [Aubi, 199]-[Buckley, 1992], [Friedmaa, 1994]-[Park, 1999], [Seikkala, 1987] I his aricle ew approaches o adequae descripio ad aalysis of ocrisp ad ucerai iformaio ad dyamics of objecs are proposed Mehods of possibiliy heory allow o esimae a eve ruh wih respec o oher eves ad o ake io accou a subjecive exper opiio For example i is very impora for progosicaio of he social-ecoomic pheomea, for medical diagosic asks, for mahemaical modelig of huma hikig process ad oher processes Prelimiaries Le ( X, A ) be a measurable space Accordig o [Piiev, 2], [Bychkov, 25], [Bychkov, 26] we cosider some defiiios A ew oaios, are offered i [Piiev, 2] We shall use hem i he furher Defiiio 1 Possibiliy scale is a semi-rig L {[ 1, ],,, }, ie [,] 1 segme wih usual order ad wo operaios: a b max{ a, b} ;

2 14 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 ab mi{ a, b} Heceforh we cosider oly A -measurable fucios f : X L Le s deoe as LX ( ) a class of fucios ha saisfies he followig codiios: f L( X), al af mi( a, f( x)) L( X) ; fg, LX ( ) f gmax( fx ( ), gx ( )) LX ( ); f g mi( f( x), g( x)) L( X); f L( X) f 1 f( x) L( X) ; If a sequece of fucios f 1,, f, L( X), he f ( ) ( ) x L X ; 1 f ( ) ( ) x L X 1 Defiiio 2 Fucio p : LX ( ) L : p(( af)( ) ( bg)( )) ( a p( f( ))) ( b p( g( ))) ; p f() pf() ; 1 1 If f ( ) 1, pf (()) 1, is called a possibiliy measure Le s oe ( X) - se of all subses X, - characerisic fucio of A A Defiiio 3 Le s defie he fucio P : ( X) L i he followig maer: PA ( ) p ( A ()) his fucio is called a possibiliy of crisp eve A he le s call a riple ( X, A, P ) a possibiliy space We will use a erm fuzzy percepio (from percepio la), i fuure, ha will o coradic wih erm Zadeh L fuzzy Defiiio 4 Le ( X, A, P ) be a possibiliy space Le s call a A -measurable fucio A : X ( Y) as fuzzy percepio se Characerisic fucio of fuzzy percepio se A 1 is calculaed as ( y) Py { Ax ( )} PA ( ( y)) A Defiiio 5 If A is a fuzzy percepio se, le s call a umber PA ( ) p ( A ()) as a possibiliy of fuzzy percepio se A he defiiio implies he followig properies: PA ( B) PA ( ) PB ( ) max( PA ( ), PB ( )); P( AB) P( A) P( B) mi( P( A), P( B)) Defiiio 6 If P( AB) P( A) P( B) he eves are called idepede Defiiio 7 Codiioal possibiliy of eve A i respec o eve B is he soluio P{ A B } of he equaio

3 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, mi PA ( B), PB ( ) PA ( B) heorem 1 [Bychkov, 27] I is possible o exed possibiliy P o all he subses of se X (le s call he exeded possibiliy P: ( X) L ) ad for all A A PA ( ) PA ( ) his heorem allows o o build he eclosed sequece of sigma algebras for he correc cosrucio of fuzzy differeial equaio soluio Defiiio 8 Wih a give possibiliy space ( X, A, P ) ad a measurable space ( Y, B) a fuzzy percepio variable is a ( AB, ) - measurable fucio : X Y heorem 1 implies ha we may defie a fucio ( y) P{ y} ha is called disribuio of a fuzzy percepio variable he possibiliy ha a fuzzy percepio variable will fall io se B ca be expressed by disribuio: P{ B} sup ( y) yb Defiiio 9 Fuzzy percepio variables ad are idepede if disribuio of percepio vecor (, ) equals o ( uv, ) mi{ ( u), ( v)} Mai resuls Modelig he ucerai dyamics Le s give some defiiios Defiiio 1 [Bychkov, 25] Fuzzy percepio variable (scalar or vecor) is ormal if is disribuio equals o ( u) ( u u ), where ( x) decreasig fucio, ha s specified for x such as ( x), ( ) 1 x Defiiio 11 [Bychkov, 25] Fuzzy percepio process is a fucio ( x, ): X R Y Defiiio 12 [Bychkov, 25]Normal fuzzy percepio process () is a process of fuzzy percepio roamig if he followig assumpios exis: Uder idepede icremes, ie for all momes of ime fuzzy percepio variables ( 2) ( 1) ad ( 4) ( 3) are idepede

4 16 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 Uder fixed is rasie possibiliy is P () x ( ) x ( ) 12 / 2 ( x x ) Defiiio 13 A mode (or modal value) of a fuzzy percepio variable is a poi i which disribuio fucio equals o 1, ie mod u; ( u) 1 Defiiio 14 -cu of fuzzy percepio variable is he followig se: [ ] y : P{ y}, ( 1,] A aleraive way of specifyig fuzzy disribuio is o specify is -cus Defiiio 15 [Piiev, 2] Le s cosider a semi-rig L [,]; 1 ; ;, where a b if b a ; a bmi( a, b) ; a b max( a, b) We ca defie a ecessiy measure f (()): LX ( ) L o L i he same maer as o L he for ay crisp se A ( X ) ecessiy of se A is NA ( ) ( A ()) For ay fuzzy percepio se ecessiy is NA ( ) ( A ()) Defiiio 16 [Bychkov, 26] Sequece of fuzzy percepio variables coverges o fuzzy percepio variable wih ecessiy 1 (N1lim ) if for each x X for which P{ x}, ( x ) ( x ) I his case N { } 1 Defiiio 17 Sequece of fuzzy percepio variables coverges o fuzzy percepio variable by possibiliy (Plim ) if for ay P c c If ay of hese limis exiss i is uique Plim implies N1lim (ulike probabiliy heory) Lemma 1 If N1lim ad for arbirary ( 1,] sequece ( x ) is coverge o ( x ) o he se [ X] { xx : P({ x}) } uiformly by x, he Plim Ad coversely, if Plim he for ay ( 1,] sequece ( x ) coverges o ( x ) uiformly by x o [ X ] Proof By defiiio of uiform covergece, for all ad c a idex (, c) N exiss wih followig propery: if (, c) ad P({ x}) he ( x ) ( x ) c Possibiliy ha differece bewee ad will exceed c may be esimaed as

5 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, P c sup : (, c) (, c) Le s prove ha ( c, ) Le s assume he corary: ( c, ) whe he such value exiss ha for ay sequece of idexes k holds he followig: k (, c), ha s impossible, because (, c) is fiie he direc lemma is prove Le s prove he coverse lemma Le s assume he corary: such c ad exis here ha for ay umber k, here exis a eleme of possibiliy space x ad a umber k, for which P{ x} ad ( x) ( x) c We foud such sequece k k covergece by possibiliy k ha P c P({ x}), ie here s o k Defiiio 18 [Bychkov, 25] Assume ha a piecewise-cosa fucio f : f () { yk : k k 1}, k N 1,, N is give, w() is a scalar process of fuzzy percepio roamig Deoe N 1 fdw () () yk( w ( k1) w ( k)) k his fuzzy percepio variable is called a iegral by a process of fuzzy percepio roamig of piecewisecosa fucio Lemma 2 For piecewise-cosa fucio f() he followig equaliy exiss: 1 1 f() dw() f() d ( ), f() d ( ) Proof Le s use mahemaical iducio For N 1 he equaliy is rue by defiiio of process of fuzzy percepio roamig Le s assume ha he equaliy is rue for N, ad prove i for N 1 Fuzzy percepio variables N fdw () () ad N 1 fdw () () are idepede herefore N N1 N N1 fdw () () fdw () () fdw () () N N N 1 1 f () d ( ), f () d ( )

6 18 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 N 1 N1 1 1 f () d ( ), f () d ( ) N N N1 N1 1 1 f () d ( ), f () d ( ) Corollary 1 Le s assume ha a sequece of piecewise-cosa fucios f () i he mea he Plim f() dw() his corollary implies he followig heorem: heorem 2 [Bychkov, 25] A measurable fucio f() is give ad wo sequeces of piecewise-cosa fucios f () ad f () coverge o i i he mea If he limi Q Plim f () dw() also exiss, ad Q Q Q Plim f () dw() exiss he he limi heorem 2 proves correcess of he followig defiiio: Defiiio 19 [Bychkov, 25] Assume ha a measurable fucio f() is give ad a sequece of piecewisecosa fucios f() coverges o i i he mea ( f () f() d ) If he sequece f () dw() coverges by possibiliy le s call is limi as a iegral of a measurable fucio by he process of fuzzy percepio roamig ad fdw () () Plim f() dw () Defiiio 2 Sequece of fucios a ( x ) is uiformly fudameal by x if for arbirary he idepede from x ( ) exiss ad such ha whe m : a ( x) a ( x) m Lemma 3 If a sequece of fucios a ( x ) is uiformly fudameal i is uiformly coverge

7 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, Proof Le s deoe lim a ( x) a( x) From uiform fudamealiy for ay such exiss ha for m a( x) am( x) occurs Hece, for all m 2 ( ) [ ( ) 2, ( ) a 2 ] m x a x a x ad ( ) [ ( ) 2, ( ) a x a 2 ] x a x herefore, ax ( ) am( x) ha proves uiform covergece Lemma 4 For crisp iegrable fucio f() -cus of he iegral also equal o 1 1 f() dw() f() d ( ), f() d ( ) Proof As we kow from fucioal aalysis such sequece of piecewise-cosa fucios f() exiss ha f() f () f () ad 1 2 f () d f() d, furhermore f () f() almos everywhere I such a case f () f() i he mea Le s prove he exisece of he iegral Le s fix a arbirary x X ad cosider sequece f () dw(, x ), he f () dw(, x ) f () dw(, x ) f () f () dw(, x ) m m Sice f () f() i he mea f () f () i he mea ad from Corollary 1 m m, Plim f () f () dw() Nlim 1 f () f () dw() m m m, m, herefore lim f() fm() dw(, x ) So, he sequece f () dw(, x ) is fudameal ad m, hece i is coverge We proved ha N1lim f( ) dw(, x) Q( x) exiss Le s prove ha Plim exiss By Lemma 3 for ay x X, for which P({ x}) ad m from choice of f ( x ) also

8 11 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, () m() (, ) () m() ( ) f f dw x f f d f f d 1 () ( ) Holds, ie sequece f () dw(, x) is uiformly fudameal by x Accordig o Lemma 3 his sequece uiformly coverges by x By Lemma 1 Plim f( ) dw( ) exiss We have prove ha all Lebesgue iegrable fucios are also iegrable by process of fuzzy roamig However, he disribuio of his iegral is ukow Le s calculae i By Lemma f() dw() f() d ( ), f() d ( ) I his segme icreases wih, is limi equals o I I f d f d 1 1 () ( ), () ( ) Le s deoe closure of I as I Obviously fdw () () I We should prove ha i is exacly I For breviy we deoe 1 (, ) f () d ( ) Mf Mf (, ) Mf (, ) Because For arbirary such 1 exiss ha for 1 f() dw() [ M( f, ); M( f, )], ha for each 1 such x X exiss ha P({ x}) ad f () dw( x,) M( f, ) By Lemma 1 (coverse par) such exiss ha for each f () dw( x,) f () dw( x,) idepedely of x I meas ha for ay Mf (, ) 3, Mf (, ) 3 fdw ( ) ( ) Mf (, ) fdw () () for ay sufficiely small, herefore I f() dw() Because P f() dw( x,) M(, f ) Hece, P f() dw( x,) M(, f ) ad he eds of I segme also belog o cu

9 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, heorem 3 For a fuzzy percepio process (, ) which has iegrable pahs ad is idepede from wx (, ), holds he followig: (, xdwx ) (, ) 1 1 sup ( x, ) d ( ), sup ( x, ) d ( ) xp : { x} xp : { x} Proof From idepedece of processes wx (, ), (, x) follows ha: Pwx { (, ) A ( x, ) B} Pwx { (, ) A} herefore, process { wx (, ) ( x, ) B} is a process of fuzzy percepio roamig ideical o wx (, ) Fixig ( x, ) ad usig Lemma 4 coclude proof Le s cosider a iiial-value problem yx (, ) y( x) ays ( ( ), sds ) bys ( ( ), sdwsx ) (, ), (1) y (, x) y (2) he soluio of he problem (1), (2) is a fuzzy percepio process y( x, ) ha urs (1), (2) io equaliies for each x X, for which P({ x}) Mai resuls A Cauchy problem for ew class of differeial equaios heorem 4 Le crisp fucios a( y, ) ad b( y, ) be coiuous by ad saisfy Lipschiz codiio by y i he regio DI[, ], I [ y1, y2], ie ay (, ) az (, ) Ly z, by (, ) bz (, ) Lyz for all xy, I, [, ] ; iiial value y ( ) x is ay fuzzy percepio value

10 112 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 If for fixed x X, for which P({ x}) he codiio [ y( x) y; y( x) y] I holds, he for his x he problem (1), (2) has a uique soluio i he segme [ ; h], where 1 1 y h mi k mi 1,,,, k ( 1, ) 2L ( ) max ax ( ) ( ) max bx ( ) Proof Le s fix x X, for which P({ x}) he pah of he fuzzy percepio process w() ad he iiial value y deped o i Le s prove ha for give x, y( x ) he uique soluio exiss i [, h] Le s cosider such space of all coiuous fucios f (), [, h] ha f ( ) y( x) Le s defie a disace i his space: (, fg) max f () g () [ ; ] We cosider he followig mappig: (()) f y ( x ) a(( f s), s) ds b( g( s), s) dw(, s x ) Le s prove ha i s coracig Lipschiz codiio implies ( ( f), ( g)) max a( f( s), s) a( g( s), s) ds b( f( s), s) b( g( s), s) dw( s, x ) [, h] max af s, sags, sds bf s, sbgs, sdws, x [, h] Lh 1 dw s x f g Lh 1 w h x w x f g h (, ) (, ) ( ( (, ) (, ))) (, ) Because of P{ x}, w hx w x h, where ( y ):[ ; ) [,] 1 is he 1 (, ) (, ) ( ) fucio ha correspods o disribuio of possibiliies of he process w () We obai: f, g 1 2 Lh 1 h (, f g) L max h, 1 h 2 (, f g)

11 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, So, wih h k mi 1,, k ( 1, ), he mappig is coracig ad accordig o a coracio 2 L 4 ( ) priciple i has a uique fixed poi Obviously, ay fixed poi of mappig is a soluio of iiial-value problem for give x A las, le s obai codiios for h, which provide ha he soluio will remai wihi he domai D Firs of all y h he wih h we ca esimae y(, x ): 1 max ax ( ) ( ) max bx ( ) y y a y( s), s) ds b y( s), s dw( s) 1 hmax a max b, ie wih hese assumpios y() I If he fucios a( y, ) ad b( y, ) are defied i a fiie domai D i is possible ha for ay h such x X exiss ha P({ x}) ad i he ime segme [, h] for x x he soluio of (1), (2) will go beyod he bouds of D I oher words, for arbirary small h i is impossible o rely o a soluio exisece for all possible x X i [, h] segme Le s prove he heorem ha is a sufficie codiio of a soluio exisece for all heorem 5 If fuzzy percepio fucios a( y,, x ) ad b( y,, x ), where y, R, x X arbirary x X are coiuous by ad saisfy local Lipschiz codiio by y o R, ie for arbirary segme I, for ayx (,, ) azx (,, ) LI ( ) y z, byx (,, ) bzx (,, ) LI ( ) y z for, xy I, where LI () is a fiie value ha depeds o chose segme I, ad growh codiio 2 2 1, byx (,, ) K1 y 2 2 ayx (,, ) K y, he for ay fuzzy y he problem (1),(2) has a uique soluio for [ ; ) Proof Crispess or fuzziess of fucios a ad b does ifluece he proof

12 114 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 Like i heorem 4, we fix x X, for which P({ x}) Le s ake arbirary grea ad prove ha o he segme [, ] soluio of he problem (1), (2) exiss Alhough heorem 4 proves ha he soluio exiss for [, h], we ca exed his soluio beyod hese limis Ideed, le s cosider he followig problem yx (, ) y ( hx, ) ays ( ( ), sds ) bys ( ( ), sdwsx ) (, ) h h Accordig o heorem 4, his problem has a soluio o [ h, h] Subsiue y( h, x) wih h1 h1 y ( h, x) y (, x) ays ( ( ), sds ) bys ( ( ), sdwsx ) (, ) 1 ad obai ha y(, x ) y(, x ) a( y( s), s) ds b( y( s), s) dw( s, x ) Hece, we go a soluio of he sysem o [, h h] segme his operaio ca be repeaed as may imes as eeded We mus show ha he soluio will o ru o ifiiy for fiie Assume ha he soluio of he sysem exiss o he segme [, ] Esimae y(, x ): y(, x) y (, x) ays ( ( ), s) ds bys ( ( ), s) dwsx (, ) Growh codiio implies ha y(, x ) y(, x ) z(, x ) y(, x ), (3) where zx (, ) is a soluio of crisp Cauchy problem zx (, ) yx (, ) ( ) Kzsx (, ) ds his problem has a soluio for all hus, havig fixed x, we fix he segme I [ y(, x) 2; y(, x) 2 ], where z (, x) y (, x) Accordig o heorem 4, he soluio of (1), (2) exiss o [, h ], where 1 1 h k mi 1, Exed his soluio o [ 2 LI ( ) 4 1 ; 2h ], ec, uil we cover he whole ( ) segme, or cross he bouds of I Le s assume ha we cross he bouds for some 1 For his 1 (3) does hold ad ha s impossible heorem 4 esures uiqueess of he soluio, ha cocludes he proof

13 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, Coclusios I his aricle, we cosider he problem of modelig o crisp dyamics A ew formalism for he desig of ucerai dyamics differeial equaio by fuzzy percepio roamig processes is offered Correcess of cosrucio of his equaio is show ad a Cauchy problem heorem is proved Local Lipschiz codiio provides uiqueess of he soluio; growh codiio esures ha i wo ru o ifiiy for fiie Bibliography [Aubi, 199] J-PAubi Fuzzy Differeial Iclusios // Problems of Corol ad Iformaio heory, 199 Vol 19, No1, pp [Buckley, 2] JJBuckley, hfeurig Fuzzy Differeial Equaios // Fuzzy Ses ad Sysems, 2 Vol11, pp [Buckley, 1992] JJBuckley, YQu Solvig fuzzy equaios: a ew soluio cocep // Fuzzy Ses ad Sysems 1992 Vol 5, pp 1-14 [Bychkov, 25] AS Bychkov A Cosrucig of Iegral by Process of Fuzzy Roamig // Bullei of he Uiversiy of Kiev, Series: Physics & Mahemaics, 25 No 4, pp19-24 (i Ukraiia) [Bychkov, 26] AS Bychkov, MG Merkuriev Exisece ad Uiqueess Soluio of Fuzzy Differeial Equaios // Bullei of he Uiversiy of Kiev, Series: Physics & Mahemaics, 26 No 1, pp27-34 (i Ukraiia) [Bychkov, 27] AS Bychkov Abou of possibiliy heory ad is applicaio // Repors of Naioal Academy of Scieces of Ukraie, 27 No 5, pp 7-12 (i Ukraiia) [Dubois, 1988] DDubois, HPrade héorie des possibiliés: applicaio à la représeaio des coaissaces e iformaique Masso, Paris, 1988 [Friedmaa, 1994] YFriedmaa, USadler, A New Approach o Fuzzy Dyamics // Proc of he 12h IAPR Ieraioal Coferece, Jerusalem, Israel, 1994 [Kaleva, 199] O Kaleva he Cauchy Problem for Fuzzy Differeial equaios // Fuzzy Ses ad Sysems, 199 Vol 35, pp [Kloede, 1982] P EKloede Fuzzy Dyamical Sysems // Fuzzy Ses ad Sysems, 1982 Vol 7, pp [Lelad, 1995] RPLelad Fuzzy Differeial Sysems ad Malliavi Calculus // Fuzzy Ses ad Sysems, 1995 Vol 7, No1, pp [Park, 1999] J Yeoul Park, H Keu Ha Exisece ad Uiqueess heorem for a Soluio of Fuzzy Differeial Equaios // Ieraioal Joural of Mahemaics ad Mahemaical Scieces, 1999 Vol 22, No2 pp [Piiev, 1989] YuP Piiev Mahemaical Mehods of Experime Ierpreaio High school, Moscow, 1989, (i Russia) [Piiev, 2] YuP Piiev he possibiliy he elemes of heory ad applicaios URSS, 2 (i Russia) [Seikkala, 1987] S Seikkala O he Fuzzy Iiial Value Problem // Fuzzy Ses ad Sysems, 1995 Vol 24, pp

14 116 Ieraioal Joural "Iformaio Models ad Aalyses" Volume 4, Number 2, 215 Auhors' Iformaio Alexei Bychkov Head of he Deparme of programmig ad compuer egieerig, Faculy of Iformaio echology, aras Shevcheko Naioal Uiversiy of Kiev; Posal Code 322O, 81A Lomoosova Sr, Kyiv, Ukraie; Major Fields of Scieific Research: aalysis of hybrid auomaa as a model of discreecoiuous processes, cosrucio of a cohere heory of opporuiies percepive fuzzy variables ad processes, mahemaical foudaios of fuzzy modelig of complex sysems Eugee Ivaov Assisa of he Deparme of programmig ad compuer egieerig, Faculy of Iformaio echology, aras Shevcheko Naioal Uiversiy of Kiev; Posal Code 322O, 81A Lomoosova Sr, Kyiv, Ukraie; ivaoveuge@gmailcom Major Fields of Scieific Research: Semaics of programmig laguages, formal mehods, Mahemaical sysems heory, Hybrid (discree-coiuous) sysem Olha Supru Associae Professor of he Deparme of programmig ad compuer egieerig, Faculy of Iformaio echology, aras Shevcheko Naioal Uiversiy of Kiev, Posal Code 322O, 81A Lomoosova Sr, Kyiv, Ukraie; osupru@gmailcom Major Fields of Scieific Research: mahemaical modelig ad compuaioal mehods, fuzzy variables ad processes, hybrid models of discree-coiuous processes