Regional Controllability of Semi-Linear Distributed Parabolic Systems: Theory and Simulation

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1 Inelligen Conol an omaion,,, hp://xoiog/46/ica7 Pblishe Online May (hp://wwwscirpog/jonal/ica) Regional Conollabiliy of Semi-Linea Disibe Paabolic Sysems: heoy an Simlaion smae Kamal, li Boolo, Sii hme Ol Beinane SI op, Depamen of Mahemaics an Compe Sciences, Facly of Sciences, Molay Ismail Univesiy, Meknes, Moocco Depamen of Mahemaics, College of Sciences, l Jof Univesiy, Sakakah, KS {askamal, booloali}@yahoof, beinane6@gmailcom Receive Mach, ; evise May, ; accepe May, BSRC he aim of his bief pape is o give seveal esls concening he egional conollabiliy of isibe sysems govene by semi-linea paabolic eqaions We concenae on he eeminaion of a conol achieving inenal an bonay egional conollabiliy he appoach is base on an exension of he Hilbe Uniqeness Meho (HUM) an Schae s fixe poin heoem We give a nmeical example evelope in inenal an bonay sb egion hese nmeical illsaions show he efficiency of he appoach an lea o conjeces Keywos: Semi-Linea Paabolic Sysems; Regional; Inenal/Bonay Conollabiliy; Fixe-Poin heoems; Disibe Sysem; HUM ppoach Inocion Many scienific an engineeing poblems can be moele by paial iffeenial eqaions, inegal eqaions, o cople oinay an paial iffeenial eqaions ha can be escibe as iffeenial eqaions in infinie-imensional spaces sing semi gops Nonlinea inegoiffeenial eqaions, wih an wiho elays, seve as an absac fomlaion fo many paial inegoiffeenial eqaions which aise in poblems connece wih hea flow in maeials wih memoy, viscoelasiciy, an ohe physical phenomena In paicla, Sobolev-ype eqaions occ in hemoynamics in he flow of fli hogh fisse ocks, in he shea of secon-oe flis, an in soil mechanics So, he sy of conollabiliy esls fo sch sysems in infinie-imensional spaces is impoan Fo he moivaion of absac sysems an he conollabiliy of linea sysems, one can efe o he books by Cain an Picha [], an by Cain an Zwa [] Fo an ealie svey on he conollabiliy of nonlinea sysems sing fixe-poin heoems, incling nonlinea elays sysems, see [] he appoximae conollabiliy of nonlinea sysems when he semigop geneae by is compac has been sie also by many ahos he esls of Zho [4] an Naio [5] give sfficien coniions on B wih infinie-imensional ange o necessay an sfficien coniions base on moe sic assmpions on B Li an Yong [6] sie he same poblem assming he appoximae conollabiliy of he associae linea sysem ne abiay pebaion in L I, L Bian [7] invesigae he appoximae conollabiliy fo a class of semi-linea sysems, [8] se he Banach fixe-poin heoem o obain a local exac conollabiliy in he case of nonlineaiies wih small Lipschiz consans Zhang [9] sie he local exac conollabiliy of semi-linea evolions sysems Naio [5] an Seimann [] se he Schae fixe-poin heoem o pove he invaiance of he eachable se ne nonlinea pebaions Klamka [-] sie sfficien coniions fo consaine exac conollabiliy in a pescibe ime ineval fo semi-linea ynamical sysems in which he nonlinea em is coninosly Feche iffeeniable ae fomlae an pove assming ha he conols ake vales in a convex an close cone wih veex a zeo he meho se coves a wie class of semi-linea absac ynamical sysems an is specially sefl fo semi-linea ones wih elays Balachanan an Sakhivel [4] sie he conollabiliy of semilinea inegoiffeenial sysems in Banach spaces by sing he Schaefe fixe-poin heoem Fabe e al [5] p pove appoximae conollabiliy in L fo p < by means of a conol which can be inenal o on he bonay an when he nonlineaiy is globally Lipschiz Ohe elae absac esls wee given by Zaza [6], Lasiecka e al [7] an Kassaa e al [8] he sy of vaios analyic conceps elae o conollabiliy an sabiliy of sch sysems is, in geneal, Copyigh SciRes

2 KML E L 47 elicae an consieing only linea moel can no be sfficien in paicla when some popeies of sysem nees o be saisfie only in some pa of he sysem evolion omain Fom pacical poin of view, i is vey naal o consie he analysis of sch sysems only in some sbegion of is evolion sysem omain his is he aim of egional analysis he egional analysis of isibe paamee sysem has ecieve an inensive sy in he las hee ecaes he em egional analysis has been se o efe o conol poblems in which he age of o inees is no flly specifie as a sae, b efes only o a smalle egion of he sysem omain his concep has been wiely evelope an ineesing esls have been obaine, in paicla, he possibiliy o each a sae only on an inenal sbegion of (El Jai e al [9]) o on a pa of he bonay of (Zeik e al []) he pincipal eason fo inocing his concep is ha, fis i makes sense fo he sal conollabiliy concep close o eal wol poblem an, secon, i can be applie o sysems which ae no conollable on he whole omain Hee we ae ineese on egional conollabiliy of semi-linea paabolic sysems Moe pecisely he qesion concens he possibiliy of egional conollabiliy fo semi-linea sysem in he case whee he esie sae is given only on an inenal sbegion of o on a pa of he bonay of he inees of his wok focse on he evelopmen of an appoach ha leas o nmeical implemenaion fo he compaion of he conol which sees he sysem fom an iniial sae o a given egional inenal an bonay sae ypical moivaing example is he case of a biological eaco, whee he poblem is o eglae he concenaion of a ssbsam a he boom of he eaco [] In Secion, fis we pesen some peliminay maeial an sae inenal egional conollabiliy poblem of semi-linea sysems Nex, we concenae on he eeminaion of a conol achieving egional inenal conollabiliy, an we evelop a nmeical appoach ha leas o a sefl algoihm an sccessflly ese hogh a iffsion pocess Secion is focse on he egional bonay age conol poblem, an an appoach is evelope ha leas o a nmeical algoihm fo he compaion of a conol which achieves egional bonay conollabiliy Nmeical illsaions show he efficiency of he appoach an lea o conjeces Regional Inenal Conollabiliy Saemen of he Poblem Le be a egla bone open se of IR n, n, wih bonay Fo a given ime >, le [, ], an [, ] We consie a semi-linea paabolic sysem excie by conols which can be applie via vaios ypes of acaos given by he following eqaion y x, y x, Ny x, B y, () y x y x, whee is a secon-oe linea iffeenial opeao, which geneaes a songly coninos semi-gop S on Hilbe space L an N a locally lipschiz coninos nonlinea opeao B IR p, L, y L an U whee p U L, ; IR y L ; whee p epesens he nmbe of acaos We enoe by U he compleion of he space U enowe wih he sana nom of L, ; Denoe by y he solion of () when i is excie by a conol, sppose ha y L, ; L Le s ecall ha an acao is convenionally efine by a cople D, f, whee D is he geomeic sppo of he acao an f is he spaial isibion of he acion on he sppo D, see [] In his case, B D f x In he case of poinwise acao (inenal o bonay), D b an f b, whee b is he Diac mass concenae a b; in his case, he acao is enoe by b, b an B x Le y b be he solion of () excie a conol an assme ha y L, ; L, see [] Fo, open, nonempy an of posiive Lebesge mease, we consie he opeao esicion : L L y y an enoes he ajoin opeao Definiion Sysem () is sai o be -exacly egionally con- ollable if fo all y L, hee exiss a conol U sch ha y y Sysem () is sai o be -appoximaely egionally conollable if fo all y L an fo all, hee exiss a conol U sch ha y y L he noion of egional conollabiliy consiee as a paicla case of op conollabiliy was inoce an evelope fo linea sysem in (El Jai e al [9], Zeik e al []) I is clea ha: If sysem () is egionally conollable on hen i is egionally conollable on any Copyigh SciRes

3 48 KML E L In he linea case, one can fin saes which ae appoximaely egionally conollable on b no conollable on he whole omain, see [9,] o sy he conollabiliy of he sysem (), we consie is coesponing linea sysem [N in ()], y x, yx, B y, () y x y x, he poblem of egional conollabiliy on fo () can be sae as follows: Poblem Fo y a esie sae, fin a conol () U sch ha y y Moe pecisely, i is aske o fin a conol which sees sysem (), a ime, o a esie sae efine in sbegion Hilbe Uniqeness ppoach he aim of his secion is o give an exension of egional conollabiliy an Hilbe niqeness meho inoce in he linea case by (El Jai e al [9]) an [4] which allows he chaaceizaion of a conol solion of () he sysem () is appoximaely conollable in an sysem () is excie by a zone acao D, f Sysem () may be ewien in he fom y x y x Ny x f x y y,,, D, x, y x (4) an he opeao veify N : L, ; L L, ; L c x L, ; L L, ; L, N x c c sch ha on \ Le gl g x, x,, x, x, (5) (6) Which has a niqe solion L, ; L see [5] Fo a given, we consie he sysem (6) an efine he mapping f, L D (7) Which is a nom on ; since he sysem () is appoximaely conollable in Consie he sysem an he associae linea sysem an y x, y x, Ny x,, f f x L D D y, yx, y x (8) y x, y x,, f f x L D D y, (9) yx, y x he sysem (8) may be ecompose in he following hee sysems x, x,, x, yx () Copyigh SciRes

4 KML E L 49 whee x, x,, f f x L D D, x, yx is he solion of (6) an x, x, N, x, yx () () We enoe he compleion of he se wih espec o he nom (7) again by Le : be efine by P whee P Now, wih he nonlinea opeao P P given by (4) Skech of he poof: he poof may be easily achieve wih he following wo seps: Sep : We pove ha K is a compac opeao an hen ece ha K is also compac Sep : pplying he Schae fixe-poin heoem, we see ha he opeao K has a fixe poin Fo moe eails we efe he eae o [6] he poblem of egional conollabiliy () ns p o solve he eqaion y P which is Remak eqivalen o he above appoach is a genealizaion of he Hilbe P P niqeness meho given in he linea case N an y P he opeao coincies wih he isomophism y KP lgoihm Smming p, in he zone case, he egional conollabiliy is obaine via he following simplifie algoihm whee K : is he opeao efine by he fomla K P hen we have y K P Sep : We ake he following iniial coniions, () y, f, D an he linea sysem (9) is appoximaely egionally Sep : Using he pseo-coe conollable in, hen is one o one see [9] Resolion of (6) an obaining pply he eqaion (), we have Resolion of () an obaining y K P Resolion of () an obaining Now, we efine he nonlinea opeao K Resolion of () an obaining by Calclaion of an obaining K K y K P Resolion of K (4) an obaining Unil K hen he poblem () of sysem (7) ns p o Sep : he conol, f L D seach a fixe poin of K, hen we have Poposiion Simlaions ssme ha (5) hols If he linea sysem (9) is appoximaely egionally conollable in, hen he he goal of his secion is o es he efficiency of he pevios algoihm he obaine esls ae elae o conol, f sees he sysem (8) o L D he consiee sbegion, he esie sae an he acy in a, whee is he solion of he ao sce Le,,, an consie he sysem (6) an is a fixe poin of he opeao K one-imensional iffsion sysem escibe by,, y y x x B y,, [,] [, ] j y j j x j y, y, [, ] yx, [,] (5) Copyigh SciRes

5 5 KML E L Zone cao In his case B D x whee D, 7 he sbegion ne consieaion is,58 Le y xxxx9x4 be he esie egional sae in Using he pevios algoihm, he simlaion gives he Fige he egional esie sae is eache wih eo an ansfe cos y y y 76 L 57 Poinwise cao In his case B b whee b 67 We consie he sbegion 8,68 Le y x xxx9 x4 be he esie egional sae on he simlaion gives he Fige he egional esie sae is eache wih eo an ansfe cos y L y y Relaion beween he Sbegion an Locaion of he Poinwise cao he following simlaion esls show he evolion of he esie sae eo wih espec o he acao locaion Fige shows ha: Fo a given sbegion an a esie sae, hee is an opimal acao locaion (opimal in he sense ha i leas o a solion which is vey close o he esie sae) When he acao is locae sfficienly fa fom he sbegion, he esimae sae eo is consan fo any locaion he wos locaions coespon o non saegic ac- aos in,, as evelope in he linea case see [9] Fige 4 shows ha, fo a given sbegion an a esie sae, hee is an opimal acao locaion in he sense ha i leas o a smalle ansfe cos he esls ae simila fo ohe ypes of acaos Fige Desie sae (coninos line) an final sae (ashe line) on he egion ω Fige Desie sae (coninos line) an final sae (ashe line) on he egion ω Regional Bonay Conollabiliy he aim of his secion is o give an exension of he conceps of egional inenal conollabiliy [6] o he case whee is a pa of he bonay of he omain he evelope meho is oiginal an leas o a nmeical algoihm illsae by simlaions Consiee Sysem an Poblem Saemen Le be a bone open omain in IR n (n =,, ) wih a egla bonay Fo, we wie Fige he evolion of he esimae sae eo wih espec o he acao locaions Copyigh SciRes

6 KML E L 5 Fige 4 he evolion of he ansfe cos wih espec o he acao locaions,,, an consie he following semi-linea paabolic sysem y x, yx, Nyx, B y, () yx, y x whee is a secon-oe linea iffeenial opeao, which geneaes a songly coninos semi-gop S on Hilbe space H N a locally lipschiz coninos nonlinea opeao p BIR,, y U p U L, ; IR y wih whee y be he solion of () excie by a conol We enoe by U he compleion of he space U en- p owe wih he sana nom of L, ; ssme ha y L, ; he conols may be applie via vaios ypes of acaos see [] he associae linea sysem is y x, y x, B y, () yx, y x Fo Γ being a egla sbse of which has posiive Lebesge mease, consie he esicion opeao : H H y y whee enoes is ajoin opeao Le s : H H whils is consiee fo he ajoin opeao We inoce he efiniion Definiion he sysem () is sai o be -exacly (esp - appoximaely) egionally conollable if fo all (esp fo all ol U sch ha y H ) hee exiss a con- y y H y y (esp ) his efiniion genealizes he sana ones of exac an appoximae conollabiliy on he whole omain Remak ) he noion of egional conollabiliy consiee as a paicla case of op conollabiliy was inoce an evelope fo linea sysem in [] ) sysem which is -exacly (esp -appoximaely) egionally conollable is -exacly (esp - appoximaely) egionally conollable fo all ) he above efiniions o no allow fo poinwise o bonay conols since, fo sch 4) sysems B IR p, an he solion y L Howeve, he exension can be caie o in a simila manne if one akes egla conols sch ha y H [7] In he seqel, we exploe he possibiliy of fining a conol which enses he ansfe of sysem () o esie y on he bonay sbegion, consie he poblem Fo y esie sae, fine a conol () U sch ha y y heoeical ppoach Fisly, he following esl povies a link beween egional inenal conollabiliy see [6] an egional bonay conollabiliy fo semi-linea sysems Consie he linea an coninos exension opeao R: H H sch ha Rg g, fo all Fo g H exension of y y H, we enoe by y o an we efine D Ry H y H V sppry y H he Le inege small, we se F B z, an z Copyigh SciRes

7 5 KML E L F V, whee Bz, is he open ball of ais an cene z, see [8] hen, we have he following esl,,, D Poposiion If he sysem () is -exacly (esp -appoximaely) egionally conollable, hen i is -exacly (esp -appoximaely) egionally conollable Poof Ry, Le y H H hen by ace heoem, hee exiss wih a bone sppo sch ha Ry y Since he sysem () is -exacly conollable, hen hee exiss a conol U sch ha hs y y y Ry is -exacly conollable Now, if he sysem () is -appoximaely conollable, fo all, hee exiss U sch ha an hen y y Conseqenly, he sysem () Ry y H an by coniniy of he ace mapping, we have heefoe y H y R y y H Conseqenly, he sysem () is -appoximaely conollable Seconly, we evelop an appoach evoe o chaaceize a conol solion of poblem (), when he sysem () is -appoximaely conollable he appoach we shall se is base on an exension of egional conollabiliy echniqes fo linea sysems evelope in (El Jai e al [9]) an Hilbe niqeness meho see [4] he sysem () is excie by a conol applie by means of a zone acao D, f whee D is he acao sppo an f L D efines he spaial isibion of he conol on D, hen he sysem () may be wien in he fom y x y x Ny x f x y, (4) yx, y x he opeao N : L, ; L, ; veify N x c x, L, ; L, ; (5) c c, is a consan Le be he se g sch ha g in \ Fo y H, we enoe by y y o y x, y x, Ny x,, f L D D f x y, yx, y x he exension of Consie he sysem z (6) z y whee is he Laplace opeao he sysem (6) has a niqe solion z in H Le z he esicion of z in H he poblem of eaching y on may hen be solve by eaching z on hen he poblem () is fomlae as follows: Fo z H, a esie sae, fin a conol (7) U sch ha y? z Fo, he sysem x, x,, (8) x, x has a niqe solion L, ; C, ; L [8] In, we efine he mapping, f L D (9) which is a nom on ; since he sysem is -appoximaely conollable see [9] Consie he sysem / () Copyigh SciRes

8 KML E L 5 an is associae linea sysem is whee K : K y P, which gives x, yx,, f L D D f x y z K P, () y x y x, he sysem () may be ecompose ino he following hee sysems an, x,, x, y x, x, x x, x,, f whee is he solion of (8) an, x, x, x, N L D D () f x () (4) We eno e he compleion of he se wih espec o he nom (9) again by Consi e he opeao : efine by P whee is he al of an P Le s now efine he nonlinea opeao which is eqivalen o P P he poblem of egional conollabiliy () ns p o solve he eqaion z P z K P P z P P is he opeao efine by (5) Since he linea sysem () is -appoximaely egionally conollable, hen is one o one see (El Jai e al [9]) pply he eqaion (5), we have z K P hen a solion of poblem () of sys em () ns p o seach a fixe poin of nonlinea opeao K efine by K z K P (6) hen we have: Poposiion If he linea sysem () is -appoximaely egionally conollable, hen he conol, f ives he sys em () o y L D in a, whee is he solion of he sysem (9) an is a fixe poin of he opeao K given by (6) Poof Sep : We pove ha K is a compac opeao B B, p in, we have Le he ball p an we se K B P B, p p,, K B P B p p Whee is solion of he sysem (4) We have S N (7) C, ; see [] an hee exiss c sch ha P c Since S is a songly coninos semi-gop on,, hen hee exiss M sch ha an fom (7), we have,, L, S M Copyigh SciRes

9 54 KML E L S N i i Mc Mc Since is solion of he sysem (), hen S y an we have M y (8) Since is solion of he sysem (), hen we have an hen hs S s s L D, f fs L D M f s s M f L D M f L D (9) M c y M c f L D Mc D By onwall s lemma, we obain M c y M c f L D e Mc hen sp p P c M c y M c p f Hence, p Le show ha p L D e Mc () K B is nifomly bone K B is elaively compac, inee: fo an h, we have h ShSN i i h S h N B whee an B B S h S N i i h B S h N Fo all, hee exiss sch ha h L, S h S,, which gives fom (8), (9) an (), we have an hs B S h S N i i c c c Mc L D B c M y M f Mc e Mc Mc e e L D B McM y Mc cm f Mc P h P whee cc M y M p f Mc e Mc L D Copyigh SciRes

10 KML E L 55 an c Mc M y M p f e Mc L D Mc Fo an Inf is B p C, ;, we obain PhP hen, K B p is elaively compac Finally, by he zelà-scoli heoem see [9,], K : Bp is a comp ac opeao, hen K : Bp is also compac Sep : Fom (6) an (), we have whee an K y P K M 4 y P 4 he consan c veify 4 Mc cm c f e L, L D Mc cm e L D L, c f I is being se he fac ha c, is small Le s p sch ha s, hen we have K s sch ha s Hence, by applying Schae s fixe poin heoem see [], he opeao K a leas one fixe poin, an he poof is complee lgoihm Wih he same hypohesis as in he las secion, we have he following algoihm Sep : we choose he iniial coniions, sbegion,, y, an he fncion f, D an Sep : sing he pseo-coe Resolion of (8) an obaining Resolion of (), () an (4) Calclaion of an obaining K Resolion of K an obaining Unil K o, f Sep : he con l Nmeical Example L D In his sbsecion, we pesen a nmeical example which illsae he pevios algoihm I shows ha hee exiss a link beween he sbegion aea an he eache sae eo, he esls ae elae o he choice of he sbegion an he esie sae o be eache Consie he wo-imensional iffsion sysem y y x, x, B x y, ij y, ij ij, i, j y y,,, yx, Zone cao We consie he acao is locae in D 6,57 8,58 () [,] [,], [,] [,] : Inen sbegion age [,4] : Bonay sbegion age z x y y : he esie sae o be 6 eache in y x x x 9 y y : he exension of esie sae y on Using he pevios algoihm in he case zone acao we have Figes 5-8 Using he pevios algoihm, he egional esie sae y is obaine wih eo an cos y 5 y L 4, Fige 5 Desie sae on he egion ω Copyigh SciRes

11 56 KML E L Relaion beween he Sbegion ea an Reache Sae Eo he eache sae eo epens on he aea of he sbegion whee he esie sae has o be given his eo gows wih he sbegion aea I means ha he lage he egion is, he geae he eo is (see able ) he esls ae simila fo ohe ypes of acao Fige 6 Final sae on he egion ω Poinwise cao In his case, we have he acao is locae in b b, b wih b = 6, b 65,,,,, 5 Inen sbegion age, 5 : Bonay sbegion age z x y y : he esie sae o be 5 eache in y x x x y y : he 5 5 exension of esie sae y on Using he pevios algoihm in he case poin wise acao we have Figes 9- able he elaion beween he sbegion aea an eache sae eo Region Fige 7 Desie an final sae on he egion ω Fige 8 ace of esie an final sae on he egion Γ Copyigh SciRes y y f L, , , , , 4 4 Fige 9 Desie sae on he egion ω

12 KML 57 E L 4 Conclsions Fige Final sae on he egion ω he wok is povie an ineesing ool o achieve egional inenal an bonay age fo a semi-linea paabolic sysem excie by acao he poblems of egional conollabiliy ae solve sing linea egional conollabiliy echniqes an by applying HUM meho an fixe poin heoems he obaine esl leas o an algoihm which was implemene nmeically Examples of vaios siaions an simlaions ae given Vaios open qesions ae sill ne consieaion Fo example, his is he case of he poblem whee we es his algoihm fo eal applicaions his case is pesenly being sie an he esls will appea in a sepaae pape he poblem of egional conollabiliy poblem fo semi-linea paabolic sysems wih ime elays is of gea inees an he wok is ne consieaion an will be he sbjec of he feae pape REFERENCES Fige Desie an final sae on he egion ω [] R F Cain an J Picha, Infinie-Dimensional Linea Sysems heoy, Lece Noes in Conol an Infomaion Sciences, Spinge Velag, Belin, 978 [] R F Cain an H Zwa, n Inocion o Infinie Dimensional Linea Sysems heoy, Spinge Velag, Belin, 995 [] K Balachanan an J P Dae, Conollabiliy of Nonlinea Sysems via Fixe-Poin heoems, Jonal of Opimizaion heoy an pplicaions, Vol 5, No, 987, pp 45-5 oi:7/bf9894 [4] H Zho, ppoximae Conollabiliy fo a Class of Semilinea bsac Eqaions, SIM Jonal on Conol an Opimizaion, Vol, No 4, 98, pp oi:7/ [5] K Naio, ppoximae Conollabiliy fo ajecoies of Semilinea Conol Sysems, Jonal of Opimizaion heoy an pplicaions, Vol 6, No, 989, pp oi:7/bf98799 [6] Li an J Yong, Opimal Conol heoy fo InfinieDimension a Sysems, Bikhase, Basel, 994 [7] W M Bian, Consaine Conollabiliy of Some Nonlinea Sysems, pplicable nalysis, Vol 7, No -, 999, pp 57-7 oi:8/ [8] N Camichael an M D inn, Fixe-Poin Mehos in Nonlinea Conol, Lece noes in Conol an Infomaion Sciences, Spinge Velag, Belin, 984 [9] Zhang, Exac Conollabiliy of Semilinea Evolion Sysems an Is pplicaion, Jonal of Opimizaion heoy an pplicaions, Vol 7, No,, pp 454 oi:/:64687 [] I Seimann, Invaiance of he Reachable Se ne Nonlinea Pebaions, SIM Jonal on Conol an Opimizaion, Vol 5, No 5, 985, pp 7-9 oi:7/564 Fige ace of esie an final sae on he egion Γ Copyigh SciRes [] J Klamka, Consaine ppoximae Conollabiliy,

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