International Journal "Information Theories & Applications" Vol.10

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1 44 Inernaional Jornal "Informaion eories & Applicaions" Vol. [7] R.A.Jonson (994 iller & Frend s Probabili and Saisics for Engineers5 ediion Prenice Hall New Jerse 763. [8] J.Carroll ( Hman - Comper Ineracion in e New illennim AC Press NY. [9] A.Law and W.D.Kelon ( Simlaion odeling and Analsis 3d ediion cgraw-hill N.Y. Aor Informaion B. Dimirov - bdimiro@eering.ed D. Green Jr. - dgreen@eering.ed V.Rov - vrov@eering.ed P. Sancev - psance@eering.ed Deparmen of Science & aemaics Keering Universi Flin icigan USA 7 Wes ird Avene Flin icigan USA A GRADIEN-YPE OPIIZAION ECHNIQUE FOR HE OPIAL CONROL FOR SCHRODINGER EQUAIONS. H. FARAG Absrac: In is paper we are considered wi e opimal conrol of a scrodinger eqaion. Based on e formlaion for e variaion of e cos fncional a gradien-pe opimizaion ecniqe ilizing e finie difference meod is en developed o solve e consrained opimizaion problem. Finall a nmerical example is given and e resls sow a e meod of solion is robs. Kewords: Opimal conrol scrodinger eqaion Exisence and niqeness eor Gradien meod. AS sbec classificaion: 49J K. Inrodcion Opimal conrol of ssems governed b parial differenial eqaions is an applicaion-driven are of maemaics involving e formlaion and solion of minimizaion problems [3]. In is paper we are considered wi e opimal conrol of a scrodinger eqaion. Based on e formlaion for e variaion of e cos fncional a gradien-pe opimizaion ecniqe ilizing e finie difference meod is en developed o solve e consrained opimizaion problem. Finall a nmerical example is given and e resls sow a e meod of solion is robs.. Problem Formlaion We consider e fncional on e form ( J( α ( f ( α (l f ( wic is o minimized nder e condiions ( i B f(x (x ( x ( x ( 3 ( x x ( l ( ( l ( 4 x x over e class ( U :(x W ( α (x α α (x were α α α l B > are given nmbers

2 Inernaional Jornal "Informaion eories & Applicaions" Vol. 45 and f ( f ( W ( φ (x W ( l are given fncions. Definiion. e problem of finding e fncion (x V ( from condiion (-(4 a given U is called e redced problem. Definiion. A fncion (x V ( is said o be a solion of e problem (-(4 if for all η η (x W ( e eqaion η η ( 5 [ i B x x η ] dx l f ( x η dx i φ η ( x dx is valid and η ( x b η is e adoin of η. Proposiion Le f (x W ( and φ(x W (l. en e problem (-(4 as a niqe solion and saisfies e following esimae ( 6 C [ φ f ] V ( W ( l W ( l is valid and C > is dos no depend on φ and f. Proposiion Le φ (x W (l. en e solion of e redced problem (-(4 (x V ( belongs o e space W ( and saisfies e following esimae ( 7 C [ φ f ] W ( L (l W (l W ( is valid and [] C > is dos no depend on φ and f. Proposiion 3 Le all e condiions Proposiion be valid. en e opimal conrol problem (-(4 as a leas one solion. 3. Variaion of e Cos Fncional 3. e Adoin Problem Resls [4] impl a e fncion (x is a solion in L ( of e adoin problem ( 8 i B (x ( x ( x

3 46 Inernaional Jornal "Informaion eories & Applicaions" Vol. ( x x ( l ( α (9 [ ( f ( ] ( x B (l α [ (l f ( ] ( x B were (x is e solion of (-(4 corresponding o U. Definiion 3. For eac U a fncion (x; is a solion of e adoin problem (8-(9 belonging o e conrol iff (I ( x ; L ( (II e inegral ideni ( η η [ i B x η ] dx α [ ( f ( ] η ( l α [ ( f ( ] η ( is valid η W ( η (x ( η x x ( η x x l. On e basis of e above assmpions and e resls [5] we ave e following proposiion: Proposiion 4. e adoin problem (8-(9as a niqe solion from L ( and e following esimae ( L ( C 3 [ Γ Γ ] were Γ ( f ( Γ ( f ( W ( W ( is valid and C 3 is a cerain consan. 3. e Gradien Formlae of Cos Fncional e sfficien differeniabili condiions of e fncional (5 and is gradien formlae will be given as follows: eorem. Le e above assmpions be saisfied. en J ( is Gao differeniable and is gradien saisfies ( δ J ( Re ( ω dx ω W (. Proof : Sppose a U and δ W ( sc a δ U and denoing δ(x (x; δ (x;. en δ (x ;δ is e solion of e bondar vale problem:

4 Inernaional Jornal "Informaion eories & Applicaions" Vol. 47 δ δ ( 3 i B ( δ δ (x δ (x x δ( δ(l ( 4 δ(x x (l ( x x and e solion of e above bondar vale problem saisfies e following esimaion ( 5 δ C 4 δ W ( W ( were C 4 is a consan and independen of δ. From (5 and sing e eorem of imbedding [6] we ave ( 6 δ ( δ (l C5 δ (x L ( L ( W ( were C 5 is a consan and independen of δ. e incremen of e fncional J ( can be expressed as: (7 δj J( θ J( α Re α Re α δ (l [ (l [ ( L f ( f ( α ( ] δ(l ] δ( δ ( L ( If we ae complex adoin for ((3 we ave (8 η η [ i B x α [ ( l f ( ] α [ ( f ( ] (9 [ i δ (x B δ η δ x dx η ( l η ( η ] dx ( δ δ ] ηdx Sbracing (3 from (9 ( from (8 and in e obained relaion we p en we ave δ insead of η η

5 48 Inernaional Jornal "Informaion eories & Applicaions" Vol. ( α Re α Re [ δ [ ( [ δ δ ( f ( δ f ( ] dx δ δ ] dx Re δ dx Re δ B sbsiing e las relaion in (7 we ave [ ] δ (l ] δ ( δ dx. ( δj Re δ δ dx Re δ δ dx. α δ( α δ(l L( L( Sppose a ( R α δ ( δ (l L α ( L( (3 R Re δ δ dx. I is clear a (4 R α δ ( α δ (l. L ( L ( From e formlae of R i is esimaed as (5 R C δ δ L(. en 6 R R o ( δ W B sbsiing (6 in ( we obain (. ( ( 7 J( θ δ J( Re ( (θ ω dx d O (θ. Hence in lig of e variaion fncional we ave ( 8 J( θ J( δj(ω lim Re( ω dxd θ θ and is proves e differeniabili of e fncional and gradien formlae of e fncion J (. is complees e proof of e eorem. Using iinov meod [7] we define e following fncional ( 9 l J m m ( J ( α (x ω (x dx. and ω(x L (.

6 Inernaional Jornal "Informaion eories & Applicaions" Vol Discree Problem We consider e se of node vales { x } J l τ N N and e following noaions [8]: τ (3 (3 ( ( x τ ( ( x x x x x Afer appling e nmerical inegraion formla [8]we ave e discerisaion of e opimal conrol problem (-(5 as follows: Le i is reqired o minimize e fncional N (3 I m ([] τ {α [ f ] α [ f ] } N ν m τ { ω ω ω } on e conrol se [] :[ ] ( α α U N ( α N nder e condiions (33 i ( B ( x x f N N ( 34 B ( 35 ( i [ ( x f ] N ( 36 B ( i [( x f ] N Now e discree gradien formlae will be given as follows: eorem e fncional J m ( is differeniable and is gradien saisfies (37 (I / ([ ] Re ( α m ( ω m were N and is e solion of discree adoin problem: (38 i ( B ( x x N ( 39 N ( 4 ( α [ i ( x [ f ] ] N B B

7 4 Inernaional Jornal "Informaion eories & Applicaions" Vol. ( 4 ( α [ i ( [ f ] ] N x B B 5. Solion of Conrol Problem 5. e Proecion Gradien eod Here we describe e proecion gradien meod [9] for e solion of e opimal conrol problem sc as: consrc a seqence n m b seing ( 4 [ ] l n m P [ ] nm ν n ( I m ([ ] nm U N were P ( U N is e proec on e se U N. In e firs we define ( nm in e form ( 43 were ( n m Ψ α α α Ψ Ψ Ψ < > α α α ( 44 Ψ [ ] nm ν n ( I l m ([ ] nm ( 45 Ψ [ ] nm ν n ( I l m ([ ] nm and N n... m... Using e above seqence we consrc e proec in e form ( 46 ( n m ( n m ( 47 ( n m Θ Θ Θ Θ ( ( Ψ nm nm Θ < Θ > Θ were Θ Θ ( n m N νn ( I l m ( τ α n m n... ([]nm Θ ( τ α n m m Nmerical Algorim Wi e gradien obained e following gradien pe algorim can en be developed for e opimal vale of * based on e proecion gradien meod (PG wic described in e above secion. e olined of e algorim for solving conrol problem are as follows: Sep : Coose an iniial conrol (n U n. (n If I / ( (n is e solion of e problem. Sep : A eac ieraion n do

8 Inernaional Jornal "Informaion eories & Applicaions" Vol. 4 Solve e sae problem en find (. (n. Solve e adoin problem for (-(3 en find (n (n (.. Find opimal conrol * sing PG. End do. (n Sep 3: es e opimali of. (n If is opimm sop e process. Oerwise go o Sep 4. Sep 4 Se (n (n n n and go o Sep. 6. Nmerical Resls Designed algorim is implemened as a FORRAN roine []. Nmerical experimen is carried o o cec is performance. e iniial daa of e problem (-(5 are aen as follows: α α α l ε.5e 3 ( f i f i ( φ (x i x. x ω (x f(x i (x (x e nmber of division of e inervals was aen as N. e comped conrol vales of 3 N e vales of relaive error are sown in ables and e 3D plos of e opimal conrol and iniial vales are presened in Figres. e opimal vale of e cos fncional is J* inf U J( J(*.4856 E 3. e comped conrol vales of 3 N.559E.595E.63E.664E.7E.7464E.774E.8E.833E.86E.9E.983E.679E.474E.55E.837E.365E.4368E.478E.434E.478E e vales of relaive error of 3 N

4 Inernaional Jornal "Informaion eories & Applicaions" Vol. Fig.. Opimal conrol (x Fig.. Iniial conrol (x REFERENCES [] A. J. Kearsle e se of opimizaion ecniqes in e solion of parial differenial eqaions from science and engineering PD Hosen exas996.

9 4 Inernaional Jornal "Informaion eories & Applicaions" Vol. Fig.. Opimal conrol (x Fig.. Iniial conrol (x REFERENCES [] A. J. Kearsle e se of opimizaion ecniqes in e solion of parial differenial eqaions from science and engineering PD Hosen exas996. [] K. Kime Finie difference approximaion of conrol via e poenial in a -D Scrodinger eqaion Jornal of Differenial eqaions (6--(. [3] N. U. Amed and K. L. eo Opimal conrol of disribed parameer ssems Nor Holland New Yor Oxford(98. [4] S. H. Farag and. H. Farag Necessar opimali condiions in conrol problems for perbolic eqaions Jornal Egp. a. Soc. 8(--(. [5] O. A. Ladzensaa Bondar vale problems of maemaical psics Naa oscow Rssian(973. [6] V. P. iailov Parial differenial eqaionsnaa oscow Rssian(983. [7] A. N. ionov and N. Ya. Arsenin eods for e solion of incorrecl posed problems Naa oscow Rssian(974. [8] A. H. Kaer A. B. Samardan. H. Farag and A. H. Abel-Hamid Analical and nmerical solions of a qasilinear parabolic opimal conrol problem Jornal Comp. and Appl. a (998. [9] J.. Bes Srve of nmerical meods for raecor opimizaion Jornal of Gidance conrol and dnamics (93--7(998. [] D. Kraf Algorim 773:OP-Forran modles for opimal conrol calclaions AC ransacions of maemaical sofware (6--8(994. Aor Informaion. H. FARAG : Deparmen of aemaics and Comper Science; Facl of Edcaion Ibri Slanae of Oman. or: Facl of Science El-inia Universi Egp. farag5358@aoo.com

Riemann Function and Methods of Group Analysis

Riemann Function and Methods of Group Analysis American Research Jornal of Mahemaics Original Aricle ISSN 378-74X Volme Isse 3 5 Riemann Fncion and Mehods of Grop Analsis Akimov Andre Chernov Igor Abdllina Rfina 3 4533 Serliamak Rssia Lenina sree 47A