In this paper, we are considered with the optimal control of a schrodinger equation. Based on the 
formulation for the variation of the cost functional, a gradient-type optimization technique utilizing the finite 
difference method is then developed to solve the constrained optimization problem. Finally, a numerical 
example is given and the results show that the method of solution is robust

Farag, M.

Bulgarian Digital Mathematics Library at IMI-BAS

International Journal "Information Theories & Applications" Vol.10  414 [7] R.A.Johnson (1994) Miller & Freund’s Probability and Statistics for Engineers,5th edition, Prentice Hall, New Jersey 07632. [8] J.Carroll (2002) Human - Computer Interaction in the New Millennium, ACM Press, NY.  [9] A,M.Law and W.D.Kelton (2000) Simulation Modeling and Analysis, 3d edition, McGraw-Hill, N.Y. Author Information B. Dimitrov - bdimitro@kettering.edu  D. Green, Jr. - dgreen@kettering.edu  V.Rykov - vrykov@kettering.edu  P. Stanchev - pstanche@kettering.edu  Department of Science & Mathematics, Kettering University, Flint, Michigan, USA 1700 West Third Avenue, Flint, Michigan 48504-4898, USA    A GRADIENT-TYPE OPTIMIZATION TECHNIQUE FOR THE OPTIMAL CONTROL FOR SCHRODINGER EQUATIONS  M. H. FARAG Abstract: In this paper, we are considered with the optimal control of a schrodinger equation. Based on the formulation for the variation of the cost functional, a gradient-type optimization technique utilizing the finite difference method is then developed to solve the constrained optimization problem. Finally, a numerical example is given and the results show that the method of solution is robust. Keywords: Optimal control, schrodinger equation, Existence and uniqueness theory, Gradient method. AMS subject classification: 49J20, 49M29, 49M30, 49K20   1.  Introduction Optimal control of systems governed by partial differential equations is an application-driven are of mathematics involving the formulation and solution of minimization problems [1,3]. In this paper, we are considered with the optimal control of a schrodinger equation. Based on the formulation for the variation of the cost functional, a gradient-type optimization technique utilizing the finite difference method is then developed to solve the constrained optimization problem. Finally, a numerical example is given and the results show that the method of solution is robust. 2.  Problem Formulation We consider the functional on the form dtT02)t(1f)t,l(y1αdtT02)t(0f)t,0(y0α)u(J)1( ∫ −+∫ −=  which is to minimized under the conditions )T,0(x)1,0(Ω)t,x(,)t,x(fyu2xy20Btyi)2( =∈=−∂∂+∂∂  )l,0(x,0)0,x(y)3( ∈=  )T,0(t,0x)t,l(yx)t,0(y)4( ∈=∂∂=∂∂ over the class  ⎭⎬⎫⎩⎨⎧ ∈∀≤≤≤∈= Ω)t,x(,2αtu,1α)t,x(u0α,)Ω(1,02W)t,x(u:uU  where 00B,T,l,02α1α,2,0k,0kα >≠+=≥ are given numbers  International Journal "Information Theories & Applications" Vol.10  415and  )l,0(12W)x(φ,)T,0(12W)t(1f,)t(0f ∈∈ , are given functions.  Definition 1. The problem of finding the function  )Ω(1,02V)t,x(y ∈  from condition (2)-(4) at given Uu∈ is called the reduced problem.  Definition 2. A function )Ω(1,02V)t,x(y ∈  is said to be a solution of the problem (2)-(4), if for all )Ω(1,12W)t,x(ηη ∈=    the equation ( )dx)0,x(ηΩl0φidtdxη)t,x(fΩdtdx]ηyuxηxy0Btηyi[5∫ ∫+=∫ −∂∂∂∂−∂∂− is valid and  ,0)T,x(η =  but η   is the adjoint of  η  .  Proposition 1 Let )Ω(1,02W)t,x(f ∈ and )l,0(12W)t,x(φ ∈ . Then the problem (2)-(4) has a unique solution and  satisfies the following estimate  ]2)l,0(Wf2)l,0(Wφ[C2)Ω(Vy)6( 1,021210,12+≤ is valid and 01C >  is dos not depend on φ  and f .  Proposition 2 Let )l,0(22W)t,x(φ ∈ . Then the solution of the reduced problem (2)-(4) )Ω(1,02V)t,x(y ∈   belongs to the space  )Ω(1,22W and satisfies the following estimate   ]2)Ω(Wf2)l,0(Wφ[C2 )l,0(Lty2)Ω(Wy)7( 1,02222222+≤+ is valid and 02C,]T,0[t >∈∀  is dos not depend on φ and f .  Proposition 3 Let all the conditions Proposition 2 be valid. Then the optimal control problem (1)-(4) has at least one solution. 3.  Variation of the Cost Functional 3.1  The Adjoint Problem Results [4] imply that the function  )u,t,x(ΦΦ=  is a solution in  )Ω(2L  of the adjoint problem )T,0(x)1,0(Ω)t,x(,0Φu2xΦ20BtΦi)8( =∈=−∂∂+∂∂ International Journal "Information Theories & Applications" Vol.10  416 )T,0(t],)t(1f)t,l(y[0B1α2x)t,l(Φ)T,0(t],)t(0f)t,0(y[0B0α2x)t,0(Φ)l,0(x,0)T,x(Φ)9(∈−=∂∂∈−−=∂∂∈= where )t,x(y  is the solution of (1)-(4) corresponding to Uu ∈ .  Definition 3. For each Uu ∈  , a function )u;t,x(Φ  is a solution of the adjoint problem (8)-(9) belonging to the control  u  iff (I) )Ω(2L)u;t,x(Φ ∈  , (II) The integral identity  dt)t,0(1η])t(0f)t,0(y[T00α2dt)t,l(1η])t(1f)t,1(y[T01α2dtdx]1ηu2x1η20Bt1ηiΩ[Φ)10(−∫+−∫−=−∂∂+∂∂∫ is valid )Ω(1,22W1η ∈∀ , .0lx|x)1η(0x|x)1η()0,x(1η =====    On the basis of the above assumptions and the results [5], we have the following proposition:  Proposition  4. The adjoint problem  (8)-(9)has a unique solution from )Ω(2L  and he following estimate   ,]2Γ1Γ[3C2)Ω(LΦ)11( 2+≤  where  2)T,0(W)t(1f)t,1(y1Γ,2)T,0(W)t(0f)t,0(y1Γ212212−=−= is valid and 3C  is a certain constant. 3.2  The Gradient Formulae of Cost Functional The sufficient differentiability conditions of the functional (5) and its gradient formulae will be given as follows:   Theorem 1. Let the above assumptions be satisfied. Then )u(J  is Gato differentiable, and its gradient satisfies  ∫ ∞∈∀−=Ω).Ω(1,0Wω,dtdxω)Φy(Re)u(Jδ)12(  Proof : Suppose that Uu∈  and )Ω(1,0Wuδ ∞∈ such that Uuδu ∈+ and denoting  ).u;t,x(y)uδu;t,x(y)t,x(yδ −+=   Then )uδ;t,x(yδ is the solution of the boundary value problem: International Journal "Information Theories & Applications" Vol.10  417,Ω)t,x(,uδ)t,x(yyδ)uδu(2xyδ20Btyδi)13( ∈=+−∂∂+∂∂)T,0(t,0x)t,l(yδx)t,0(yδ,)l,0(x,0)0,x(yδ)14( ∈=∂∂=∂∂∈=  and the solution of the above boundary value problem satisfies the following estimation 2)Ω(Wyuδ4C2)Ω(Wyδ)15( 1,020,22≤  where 4C  is a constant and independent of uδ . From (15) and using the theorem of imbedding [6], we have  )Ω(W)t,x(yuδ5C)T,0(L)t,l(yδ)T,0(L)t,0(yδ)16( 1,0222≤+  where 5C  is a constant and independent of .uδ  The increment of the functional )u(J  can be expressed as: 2)T,0(L)t,0(yδ0α22)T,0(L)t,l(yδ1αdt)t,0(yδ])t(0f)t,0(y[T0Re0α2dt)t,l(yδ])t(1f)t,l(y[T0Re1α2)u(J)uθu(JJδ)17(22++−∫+−∫=−+= If we take complex adjoint for (10),(13), we have dt)t,0(1η])t(0f)t,0(y[T00α2dt)t,l(1η])t(1f)t,l(y[T01α2dtdx]1ηu2x1η20Bt1ηiΩ[Φ)18(−∫+−∫−=−∂∂+∂∂∫ ,dtdxηuδ)t,x(yΩdtdxη]yδ)uδu(2xyδ20Btyδi[Ω)19(∫=+−∂∂+∂∂∫ Subtracting (13) from (19), (10) from (18) and in the obtained relation we put yδ,Φ  instead of  1η,η  , then we have  International Journal "Information Theories & Applications" Vol.10  418 ∫−∫−=+∫−+∫−=−∫+−∫Ω.dtdxuδΦyδReΩdtdxuδΦyRedtdx]yδΦuδΦyδuδΩ[21dtdx]ΦyuδyΦuδΩ[21dt)t,0(yδ])t(0f)t,0(y[T0Re0α2dt)t,l(yδ])t(1f)t,1(y[T0Re1α2)20( By substituting the last relation in (17), we have  2)T,0(L)t,l(yδ1α2)T,0(L)t,0(yδ0αΩ.ΩdtdxuδΦyδRedtdxuδyδΦyReJδ)21(22++∫ ∫−−= Suppose that  .ΩdtdxuδΦyδRe2R)23(2)T,0(L)t,l(yδ1α2)T,0(L)t,0(yδ0α1R)22( 22∫−=+= It is clear that,  .2 )T,0(L)t,l(yδ1α2)T,0(L)t,0(yδ0α1R)24( 22+≤  From the formulae of  2R  ,it is estimated as 2)Ω(LuδyδC2R)25( 2≤ .  Then  ).)Ω(Wuδ(o2R1R)26( 1,0∞=+  By substituting (26) in (21), we obtain ).θ(Otddx)ωθ(Ω)Φy(Re)u(J)uδθu(J)27( +∫−=−+  Hence, in light of the variation functional, we have ddxωΩ)Φy(Reθ  )u(J)uθu(J0θlim)ω,u(Jδ)28( ∫−=−+→=  and this proves the differentiability of the functional  and gradient formulae of the function )u(J . This completes the proof of the theorem. Using Tikhinov method [7], we define the following functional . T0dtdx2)t,x(ω)t,x(ul0mα)u(J)u(mJ)29( ∫ −∫+=  and ).Ω(2L)t,x(ω ∈  International Journal "Information Theories & Applications" Vol.10  4194.  Discrete Problem We consider the set of node values   { } M,0j,hJ0xjx,kt,jx =+=  τTN,hlM,N,0k,τk0tkt ===+= and the following notations [8]: hk1jykjy2k1jyxx)kjy(,τ1kjykjyt)kjy()31(,hkjyk1jyx)kjy(,hk1jykjyx)kjy()30(−+−+=−−=−+=−−= After applying the numerical integration formula [8],we have the discertisation of the optimal control problem (1)-(5) as follows:  Let it is required to minimize the functional  }2kjωkju212kMωkMu212k0ωk0u1M1jh{N1kτmν}2]k1fkMy[1α2]k0fk0y[0α{N0kτ])u([mI)32(−+−+−∑−=∑=+−−−∑== on the control set  ⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧==≤==≤≤==N,2k,M,0j,2αt)kju(,N,0k,M,0j,1αkju0α),kju(]u[:]u[MNU  under the conditions   N,1k,1M,1j,kjfkjykju)kjy(0Bt)kjy(i)33( xx=−==−+  ,M,0j,00jy)34( ==  N,1k,]k0yk0ut)k0u([ik0fx)k0y(h0B2)35( =−−=  N,1k,]kMykMut)kMu([ikMfx)kMy(h0B2)36( =−−=−  Now, the discrete gradient formulae will be given as follows:   Theorem  2 The functional  )u(mJ  is differentiable, and its gradient satisfies  , )kjωkju(mα2)kjΦkjy(Rekj])u([/I()37(m−+−=  where N,1k,1M,0j =−=  and kjΦ  is the solution of discrete adjoint problem: 1N,1k,1M,1j,0kjΦkju)kjΦ(0Bt)kjΦ(i)38( xx−=−==−+  ,M,0j,0NjΦ)39( ==  1N,1k,]t)k0Φ(ik0Φk0u[0Bh]k0fk0y[0B0α2x)kjΦ()40( −=−=−+  International Journal "Information Theories & Applications" Vol.10  420 1N,1k,]t)kMΦ(ikMΦkMu[0Bh]k1fkMy[0B0α2x)kjΦ()41( −=−=−+  5.  Solution of Control Problem 5.1  The Projection Gradient Method Here we describe the projection gradient method [9] for the solution of the optimal control problem such as: construct a sequence   m1nu +  by setting  ⎭⎬⎫⎩⎨⎧ −=+ ))nm]u([lmI(nνnm]u[UPm1n]u[)42( MN  where )u(UP MN is the project on the set MNU .   In the first we define  nm)kju(  in the form  1α2Ψ0α2Ψ1α2Ψ0α1α0α1Ψm1n)kju()43( ><≤≤⎪⎩⎪⎨⎧=+  where  kj))nm]u([lmI(nνnm]kju[1Ψ)45(kj))nm]u([lmI(nνnm]kju[2Ψ)44(+=−= and   ...,1,0m,...,1,0n,N,1k,M,0j ====  Using the above sequence we construct the project in the form m1n)1ju(m1n)1ju()46( +=+  2Θnm)kju(1Θnm)kju(2Θ2Ψ1Θ2Θ1Θ0Θm1n)kju()47(><≤≤⎪⎩⎪⎨⎧=+  where  ...,1,0m,...,1,0n,N,2k,M,0jm1n)1kju(2ατ2Θ,m1n)1kju(2ατ1Θkj))nm]u([lmI(nνm1n)kju(0Θ====+−+=+−+−=++= 5.2  Numerical Algorithm With the gradient obtained, the following gradient type algorithm can then be developed for the optimal value of  *u based on the projection gradient method (PGM )which described in the above section.  The outlined of the algorithm for solving control problem are as follows: Step 1: Choose an initial control  0n,U)n(u =∈ .              If )n(u,0))n(u(/I = is the solution of the problem. Step 2 : At each iteration n  do International Journal "Information Theories & Applications" Vol.10  421    Solve the state problem, then find   ))n(u,(.y .          Solve the adjoint problem for (1)-(3), then find                       ))n(u,.(Φ .  Find optimal control )1n(*u+using  PGM.     End do. Step 3:  Test the optimality of )1n(u + .       If  )1n(u +  is optimum, stop the process.       Otherwise, go to Step 4. Step 4    Set 1nn,)n(u)1n(u +==+  and go to Step 2. 6.  Numerical Results Designed algorithm is implemented as a FORTRAN routine [10]. Numerical experiment is carried out to check its performance. The initial data of the problem (1)-(5) are taken as follows: )1t2x()tx(i1)t,x(f,2tx1)t,x(ω0.1)0(u,xi)x(φ,)t1(i1f,ti0f03E5.0ε,1Tl2α1α0α+++−−=++===+==−====== The number of division of the intervals was taken as  20MN ==  . The computed control values of  N,0j,13ju = the values of relative error are shown in Tables 1 ,2 and the 3D plots of the optimal control and initial values are presented in Figures 1,2 . The optimal value of the cost functional is   03E48526.0)*u(J)u(JUuinf*J −==∈= .    The computed control values of  N,0j,13ju =  0.15592E+01 0.15950E+01 0.16301E+01 0.16641E+01 0.17221E+01 0.17464E+01 0.17714E+01 0.18021E+01 0.18332E+01 0.18602E+01 0.19112E+01 0.19830E+01 0.20679E+01 0.21474E+01 0.22155E+01 0.22837E+01 0.23625E+01 0.24368E+01 0.24078E+01 0.24324E+01 0.24718E+01     The values of relative error of  N,0j,13ju =  0.025528 0.004683 0.012478 0.02563 0.050038 0.030459 0.048194 0.046236 0.041988 0.032024 0.033075 0.042328 0.055069 0.061758 0.060028 0.056052 0.054668 0.049211 0.000924 0.028029 0.049297     International Journal "Information Theories & Applications" Vol.10  422  Fig.  1.  Optimal control  )t,x(u∗   Fig.  2.  Initial control  )t,x(0u  REFERENCES [1]  A. J. Kearsley, The use of optimization  techniques  in the solution of partial differential equations from science and engineering, PhD, Housten, Texas,1996. [2] K. Kime, Finite difference approximation of control via the potential in a 1-D Schrodinger equation, Journal of  Differential equations, 2000(26),1--10(2000). [3] N. U. Ahmed and K. L. Teo, Optimal control of distributed parameter systems, North Holland New York, Oxford(1981). [4] S. H. Farag and M. H. Farag, Necessary optimality conditions in control problems for hyperbolic equations, Journal  Egypt. Math. Soc., 8(2),1--10(2000). [5]  O. A. Ladyzhenskaya, Boundary value problems of mathematical physics ,Nauka, Moscow, Russian(1973). [6] V. P. Mikhailov, Partial differential equations,Nauka, Moscow, Russian(1983). [7] A. N. Tikhonov and N. Ya. Arsenin, Methods for the solution of incorrectly posed problems, Nauka, Moscow, Russian(1974). [8] A. H. Khater, A. B. Shamardan, M. H. Farag and A. H. Abel-Hamid, Analytical and numerical solutions of a quasilinear parabolic optimal control problem, Journal Comput. and Appl. Math., 95,29--43(1998). [9] J. T. Betts, Survey of numerical methods for trajectory optimization, Journal of Guidance control and dynamics, 21(2),193--207(1998). [10] D. Kraft, Algorithm 773:TOMP-Fortran modules for optimal control calculations, ACM Transactions of mathematical software, 20(2),262--281(1994). Author Information M. H. FARAG : Department of Mathematics and Computer Science; Faculty of Education, Ibri, Sultanate of Oman.        or:    Faculty of Science ,El-Minia University, Egypt.    E-mail: farag5358@yahoo.com  

A Gradient-Type Optimization Technique for the Optimal Control for Schrodinger Equations

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A Gradient-Type Optimization Technique for the Optimal Control for Schrodinger Equations

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