We consider the optimal boundary control of a linear parabolic boundary value problem. Firstly, the problem is formulated as an optimization problem with the system state governed by a parabolic partial differential equation. Based on the formulation for the variation of the cost functional, a gradient-type optimization technique utilizing the finite element 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

Chuedoung, M.

Siew, P. F.

Wu, Y. H.

English

Australian Mathematical Society (AustMS): E-Journals

ANZIAM J. 44 (E) ppC836–C850, 2003 C836Optimal boundary control of a linearparabolic evolution systemY. H. Wu M. Chuedoung P. F. Siew∗(Received 16 July 2001; revised 26 September 2002)AbstractWe consider the optimal boundary control of a linearparabolic boundary value problem. Firstly, the problem isformulated as an optimization problem with the system stategoverned by a parabolic partial differential equation. Basedon the formulation for the variation of the cost functional,a gradient-type optimization technique utilizing the finiteelement method is then developed to solve the constrainedoptimization problem. Finally, a numerical example is givenand the results show that the method of solution is robust.Contents1 Introduction C837∗Department of Mathematics and Statistics, Curtin University ofTechnology, GPO Box U1987, W.A. 6845, Australia.mailto:yhwu@maths.curtin.edu.au0See http://anziamj.austms.org.au/V44/CTAC2001/Wu00 for this article,c© Austral. Mathematical Soc. 2003. Published 1 April 2003. ISSN 1446-87351 Introduction C8372 Variation of the cost functional C8393 Numerical algorithm C8414 Solution of the state system C8425 Numerical results C845References C8491 IntroductionMany natural and industrial process involve diffusion. Typical ex-amples are the transient transfer of heat and the diffusion of chem-icals. A diffusion process is governed by a parabolic type partialdifferential equation subject to certain initial and boundary condi-tions [3, 4], and the behavior of the process can be controlled bythe condition imposed on the boundary. In this paper, we are con-cerned with the boundary control of a diffusion process governed bya linear parabolic partial differential equation.Let T be the system state, t be time, x = (x1, x2) be the positionvector, Ω be the region under consideration, Γ be the boundary of Ω,Σ = Γ × (0, tA] , and Q = Ω × (0, tA] . Then the boundary valueproblem governing the state of a typical linear parabolic evolutionsystem is the bvp:∂T (x, t;u)∂t−∇2T (x, t;u) = f(x, t) , (x, t) ∈ Q ,T (x, 0) = T 0(x) , x ∈ Ω ,∂T (x, t)∂n= u(x, t) , (x, t) ∈ Σ , (1)1 Introduction C838xxQΩ21tFigure 1: Diagram showing the region Ω and the domain Q .where T = T (x, t;u) and the inclusion of u is to indicate that thestate T (x, t) depends on the boundary control u.Let z(x, t) be the desired target state of T and U be the admissi-ble space for u. Then the problem of finding the boundary control uto achieve the desired target state is cast in the least square senseby the following constrained optimization problem (cop) [1, 6, 7]:minv∈UJ(v) =∫Q[T (x, t; v)− z(x, t)]2 dx dt , (2)subject to T being the solution of the bvp (1).Theorem 1 The cost functional J(v) in (2) can be expressed asJ(v) = a(v, v)− 2I(v) + k , (3)2 Variation of the cost functional C839where a(v, v) and I(v) are respectively the continuous bilinear andlinear functionals defined bya(u, v) =∫Q[T (x, t;u)− T (x, t; 0)] [T (x, t; v)− T (x, t; 0)] dx dt ,I(v) = −∫Q[T (x, t; v)− T (x, t; 0)] [T (x, t; 0)− z(x, t)] dx dt ,k =∫Q[T (x, t; 0)− z(x, t)]2 dx dt . (4)Proof: by substituting (4) into (3). ♠Various attempts have been made to solve this type of con-strained optimization problem [8, 12, 5, 11]. However, there doesnot appear to be any efficient numerical algorithm available. In thispaper, we present and test an efficient gradient type numerical tech-nique for the solution of the problem based on previous work in thefield [7, 8, 12, 5, 11] and utilizing the finite element method for theassociated direct boundary value problems.2 Variation of the cost functionalConsider the cost functional J(u) as defined in (3). Let h be the vari-ation of u, then the corresponding increment of the functional J(u)is∆J(u) = J(u+ h)− J(u) .From Theorem 1, we have∆J(u) = a(u+ h, u+ h)− 2I(u+ h)− a(u, u) + 2I(u)= 2a(u, h)− 2I(h) + a(h, h) . (5)2 Variation of the cost functional C840Neglecting the higher order term of h, the variation of the func-tional J(v), which is the principal linear part of the functional in-crement, isδJ(u) = 2a(u, h)− 2I(h) .Let h = 12v , we haveδJ(u) = a(u, v)− I(v) . (6)Theorem 2 The variation of the cost functional can be determinedbyδJ(u) =∫Σp(u)v dx dt ,where p(u), denoting p(x, t;u) , is defined by the following adjointinitial boundary value problem−∂p(x, t;u)∂t−∇2p(x, t;u) = T (x, t;u)− z(x, t) , (x, t) ∈ Q ,∂p∂n= 0 , (x, t) ∈ Σ ,p(x, tA) = 0 , (x, t) ∈ Ω . (7)Proof: For simplicity in notation, we denote T (x, t;w) by T (w)throughout the proof. Now, by the definition of a and I as givenin (4), we have from (6)δJ(u) =∫Q[T (u)− z] [T (v)− T (0)] dx dt , (8)which, on using (7)1 (the first equation of (7)), becomesδJ(u) =∫Q{−∂p(u)∂t−∇2p(u)}[T (v)− T (0)] dx dt . (9)3 Numerical algorithm C841Using Green’s Theorem and making use of equations (1)1, (1)3and (7)2, we have−∫Q∇2p(u) [T (v)− T (0)] dx dt=∫Qp(u)[∂T (0)∂t− ∂T (v)∂t]dx dt+∫Σp(u)v dx dt . (10)Further, by using the product rule for differentiation and notingequation (7)3 and utilizing T (x, 0; v) = T (x, 0; 0) , we have−∫Q∂p(u)∂t[T (v)− T (0)] dx dt =∫Qp(u)[∂T (v)∂t− ∂T (0)∂t]dx dt .(11)Substituting (10) and (11) into (9), we haveδJ(u) =∫Σp(u)v dx dt . (12)♠3 Numerical algorithmThe solution of the cop problem (2) is difficult, as it involves thedetermination of the boundary control u(x, t) at an infinite numberof points on Σ. Thus, in order to find the numerical solution ofthe problem, we approximate the problem by a finite dimensionaloptimization problem. For this purpose, we firstly discretize Σ asN equal-sized subregions ∆Σi with Σ =⋃Ni=1∆Σi and then approx-imate u(x, t) as a piecewise continuous function, namelyu(x, t) = ui , ∀(x, t) ∈ ∆Σi , i = 1, 2, 3, . . . , N .4 Solution of the state system C842Thus the problem becomes to find u = [u1, u2, u3, . . . , uN ]T suchthatJ(u) = infvi∈UJ(v) .As the cost functional now depends on u1, u2, u3, . . . , uN , we definethe gradient of the cost functional byG =[∂J∂u1,∂J∂u2, . . . ,∂J∂uN,]T,where∂J∂ui=1∆ui[J(u1, u2, . . . , ui +∆ui, . . . , uN)− J(u1, u2, . . . , ui, . . . , uN)] .Using (12) with12v(x, t) = h(x, t) ={∆ui , on ∆Σi ;0 , otherwise ;we have by using the one-point quadrature rule∂J∂ui≈ 1∆ui[2 |∆Σi| pi∆ui] = 2 |∆Σi| pi ,where pi is the solution of the adjoint system (7) correspondingto ∆Σi and |∆Σi| is the area of the subregion ∆Σi. With the gradi-ent obtained, the gradient type algorithm of Table 1 determines theoptimal value of u based on the Fletcher-Reeves method [10, 2, 9].4 Solution of the state systemTo determine the gradient of the cost functional, we need to solve thestate system for T (x, t;u) and then the adjoint system for p(x, t;u).4 Solution of the state system C843Table 1: Numerical Algorithm1. Choose an initial boundary control u0. If G(u0) = 0 , u0 isthe solution of the problem.2. Set the first searching direction S0 = −G(u0) .3. Set u1 = u0+α0S0 , with α0 being the optimal step length inthe searching direction S0. Set i = 1 and go to step 4.4. Find G(ui) by solving the state and adjoint systemsand then set Si = −G(ui) + βiSi−1 , with βi =[G(ui),G(ui)]/[G(ui−1),G(ui−1)] .5. Compute the optimum step length αi in the searching direc-tion Si and update u by ui+1 = ui + αiSi .6. Test the optimality of ui+1. If ui+1 is optimum, stop theprocess. Otherwise, set i = i+ 1 and go to step 4.4 Solution of the state system C844Both systems are parabolic type initial boundary value problems.Various numerical methods, such as the finite element method andthe finite difference method can be used to solve these problems.In the present work, the finite element method [13] is used for thesolution. To keep details to a minimum, in the following, we brieflydescribe only the numerical technique for the solution of the statesystem. The adjoint system is solved similarly.To find the numerical solution of the state system (1), we firstlymultiply both sides of equation (1)1 by an arbitrary function Φ andintegrate over the domain Ω to yield∫ΩΦ(∂T∂t−∇2T)dΩ =∫ΩΦ [f(x, t)] dΩ . (13)Noting the boundary condition and using integration by parts andthe divergence theorem, we have∫ΩΦ∂T∂tdΩ +∫Ω∂Φ∂xj∂T∂xjdΩ =∫ΩΦ [f(x, t)] dΩ +∫ΓΦu dΓ . (14)To solve the initial boundary value problem, the domain Ω is dividedinto a finite number of simple shaped regions Ωe (e = 1, 2, . . . , E)called elements. Consequently, the boundary Γ of the domain Ω isdivided into a number of boundary segments Γb (b = 1, 2, . . . , B).Within each element, the coordinate dependent variables T and Φare interpolated by functions of compatible order, in terms of valuesto be determined at a set of nodal points. Denoting T e and Φe asthe column vectors of the element nodal point values of T and Φrespectively, and N(x) as the interpolation function, then T and Φwithin each element areT = NTT e , Φ = ΦTeN . (15)Substituting (15) into (14), we haveE∑e=1ΦTe{ce∂T e∂t+ keT e}=E∑e=1ΦTe f e +B∑b=1ΦTb f b , (16)5 Numerical results C845wherece =∫ΩeNNT dΩ , ke =∫Ωe∂N∂xj∂NT∂xjdΩ ,f e =∫Ωef(x, t)N dΩ , f b =∫ΓbuN dΓ . (17)Using a standard finite element assembling procedure, equation (16)is represented in matrix formΦT{C(∂T∂t)+KT}= ΦTF . (18)Further, due to the arbitrary nature of Φ, we have from equa-tion (18)C(∂T∂t)+KT = F , (19)which constitutes a system of N first-order ordinary differentialequations with N unknown values of T and is solved by using astandard time stepping scheme.5 Numerical resultsTo test the numerical algorithm developed, consider the followingexample. Find u(x, t) such thatJ(u) =∫Q[T (x, t;u)− z(x, t)]2 dx dt , (20)is minimized subject to T (x, t;u) being governed by∂T (x, t;u)∂t−∇2 [T (x, t;u)] = f(x, t) , in QT (x, 0) = T 0(x) , in Ω∂T (x, t)∂n= u(x, t) , on Σ , (21)5 Numerical results C8460 20 40 60 80024681012a)  Iteration  0−80IterationJ / Q0 20 40 60 8000.010.020.030.040.050.060.070.08b)  Iteration  5−80IterationJ / QFigure 2: Value of the scaled cost functional (J/Q) versus itera-tion.where z(x, t) is the target system state given byz(x, t) = 50et +(x21 + x22 + 0.5x21x22)t .Ω is the square region [0, 4]×[0, 4], Q = Ω×(0, tA] , Γ is the boundaryof Ω, Σ denotes Γ× (0, tA], tA = 0.5 , T 0(x) = 50 and the functionf(x, t) = 50et + (1− t)(x21 + x22) +x21x222− 4t .For this particular problem, the exact solution for u isu(x, t)exact =8t+ 4x22t , on x1 = 4 ;8t+ 4x21t , on x2 = 4 ;0 , on x1 = 0 and x2 = 0 .To validate the numerical algorithm, we use it to solve the prob-lem and then compare the numerical results with the exact solution.5 Numerical results C847024024495051024024495051024024608010002402450100150024024501001500240240100200024024495051024024501001500240240100200024024495051024024501001500240240100200t = 0.00 t = 0.25 t = 0.5 a) Iteration 0 b) Iteration 1 c) Iteration 80 d) Target State Figure 3: Comparison between computed states and the targetstate .Figure 2 shows the variation of the value of the scaled cost func-tional (J/Q) in the iteration process. Figure 3 shows the computedsystem state T (x, ti,u) at various stages of the iteration processagainst the target state. It is noted that, the system state convergesto the target state. Figure 4 shows the variation of the computedboundary control u during the iteration process at various timesteps. The results show that the numerical algorithm is robust.5 Numerical results C8480 20 40 60 80−20246Iterationa)  at  t = 0.05Control  u0 20 40 60 800246IterationControl  ub)  at  t = 0.200 20 40 60 80051015Iterationc)  at  t = 0.35Control  u0 20 40 60 80051015IterationControl  ud)  at  t = 0.40Figure 4: Comparison between the computed boundary controlat the typical point (2, 4) and the exact solution: —— Computedresult; - - - Exact result.References C849References[1] Engl, H. W. and Langthaler, T. (1978), “On an inverseproblem for a nonlinear heat equation connected withcontinuous casting of steel”, Optimal control of differentialequations, Birkha¨user Verlag, Boston. C838[2] Gabay, D. (1982), “Reduced quasi-Newton methods withfeasibility improvement for nonlinearly constrainedoptimization”, Mathematical Programming Study, Vol. 16,pp. 18–44. C842[3] Hill, J. M and Wu, Y. H. (1994), “On a nonlinear Stefanproblem arising in continuous casting of steel”, Actamachanica, Vol. 107, pp. 183–198. C837[4] Latinen E., and Neittaanmaki, P. (1998), “On numericalsimulation of the continuous casting process”, J. Eng. Math,Vol. 22, pp. 164–168. C837[5] Murray W., Pricet. J.P. (1995), “A sequential quadraticprogramming algorithm using an incomplete solution of thesubproblem”, Numer. SIAM J. Optim., Vol. 5, pp. 590–640.C839[6] Neittaanma¨ki, P. (1978), “On the control of the secondarycooling in the continuous casting process”, Optimal control ofdifferential equations, Birkha¨user Verlag, Boston C838[7] Pawlow, I;, Shindo Y. and Satawa, T. (1985), “Numericalsolution of multidimension two - phase Stefan problem”,Numer. Funct. Anal. Optimiz., Vol. 8, pp. 55–84. C838, C839References C850[8] Phillipson, G.A. (1971), Identification of Distributed Systems,American Elsevier Publishing Company Inc., New York.C839[9] Rao, S. S. (1996), Engineering optimization — Theory andPractice, John Wiley & Sons, New York. C842[10] Shanno D. F. (1983), “Recent advances in gradient basedunconstrained optimization techniques for large problem”,ASME J. of Mechanisms, Transmission, and Automation inDesign, Vol. 105, pp. 155–159. C842[11] Spellucci, P.(1998), “An SQP method for general nonlinearprograms using only equality constrained subproblems”,Mathematical Programming, Vol. 82, pp. 413–448. C839[12] Tro¨ltzech, T. (1989), “On the semigroup approach for theoptimal control of semilinear parabolic equations includingdistributed and boundary control”, Zeitschr. F. Analysis andihre Anwedungen, Vol. 8, pp. 431–443. C839[13] Wu, Y. H., Hill, J. M. and Flint, P. (1994), “A novel finiteelement method for heat transfer in the continuous caster”,J. Austral. Math. Soc., Ser. B, Vol. 35, pp. 263–288. C844

Optimal boundary control of a linear parabolic evolution system

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Optimal boundary control of a linear parabolic evolution system

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