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Finite element methods for optimal control problems governed by integral equations and integro-differential equations. (English) Zbl 1076.65057

The aim of this paper is to investigate the optimal control problem \[ \min_{u \in K} \biggl\{{1\over 2} ||y-y_0||^2_{0, \Omega}+ \frac{\alpha}{2}||u||^2_{0, \Omega} \biggr\}, \] governed either by a Fredholm integral equation, \[ y- \int_{\Omega} G(s,t)y(s)\,ds=f+Bu \quad\text{in } \Omega , \] or by a Fredholm integro-differential equation, \[ - \text{div} (A \nabla y)- \int_{\Omega} G(s,t)y(s)\,ds=f+Bu\quad \text{in } \Omega , \] \(y=0\) on \(\partial \Omega\). Here \(A\) is symmetric positive definite matrix, \(G(.,.)\) is at least in \(L_2(\Omega \times \Omega), f \in L_2(\Omega)\), and \(K\) denotes a convex subset in the space \(U:=L^2(\Omega_U), \Omega \in \mathbb R^2\) is a bounded domain on which the state equations are defined [cf. W. Alt, Appl.Math. Optimization 12, 15–27 (1984; Zbl 0567.49015); K. E. Atkinson, The numerical solution of integral equations of the second kind (1997; Zbl 0899.65077)], and \(\Omega_U \in R^2\) is another bounded domain on which the control \(u\) can be applied so as to optimize the system [cf. M. Ainsworth and J. T. Oden, Comput. Methods Appl. Mech. Eng. 142, No. 1–2, 1–88 (1997; Zbl 0895.76040), W. Alt (loc. cit.), or K. E. Atkinson (loc. cit.)].
The authors analyze finite-element Galerkin discretizations for a class of constrained control problems. Main result: A priori and a posteriori error estimates for the optimal control problems are derived. The a posteriori error estimates in the Theorems (the precise proofs are proposed) are given in terms of the quantity \(\eta^2_1+\eta^2_2+\eta^2_3\), where the \(\eta_i\) are defined by means of the residuals associated with the equations [cf. M. Ainsworth and J. T. Oden (loc. cit.); W. Alt (loc. cit.)]. The analogous analysis for the optimal control problem [cf. M. Ainsworth and J. T. Oden (loc. cit.); K. E. Atkinson (loc. cit.)] is presented.

MSC:

65K10 Numerical optimization and variational techniques
49J22 Optimal control problems with integral equations (existence) (MSC2000)
49M15 Newton-type methods
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[1] Ainsworth, M., Oden, J.T.: A posteriori error estimators in finite element analysis. Comput. Meth. Appl. Mech. Engrg., 142, 1–88 (1997) · Zbl 0895.76040 · doi:10.1016/S0045-7825(96)01107-3
[2] Alt, W.: On the approximation of infinite optimisation problems with an application to optimal control problems. Appl. Math. Optim., 12, 15–27 (1984) · Zbl 0567.49015 · doi:10.1007/BF01449031
[3] Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge; 1997 · Zbl 0899.65077
[4] Babuška, I., Aziz, A.K.: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations. Academic Press, New York, 1972
[5] Babuška, I., Miller, A. D.: A feedback finite element method with a posteriori error estimation Part 1. Comput. Methods Appl. Mech. Engrg., 61, 1–40 (1987) · Zbl 0593.65064 · doi:10.1016/0045-7825(87)90114-9
[6] Becker, R., Kapp, H., Rannacher, R.: Adaptive finite element methods for optimal control of partial differential equations: Basic concept. SIAM J. Control Optimization, 39, 113–132 (2000) · Zbl 0967.65080 · doi:10.1137/S0363012999351097
[7] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978 · Zbl 0383.65058
[8] Dreyer, T., Maar, B., Schulz, V.: Multigrid optimization in application. J. Comput. Appl. Math., 120, 67–84 (2000) · Zbl 0955.65043 · doi:10.1016/S0377-0427(00)00304-6
[9] Falk, F.S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl., 44, 28–47 (1973) · Zbl 0268.49036 · doi:10.1016/0022-247X(73)90022-X
[10] French, D.A., King, J.T.: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Appl., 12, 299–315 (1991) · doi:10.1080/01630569108816430
[11] Hackbusch, W., Will, Th.: A numerical method for parabolic bang-bang problem. In: Hoffmann, K.H., Krabs, W. (eds.) Optimal control of partial differential equations, Vol. 68 of ISNM, 20–45, Birkhäuser, Basel, 1982
[12] Hackbusch, W., Will, Th.: A numerical method for parabolic bang-bang problem. Control Cybernet, 12, 99–116 (1983) · Zbl 0583.49022
[13] Huang, Y.C., Li, R., Liu, W.B., Yan, N.: Efficient discretization for finite element approximation of constrained optimal control problem. to appear
[14] Krasnosel’skii, M.A., Zabreiko, P.P., et al.: Integral Operators in Spaces of Summable Functions. Noordhoff International Publishers, Leyden, 1976
[15] Kress, R.: Linear Integral Equations (2nd Edition). Springer-Verlag, New York, 1999 · Zbl 0920.45001
[16] Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin, 1971 · Zbl 0203.09001
[17] Lions, J.L.: Some Methods in the Mathematical Analysis of Systems and Their Control. Science Press, Beijing, 1981 · Zbl 0542.93034
[18] Liu, W., Yan, N.: A posteriori error estimates for optimal boundary control. SIAM J. Numer. Anal., 39, 73–99 (2001) · Zbl 0988.49018 · doi:10.1137/S0036142999352187
[19] Liu, W.B., Yan, N.: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math., 15, 285–309 (2002) · Zbl 1008.49024 · doi:10.1023/A:1014239012739
[20] Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Marcel Dekker, New York, 1994
[21] Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54, 483–493 (1990) · Zbl 0696.65007 · doi:10.1090/S0025-5718-1990-1011446-7
[22] Tiba, D.: Lectures on the optimal control of elliptic equations. University of Jyvaskyla Press, Finland, 1995 · Zbl 0832.65130
[23] Verfurth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement. Wiley-Teubner, 1996 · Zbl 1189.76394
[24] Zabreiko, P.P., et al.: Integral Equations – A Reference Text. Noordhoff International Publishers, Leyden, 1975
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