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.


65K10 Numerical optimization and variational techniques
49J22 Optimal control problems with integral equations (existence) (MSC2000)
49M15 Newton-type methods
Full Text: DOI


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