Abstract
In this paper, we present an a posteriori error analysis for finite element approximation of distributed convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for control problems. Explicit estimates are obtained for some model problems which frequently appear in real-life applications.
Similar content being viewed by others
References
M. Ainsworth, J.T. Oden and C.Y. Lee, Local a posteriori error estimators for variational inequalities, Numer. Methods Partial Differential Equations 9 (1993) 22–33.
M. Ainsworth and J.T. Oden, A posteriori error estimators in finite element analysis, Comput.Methods Appl. Mech. Engrg. 142 (1997) 1–88.
P. Alotto et al., Mesh adaption and optimisation techniques in magnet design, IEEE Trans. Magn. 32 (1996).
W. Alt and U. Mackenroth, Convergence of finite element approximation to state constrained convex parabolic boundary control problems, SIAM J. Control Optim. 27 (1989) 718–736.
I. Babuska and W.C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 5 (1978) 736–754.
N.V. Banichuk et al., Mesh refinement for shape optimisation, Structural Optimisation 9 (1995) 45–51.
R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp. 44 (1985) 283–301.
J. Baranger and H.E. Amri, A posteriori error estimators in finite element approximation of quasi-Newtonian flows, M2AN 25 (1991) 31–48.
V. Barbu, Optimal Control for Variational Inequalities, Research Notes in Mathematics, Vol. 100 (Pitman, London, 1984).
J.W. Barrett and W.B. Liu, Finite Element Approximation of Some Degenerate Quasi-linear Problems, Lecture Notes in Mathematics, Vol. 303 (Pitman, 1994) pp. 1–16.
R. Becker and H. Kapp, Optimization in PDE models with adaptive finite element discretization, in: Proc. ENUMATH'97, Heidelberg, September 29-October 3, 1997 (World Scientific, 1998) (in press).
R. Becker, H. Kapp and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept, SFB 359, University of Heidelberg (1998).
C. Bernardi and V. Girault, A local regularisation oprator for triangular and quadrilateral finite element, SIAM J. Numer. Anal. 35(5) (1998) 1893–1916.
P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North Holland, Amsterdam, 1978).
Z.H. Ding, L. Ji and J.X. Zhou, Constrained LQR problems in elliptic distributed control systems with point observations, SIAM J. Control Optim. 34 (1996) 264–294.
T. Dreyer, B. Maar and V. Schulz, Multigrid optimization in applications, J. Comput. Appl. Math. 120 (2000) 67–84.
R.D. Duran et al., On the asymptotic exactness of error estimators for linear triangular finite elements, Numer. Math. 59 (1991) 107–127.
F.S. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl. 44 (1973) 28–47.
D.A. French and J.T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Appl. 12 (1991) 299–315.
T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numer. 13 (1979) 313–328.
R. Glowinski, J.L. Lions and R. Tremolieres, Numerical Analysis of Variational Inequalities (North-Holland, The Netherlands, 1976).
M.D. Gunzburger, L. Hou and Th. Svobodny, Analysis and finite element approximation of optimal control problems for stationary Navier-Stokes equations with Dirichlet controls, RAIRO Model. Math. Anal. Numer. 25 (1991) 711–748.
J. Haslinger and P. Neittaanmaki, Finite Element Approximation for Optimal Shape Design (Wiley, Chichester, 1989).
C. Johnson, Adaptive finite element methods for the obstacle problem, Math. Models Methods Appl. Sci. 2 (1992) 483–487.
R. Kornhuber, A posteriori error estimates for elliptic variational inequalities, Comput. Math. Appl. 31 (1996) 49–60.
G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20 (1982) 414–427.
A. Kufner, O. John and S. Fucik, Function Spaces (Nordhoff, Leyden, The Netherlands, 1977).
I. Lasiecka, Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, SIAM J. Control Optim. 22 (1984) 744–750.
R. Li, T. Tang and P.-W. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps (2000) (submitted).
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer-Verlag, Berlin, 1971).
W.B. Liu and J.W. Barrett, Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality, Adv. Comput. Math. 1(2) (1993) 223–239.
W.B. Liu and J.W. Barrett, Quasi-norm error bounds for the finite element approximation of degenerate quasilinear elliptic variational inequalities, RAIRO Numer. Anal. 28 (1994) 725–744.
W.B. Liu, H.P. Ma and T. Tang, On mixed error estimates for elliptic obstacle problems, Adv. Comput. Math. (accepted).
W.B. Liu and J. Rubio, Optimality conditions for strongly monotone variational inequalities, Appl. Math. Optim. 27 (1993) 291–312.
W.B. Liu and D. Tiba, Error estimates for the finite element approximation of a class of nonlinear optimal control problems, J. Numer. Funct. Optim. (1999) (accepted).
W.B. Liu and N. Yan, A posteriori error estimates for a model boundary optimal control problem, J. Comput. Appl. Math. 120 (2000) 491–506.
W.B. Liu and N. Yan, A posteriori error estimators for a class of variational inequalities, J. Sci. Comput. 35 (2000) 361–393.
W.B. Liu and N. Yan, A posteriori error estimates for convex boundary optimal control problems, SIAM J. Numer. Anal. 39 (2001) 73–99.
K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems, Appl. Math. Optim. 8 (1982).
R.S. McKnight and Jr. Borsarge, The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim. 11 (1973) 510–524.
P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms and Applications (M. Dekker, New York, 1994).
O. Pironneau, Optimal Shape Design for Elliptic Systems (Springer-Verlag, Berlin, 1984).
L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990) 483–493.
D. Tiba and F. Troltzsch, Error estimates for the discretization of state constrained convex control problems, Numer. Funct. Anal. Optim. 17 (1996) 1005–1028.
F. Troltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal control, Appl. Math. Optim. (1994).
R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh Refinement Techniques (Wiley-Teubner, 1996).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liu, W., Yan, N. A Posteriori Error Estimates for Distributed Convex Optimal Control Problems. Advances in Computational Mathematics 15, 285–309 (2001). https://doi.org/10.1023/A:1014239012739
Issue Date:
DOI: https://doi.org/10.1023/A:1014239012739