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Finite Element Approximation of Semilinear Parabolic Optimal Control Problems

  • Hongfei Fu (a1) and Hongxing Rui (a2)

Abstract

In this paper, the finite element approximation of a class of semilinear parabolic optimal control problems with pointwise control constraint is studied. We discretize the state and co-state variables by piecewise linear continuous functions, and the control variable is approximated by piecewise constant functions or piecewise linear discontinuous functions. Some a priori error estimates are derived for both the control and state approximations. The convergence orders are also obtained.

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Corresponding author

Corresponding author.Email address:hongfeifu@upc.edu.cn
Corresponding author.Email address:hxrui@sdu.edu.cn

References

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[1]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002.
[2]Brenner, S.C. and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York-Berlin-Heidelberg, 2002.
[3]Geveci, T., On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numer., 13 (1979), pp. 313328.
[4]Alt, W. and Mackenroth, U., Convergence of finite element approximation to state constraint convex parabolic boundary control problems, SIAM J. Control Optim., 27 (1989), pp. 718736.
[5]Becker, R., Kapp, H. and Rannacher, R., Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), pp. 113132.
[6]Duvaut, G. and Lions, J. L., The Inequalities in Mechanics and Physics, Springer, Berlin, 1973.
[7]Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, SpringerVerlag, Berlin, 1971.
[8]Tiba, D., Lectures on the Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Finland, 1995.
[9]Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems: Theorey, Algorithms and Applications, M. Dekker, New York, 1994.
[10]Pironneau, O., Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984.
[11]Falk, F. S., Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 2847.
[12]Liu, W. and Tiba, D., Error estimates for the finite element approximation of nonlinear optimal control problems, Numer. Func. Anal. Optim., 22 (2001), pp. 953972.
[13]Liu, W. and Yan, N., A posteriori error estimates for control problems governed by nonlinear elliptic equations, Appl. Numer. Math., 47 (2003), pp. 173187.
[14]Fu, H. and Rui, H., A priori error estimates for optimal control problems governed by transient advection-diffusion equations, J. Sci. Comput., 38 (2009), pp. 290315.
[15]Fu, H., A characteristic finite element method for optimal control problems governed by convection-diffusion equations, J. Comput. Appl. Math., 235 (2010), pp. 825836.
[16]Meidner, D. and Vexler, B., A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 13011329.
[17]Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), pp. 970985.
[18]Li, R., Liu, W., Ma, H. and Yan, N., Adaptive finite elememt approximation for distributed optimal control governed by parabolic equations, submitted.
[19]Wheeler, M. F., A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10 (1973), pp. 723759.

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Finite Element Approximation of Semilinear Parabolic Optimal Control Problems

  • Hongfei Fu (a1) and Hongxing Rui (a2)

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