A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

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References

Adams, R. [1975] Sobolev Spaces (Academic, New York). Google Scholar
Babuska, I., Banerjee, U. and Osborn, J. E. [2004] “ Generalized finite element methods–main ideas, results and perspective,” Int. J. Comput. Methods 1(1), 67–103. Link, Web of Science, Google Scholar
Babuska, I. and Chatzipantelidis, P. [2002] “ On solving elliptic stochastic partial differential equations,” Comput. Methods Appl. Mech. Eng. 191, 4093–4122. Crossref, Web of Science, Google Scholar
Babuska, I., Liu, K. and Tempone, R. [2003] “ Solving stochastic partial differential equations based on the experimental data,” Math. Models Methods Appl. Sci. 13(3), 415–444. Link, Web of Science, Google Scholar
Babuska, I., Tempone, R. and Zouraris, G. E. [2004] “ Galerkin finite element approximations of stochastic elliptic partial differential equations,” SIAM J. Numer. Anal. 42(2), 800–825. Crossref, Web of Science, Google Scholar
Barbu [1984] Optimal Control for Variational Inequalities, Research Notes in Mathematics, Vol. 100 (Pitman, London). Google Scholar
Chen, C. S., Fan, C. M. and Wen, P. H. [2011] “ The method of approximate particular solutions for solving elliptic problems with variable coefficients,” Int. J. Comput. Methods 8(3), 545–559. Link, Web of Science, Google Scholar
Ciarlet, P. G. [2002] The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, Vol. 40 (SIAM, Philadelphia). Crossref, Google Scholar
Cohen, A., DeVore, R. and Schwab, C. [2010] “ Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs,” Found Comput Math. 10, 615–646. Crossref, Web of Science, Google Scholar
Deb, M. K., Babuska, I. and Oden, J. T. [2001] “ Solution of stochastic partial differential equations using Galerkin finite element techniques,” Comput. Methods Appl. Mech. Eng. 190, 6359–6372. Crossref, Web of Science, Google Scholar
Fursikov, A. V. [2000] Optimal Control of Distributed Systems, Theory and Applications (American Mathematical Society Providence, Rhode Island). Google Scholar
Ghanem, R. and Spanos, P. D. [1991] Stochastic Finite Elements: A Spectral Approach (Springer-Verlag, Berlin). Crossref, Google Scholar
Glowinski, R. and Lions, J. L. [1996] Exact and Approximate Controllability for Distributed Parameter Systems (Cambridge University Press). Google Scholar
Grisvard, P. [1986] Elliptic Problems in Nonsmooth Doamin (Longman Higher Education). Google Scholar
Gunzburger, M. D., Lee, H. C. and Lee, J. [2011] “ Error estimates of stochastic optimal Neumann boundary control problems,” SIAM J. Numer. Anal. 49(4), 1532–1552. Crossref, Web of Science, Google Scholar
Hou, L. S., Lee, J. and Manouzi, H. [2011] “ Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs,” J. Math. Anal. Appl. 384, 87–103. Crossref, Web of Science, Google Scholar
Lee, H. C. and Lee, J. [2013] “ A stochastic Galerkin method for stochastic control problems,” Commun. Comput. Phys. 14(1), 77–106. Crossref, Web of Science, Google Scholar
Li, M. W., Peng, H. J. and Wu, Z. G. [2015] “ Symplectic irregular interpolation algorithms for optimal control problems,” Int. J. Comput. Methods 12(6), doi: 10.1142/S0219876215500401. Link, Web of Science, Google Scholar
Lions, J. L. [1971] Optimal Control of Systems Governed by Partial Differential Equations (Springer-Verlag, Berlin). Crossref, Google Scholar
Liu, G. R. [2008] “ A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods,” Int. J. Comput. Methods 5(2), 199–236. Link, Web of Science, Google Scholar
Liu, W. B. and Tiba, D. [2001] “ Error estimates for the finite element approximation of a class of nonlinear optimal control problems,” J. Numer. Func. Optim. 22, 953–972. Crossref, Web of Science, Google Scholar
Liu, W. B. and Yan, N. N. [2001] “ A posteriori error estimates for convex boundary control problems,” SIAM Numer. Anal. 39, 73–99. Crossref, Web of Science, Google Scholar
Liu, W. B. and Yan, N. N. [2008] Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Series in Information and Computational Science, Vol. 41 (Science Press, Beijing). Google Scholar
Liu, W. B., Yang, D. P., Yuan, L. and Ma, C. Q. [2010] “ Finite element approximation of an optimal control problem with integral state constraint,” SIAM J. Numer. Anal. 48(3), 1163–1185. Crossref, Web of Science, Google Scholar
Nobile, F. and Tempone, R. [2009] “ Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients,” Int. J. Numer. Meth. Eng. 80, 979–1006. Crossref, Web of Science, Google Scholar
Rosseel, E. and Wells, G. N. [2012] “ Optimal control with stochastic PDE constrains and uncertain controls,” Comput. Methods Appl. Mech. Eng. 213–216, 152–167. Crossref, Web of Science, Google Scholar
Shen, W. F., Sun, T. J., Gong, B. X. and Liu, W. B. [2015] “ Stochastic Galerkin method for constrained optimal control problem governed by an elliptic integro-differential PDE with stochastic coefficients,” Int. J. Numer. Anal. Model. 12(4), 593–616. Web of Science, Google Scholar
Sun, T. J. [2010] “ Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem,” Int. J. Numer. Anal. Mod. 7(1), 87–107. Web of Science, Google Scholar
Sun, T. J., Ge, L. and Liu, W. B. [2013] “ Equivalent a posteriori error estimates for a constrained optimal control problem governed by parabolic equations,” Int. J. Numer. Anal. Mod. 10(1), 1–23. Web of Science, Google Scholar
Sun, T. J., Shen, W. F., Gong, B. X. and Liu, W. B. [2015] “A priori error estimate of stochastic Galerkin method for optimal control problem governed by stochastic elliptic PDE with constrained control,” accepted by J.S.C., doi:10.1007/s10915-015-0091-7. Google Scholar
Tiba, D. [1995] Lectures on the Optimal Control of Elliptic Equations (University of Jyvaskyla Press, Finland). Google Scholar
Todor, R. A. and Schwab, C. [2007] “ Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients,” IMA J. Numer. Anal. 27, 232–261. Crossref, Web of Science, Google Scholar
Wiener, N. [1938] “ The homogeneous chaos,” Amer. J. Math. 60, 897–936. Crossref, Google Scholar
Xiu, D. and Karniadakis, G. E. [2002] “ The Wiener-Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644. Crossref, Web of Science, Google Scholar