### HETEROGENEOUS MULTISCALE METHOD FOR OPTIMAL CONTROL PROBLEM GOVERNED BY ELLIPTIC EQUATIONS WITH HIGHLY OSCILLATORY COEFFICIENTS

Liang Ge1, Ningning Yan2, Lianhai Wang3, Wenbin Liu4, Danping Yang5

1. 1. School of Mathematical Sciences, University of Jinan, Jinan 250022, China;
2. Institute of Systems Science, Academy of Mathematics and System Science, Chinese Academy of Science, Beijing 10080, China;
3. Shandong Provincial Key Laboratory of Computer Networks, Shandong Computer Science Center, Jinan 250014, China;
4. KBS, University of Kent, CT2 7PE, UK;
5. Department of Mathematics, East China Normal University, Shanghai 200062, China
• Received:2015-10-28 Revised:2017-01-09 Online:2018-09-15 Published:2018-09-15

Liang Ge, Ningning Yan, Lianhai Wang, Wenbin Liu, Danping Yang. HETEROGENEOUS MULTISCALE METHOD FOR OPTIMAL CONTROL PROBLEM GOVERNED BY ELLIPTIC EQUATIONS WITH HIGHLY OSCILLATORY COEFFICIENTS[J]. Journal of Computational Mathematics, 2018, 36(5): 644-660.

In this paper, we investigate heterogeneous multiscale method (HMM) for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the wellknown Lions' Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both L2 and H1 norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.

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