Abstract
This paper studies the convergence and error estimates of approximate solutions to an optimal control problem governed by semilinear elliptic equations with non-convex cost function and non-convex mixed pointwise constraints, and unbounded constraint set. We discretize the optimal control problems by the finite element method in order to obtain a sequence of mathematical programming problems in finite-dimensional spaces. We show that under certain conditions, the optimal solutions of the obtained mathematical programming problems converge to an optimal solution of the original problem. In particular, if the original problem satisfies the so-called no-gap second-order conditions, then some error estimates of approximate solutions are obtained.
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Acknowledgements
This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01\(-\)2019.308. A part of this paper was completed at Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for their financial support and hospitality. The authors would like to thank the anonymous referees for their useful suggestions and comments which improved the manuscript greatly.
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Kien, B.T., Rösch, A., Son, N.H. et al. FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints. J Optim Theory Appl 197, 130–173 (2023). https://doi.org/10.1007/s10957-023-02187-3
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DOI: https://doi.org/10.1007/s10957-023-02187-3
Keywords
- Finite element method
- Optimal control
- Semilinear elliptic equation
- First-and second-order optimality conditions
- Convergence
- Error estimate