SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 2354-2382, January 2020.
Penalization of Dirichlet boundary controlled nonstationary magneto-hydrodynamic equations is considered. Asymptotic behavior of solutions of a penalized control problem with respect to the penalty parameter is investigated. It is proved that solutions of the penalized boundary control problem converge to the solutions of the Dirichlet control problem as penalty parameter $\epsilon$ goes to zero. The existence of optimal solutions as well as the characterization of such solutions by first order necessary conditions are established. A second order sufficient optimality condition for the optimal control problem is also developed. Numerical results are provided showing the feasibility and effectiveness of the penalty method.

中文翻译：

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非平稳磁流体动力学的Dirichlet边界控制的惩罚

SIAM控制与优化杂志，第58卷，第4期，第2354-2382页，2020年1月。
考虑了Dirichlet边界控制的非平稳磁流体动力学方程的惩罚。研究了惩罚控制问题解关于惩罚参数的渐近行为。证明了当惩罚参数$ε为零时，惩罚边界控制问题的解收敛到Dirichlet控制问题的解。建立了最优解的存在性以及通过一阶必要条件对这种解的表征。还开发了用于最优控制问题的二阶充分最优条件。数值结果表明了惩罚方法的可行性和有效性。