SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 1961-1983, January 2020.
An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection technique. First-order necessary optimality conditions are derived and two types of second-order sufficient optimality conditions are established. A first theorem invokes a well-known assumption on the set of zeros of the switching function. A second relies on coercivity of the second derivative of the reduced objective functional. The results are applied to the convergence of optimal state functions for a finite element discretizion of the control problem.

中文翻译：

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椭圆状态方程中非线性控制的最优控制问题

SIAM控制与优化杂志，第58卷，第4期，第1961-1983页，2020年1月。
讨论了半线性椭圆方程的最优控制问题，其中控制非线性出现在状态方程中，但不包含在目标函数中。最优控制的存在通过可测量的选择技术得以证明。推导了一阶必要最优条件，并建立了两种二阶充分最优条件。第一定理在切换函数的零集合上调用众所周知的假设。二阶依赖于简化目标函数的二阶导数的矫顽力。将结果应用于控制问题的有限元离散的最佳状态函数的收敛。