In this paper, we carry out the numerical analysis of a nonsmooth quasilinear elliptic optimal control problem, where the coefficient in the divergence term of the corresponding state equation is not differentiable with respect to the state variable. Despite the lack of differentiability of the nonlinearity in the quasilinear elliptic equation, the corresponding control-to-state operator is of class $C^1$ but not of class $C^2$. Analogously, the discrete control-to-state operators associated with the approximated control problems are proven to be of class $C^1$ only. By using an explicit second-order sufficient optimality condition, we prove a priori error estimates for a variational approximation, a piecewise constant approximation, and a continuous piecewise linear approximation of the continuous optimal control problem. The numerical tests confirm these error estimates.

翻译：本文对一类非光滑拟线性椭圆最优控制问题进行数值分析，其中相应状态方程散度项中的系数关于状态变量不可微。尽管拟线性椭圆方程的非线性项缺乏可微性，但对应的控制到状态算子属于$C^1$类而非$C^2$类。类似地，与近似控制问题相关的离散控制到状态算子被证明仅属于$C^1$类。通过使用显式的二阶充分最优性条件，我们证明了连续最优控制问题的变分逼近、逐片常数逼近和连续逐片线性逼近的先验误差估计。数值实验验证了这些误差估计。