In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

翻译：本文推导了二阶拟线性偏微分方程最优控制问题中局部极小点的显式二阶必要与充分最优性条件，该方程的最高阶项包含分段光滑但不可微的非线性项。核心论证基于状态水平集的分析。具体而言，我们证明：若函数在边界上为零且其梯度在某个水平集上非零，则该水平集可分解为有限条闭合简单曲线。此外，水平集连续依赖于定义这些集的函数。我们还证明了水平集上积分的连续性。特别地，Green第一恒等式被证明适用于由两个梯度非零函数确定的开集。在本文后续部分，显式充分二阶条件将用于推导控制问题有限元离散化的误差估计。