Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

¹ Faculty of Mathematics, University of Duisburg-Essen, 45117 Essen, Germany
² Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
³ Faculty of Fundamental Sciences, PHENIKAA University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam
⁴ Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

^* Corresponding author: nhu.vuhuu@phenikaa-uni.edu.vn

Received: 30 March 2022
Accepted: 8 April 2024

Abstract

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.