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

TL;DR

This paper develops explicit second-order optimality conditions for a nonsmooth quasilinear elliptic control problem, analyzing level sets and their properties, and applies these results to finite-element discretization error estimates.

Contribution

It introduces explicit second-order necessary and sufficient conditions for nonsmooth control problems involving quasilinear PDEs, with a detailed analysis of level set properties.

Findings

Level sets decompose into finitely many simple curves when the gradient is nonzero.

Continuity of level sets and integrals on them is established.

Green's identity is extended to certain nonsmooth contexts.

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.