Optimal control of a non-smooth semilinear elliptic equation

Abstract

This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method.

Sensitivity Analysis and Optimal Control of Obstacle-Type Evolution Variational Inequalities

“…This makes sense since (5.14) yields information about the behavior ofp on the bi-active set (i.e., the intersection of the zero level sets ofȳ andφ) where regularization approaches are known to be imprecise. Note that our results are in particular consistent with the observations made in [31,Theorems 6.6,6.8] and [11,Section 4] for optimal control problems governed by non-smooth partial dierential equations.…”

“…[1,4,5,9,19,23,25,28]. The aim of this paper is to provide dierentiability results for evolution variational inequalities with unilateral constraints that allow to avoid regularization and that may serve as a point of departure for the development of solution algorithms for optimal control problems that take into account the non-smooth behavior of the governing EVI, cf., e.g., the approaches in [11,12,38]. We further demonstrate that our dierentiability results give rise to strong stationarity conditions that resemble those derived by Mignot and Puel for the timeindependent classical obstacle problem in [32,33] and that contain information which is not obtainable with regularization techniques.…”

“…To obtain a strong stationarity system analogous to that in [32,33] for the optimal control problem (O), we proceed similarly to [11,31] and note the following:…”

Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem

“…Iterative regularization for a non-smooth forward operator. In this section, we study the solution operator to (1.2) based on previous results from [2,3]. In particular, we show that this operator together with one of its Bouligand subderivatives satisfies the assumptions in Section 2. where y + (x) := max(y(x), 0) for all x ∈ Ω.…”

“…We shall use the standard continuous piecewise linear finite elements (FE) (see, e.g., [5,17]) to discretize the non-smooth semilinear elliptic equation (3.1) as well as the linear system (3.13). In [2,3], the discrete version of (3.1) as well as its equivalent nonlinear algebraic system were obtained by employing a mass lumping scheme for the non-smooth nonlinearity. We shall use the same technique to discretize the system (3.13).…”