Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind

Abstract

This paper is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. So-called strong stationarity conditions are derived in an abstract framework. Strong stationarity conditions are regarded as the most rigorous ones, since they imply all other types of stationarity concepts and are equivalent to purely primal optimality conditions. The abstract framework is afterwards applied to four application-driven examples.

A Priori Error Estimates for an Optimal Control Problem Governed by a Variational Inequality of the Second Kind

“…More precisely, using the Hadamard directional differentiability of S, one can follow the lines of [29,30] to derive an optimality system, which is equivalent to the purely primal optimality condition f (ū; h) ≥ 0 for all h ∈ L 2 (Ω). For the problem under consideration this has been carried out in detail in [13]. The optimality conditions obtained in this way read as follows:…”

“…Theorem 5.1 (Strong Stationarity, [13,Theorem 3.10]). Let ū ∈ L 2 (Ω) be locally optimal and denote the associated state by ȳ = S(ū).…”

“…• As already mentioned, the stationarity conditions in (5.2) are equivalent to purely primal optimality conditions (see [13] for details). They can thus be regarded as the most rigorous stationarity conditions, in particular they are sharper than the ones in [14], the latter obtained via regularization.…”

“…Moreover, based on strong stationarity second order sufficient conditions can be established which ensure a quadratic growth condition, see [25,7]. For variational inequalities of the second kind strong stationarity conditions are established in [15,10,13]. Second order sufficient conditions ensuring local optimality are however not yet known for this class of problems.…”

Strong Stationarity Conditions for Optimal Control Problems Governed by a Rate-Independent Evolution Variational Inequality

“…Because of these advantageous properties, strong stationarity conditions have come to play a distinct role in the field of optimal control of nonsmooth systems and have received considerable attention in the recent past. See, e.g., [4,11,13,18,25,26,48,51] for contributions on strong stationarity conditions for optimal control problems governed by various elliptic variational inequalities of the first and the second kind, [3,14,16,37] for extensions to optimal control problems governed by nonsmooth semi-and quasilinear PDEs, and [15] for a generalization to the multiobjective setting. Note that all of these works on the concept of strong stationarity have in common that they are only concerned with elliptic variational inequalities or PDEs involving nonsmooth terms.…”