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

Abstract

This paper is concerned with the dierential sensitivity analysis and the optimal control of evolution variational inequalities (EVIs) of obstacle type. We demonstrate by means of a counterexample that the solution map S of an EVI with a unilateral constraint is typically not (weakly) directionally dierentiable or Lipschitz continuous in any of the spaces H s (0, T ; H), s ≥ 1/2, where (0, T) is the time interval and H is the pivot space of the underlying Gelfand triple V → H → V *. We further establish that, despite this negative result, the solution operator is always strongly Hadamard directionally dierentiable as a function S : L 2 (0, T ; H) → L q (0, T ; H) for all 1 ≤ q < ∞, weakly-directionally dierentiable as a function S : L 2 (0, T ; H) → L ∞ (0, T ; H), and weakly directionally dierentiable as a function S : L 2 (0, T ; H) → L 2 (0, T ; V). Using the dierentiability properties of the map S, we derive strong stationarity conditions for optimal control problems that are governed by EVIs of obstacle type. The resulting optimality system is compared with that obtained by regularization.

New regularity results and finite element error estimates for a class of parabolic optimal control problems with pointwise state constraints

“…The unique solvability of the PDE (2.3), the C([0, T ]; H 1 0 (Ω))-regularity of the solution y and the Lipschitz estimate (2.4) follow from ( [4], Thm. 4.3), ( [15], Thm. 2.3) and the regularity results for the operator A in ( [33], Thm.…”

On the nonuniqueness and instability of solutions of tracking-type optimal control problems

“…on a set of positive surface measure. Indeed, if the latter was not the case for an a.e.-positive control u, then the variational inequality in (8) and the inclusion H 1 0 (Ω) ⊂ K would imply that y = S(u) ∈ H 1 (Ω) is also the solution of…”

“…in Ω, ∂ t is the time derivative in the Sobolev-Bochner sense, ∆ is the distributional Laplacian, B denotes the canonical embedding of L 2 (0, T ; L 2 (D)) into L 2 (0, T ; L 2 (Ω)) obtained from an extension by zero, and •, • denotes the dual pairing in H 1 0 (Ω). Then, using [3, Theorem 1.13, Equation (1.70)], [8,Theorem 2.3], and the lemma of Aubin-Lions, see [31,Theorem 10.12], it is easy to check that the variational inequality in (10) possesses a well-defined, weak-to-weak continuous solution map G : L 2 (0, T ; L 2 (D)) → H 1 (0, T ; L 2 (Ω)), u → y. (Note that, in order to apply [3,Theorem 1.13], one has to define the function Φ appearing in this theorem as in [3,Equation (4.9)]. )…”

Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities

“…known from the analysis of elliptic variational inequalities, see [23,27,36], but rather by considering pointwise curvature properties similar to those already exploited in Theorem 3.2. This approach to the sensitivity analysis of nonsmooth systems has already been used in [13,14] to establish directional differentiability results for solution operators of obstacle-type evolution variational inequalities and is very natural for problems of the form (F) whose solvability can also be discussed with pointwise arguments, see subsection 2.2. As in the last section, we require some additional assumptions for our analysis.…”