Abstract
This paper is concerned with the differential 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 differentiable or Lipschitz continuous in any of the spaces $H^s(0, T; H)$, $s \geq 1/2$, where $(0, T)$ is the time interval and $H$ is the pivot space of the underlying Gelfand triple $V \hookrightarrow H \hookrightarrow V^*$. We further establish that, despite this negative result, the solution operator is always strongly Hadamard directionally differentiable as a function $S : L^2(0, T; H) \to L^q(0, T; H)$ for all $1 \leq q < \infty$, weakly-$\star$ directionally differentiable as a function $S : L^2(0, T; H) \to L^\infty(0, T; H)$, and weakly directionally differentiable as a function $S : L^2(0, T; H) \to L^2(0, T; V)$. Using the differentiability 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.
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