We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable as well. We also study the optimal control of the quasi-variational inequality and provide necessary optimality conditions of strongly stationary type.

中文翻译：

---

小假设下的椭圆拟变分不等式：唯一性，微分稳定性和最优控制

我们考虑由移动集控制的拟变分不等式。我们假设集合的运动具有小的Lipschitz常数。在此要求下，我们证明了拟变分不等式具有一个唯一的解，该解连续取决于Lipschitz取决于源项。如果问题的数据是（方向）可区分的，则解决方案图也是方向可区分的。我们还研究了拟变分不等式的最优控制，并提供了强平稳型的必要最优性条件。