Numerical Solution of bilateral obstacle optimal control Problem

In this work we consider the numerical resolution of the bilateral obstacle optimal control problem given in Bergounioux et al [7]. Where the main feature of this problem is that the control and the obstacle are the same. Variational inequalities and related optimal control problems have been recognized as suitable mathematical models for dealing with many problems arising in different fields, such as shape optimization theory, image processing and mechanics, (see for example [4], [5], [10], [19], [23]). Optimal control problem governed by variational inequalities has been studied extensively during the last years by many authors, such as [2], [21], [22]. These authors have studied optimal control problems for obstacle problems (or variational inequalities) where the obstacle is a known function, and the control variables appear as variational inequalities. In other words, controls do not change the obstacle and, on the other hand , in [1], [7], [11] the authors have studied another class of problems where the obstacle functions are unknowns and are considered as control functions. In this paper, we investigate optimal control problems governed by variational inequalities of obstacle type. This kind of problem is very important and it can lead to the shape optimization problem governed by variational inequality, it may concern the optimal shape of dam [9], for which the obstacle gives the shape to be designed such that the pressure of the fluid inside the dam is close to a desired value. Besides, if we want to design a membrane having an expected shape, we need to choose a suitable obstacle. In this case, the obstacle can be considered as a control, and the membrane as the state (see for example [15]). It should be pointed out that, in the optimal control problem of a variational inequality, the main difficulty comes from the fact that the mapping T between the control and the state (control-to-state operator) is not Gateaux differentiable as pointed it out in [22], [21