Use of Augmented Lagrangian Methods for the Optimal Control of Obstacle Problems

Abstract

Abstract. We investigate optimal control problems governed by variational inequalities involving constraints on the control, and more precisely the example of the obstacle problem. In this paper we discuss some augmented Lagrangian algorithms to compute the solution.

The existence results for optimal control problems governed by a variational inequality

“…The optimal control problem governed by a semilinear elliptic variational inequality was studied by many authors in different aspects. For example, see [1,3,8,15,17] and the references cited therein. The optimal control problem governed by a quasilinear elliptic variational inequality was investigated in [16,21].…”

“…As Lions [13] points out, finding necessary and sufficient conditions for the optimal control and constructing algorithms amenable to numerical computation for the approximation of the optimal control are two important objectives of the optimal control theory. Necessary and sufficient conditions for optimal control problems governed by variational inequalities have been investigated by a number of authors (see Lions [12][13][14], Adams and Lenhart [1], Barbu [2], Mignot and Puel [17], He [8], Bergounioux [3] and Ye [20]). We may consider these problems from many methods.…”

“…Research in optimal obstacle control is also useful in these theories and applications. For example, see [3,15,16,21].…”

A constraints relaxation technique for solving an identification of coefficients problem issued from hydrodynamic lubrication

“…The optimality system can be solved with some Lagrangian or augmented Lagrangian algorithms (see for instance [10,15]), since [θ ν , h ν , p ν , q ν , λ ν ] may be interpreted as stationary point for the Lagrangian L defined by…”

“…In order to recover discontinuous thickness, we introduce the space BV (Ω) of bounded variation functions and suppose that the total variation of control variables is uniformly bounded to ensure a compactness result. The constraint relaxation technique, that we use here, has been initiated by M. Bergounioux [10], and considered later by M. Bergounioux [11], M. Bergounioux and D. Tiba [12]to solve optimal control of problems governed by elliptic variational inequalities with state constraints. The optimality conditions obtained by the authors are easy to exploit in the numerical experiments.…”