Augemented Lagrangian Techniques for Elliptic State Constrained Optimal Control Problems
SIAM Journal on Control and Optimization
Homogenization of an Optimal Control Problem
SIAM Journal on Control and Optimization
Use of augmented Lagrangian methods for the optimal control of obstacle problems
Journal of Optimization Theory and Applications
Optimal Control of PDEs with Regularized Pointwise State Constraints
Computational Optimization and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
Lagrange Multiplier Approach to Variational Problems and Applications
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First, let u g be the unique solution of an elliptic variational inequality with source term g. We establish, in the general case, the error estimate between $u_{3}(\mu)=\mu u_{g_{1}}+ (1-\mu)u_{g_{2}}$ and $u_{4}(\mu)=u_{\mu g_{1}+ (1-\mu) g_{2}}$ for μ驴[0,1]. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy g for each positive heat transfer coefficient h given on a part of the boundary of the domain. For a given cost functional and using some monotony property between u 3(μ) and u 4(μ) given in Mignot (J. Funct. Anal. 22:130---185, 1976), we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter h goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot's conical differentiability) which is a great advantage with respect to the proof given in Gariboldi and Tarzia (Appl. Math. Optim. 47:213---230, 2003), for optimal control problems governed by elliptic variational equalities.