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Convergence of distributed optimal control problems governed by elliptic variational inequalities

机译：椭圆变分不等式控制的分布式最优控制问题的收敛性

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摘要

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 u3(m)=mug1+ (1-m)ug2u_{3}(mu)=mu u_{g_{1}}+ (1-mu)u_{g_{2}} and u4(m)=umg1+ (1-m) g2u_{4}(mu)=u_{mu g_{1}+ (1-mu) g_{2}} for μ∈[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.
机译：首先，让u g 为具有源项g的椭圆变分不等式的唯一解。在一般情况下，我们建立u 3 （m）= mu g 1 +（1-m）u < sub> g 2 u_ {3}（mu）= mu u_ {g_ {1}} +（1-mu）u_ {g_ {2}}和u 4 （m）= u mg 1 +（1-m）g 2 u_ {4}（mu）= u_ { μμ[0,1]为mu g_ {1} +（1-mu）g_ {2}}。其次，我们考虑了在域边界的一部分上给出的每个正传热系数h，由内能量g上的椭圆变分不等式控制的一族分布式最优控制问题。对于给定的成本函数，并使用Mignot中给出的u 3 （μ）和u 4 （μ）之间的某些单调性（J. Funct。Anal。22：130– 185，1976），我们证明了与由椭圆变分不等式控制的该分布最优控制问题族相关的最优控制和状态的强收敛性，也由椭圆变分不等式控制了极限Dirichlet分布最优控制问题。 h达到无穷大。我们无需使用伴随状态问题（或Mignot的圆锥可微性）即可获得这种收敛，这相对于Gariboldi和Tarzia（Appl。Math。Optim。47：213–230，2003）中给出的证明是一个很大的优势椭圆变分等式控制问题。

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