Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind

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• 作者：Mahdi Boukrouche^a ; ^{Mahdi.Boukrouche@univ-st-etienne.fr"" rel=""nofollow} ; Domingo A. Tarzia^b ; ¹ ; ^{DTarzia@austral.edu.ar"" rel=""nofollow}
• 关键词：Parabolic variational inequalities of the second kind ; Convex combination of solutions ; Monotony property ; Regularization method ; Dependency of the solutions on the data ; Strict convexity of cost functional ; Optimal control problems
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2011
• 出版时间：August 2011
• 年：2011
• 卷：12
• 期：4
• 页码：2211-2224
• 全文大小：325 K

文摘

Let u_g be the unique solution of a parabolic variational inequality of second kind, with a given g. Using a regularization method, we prove, for all g₁ and g₂, a monotony property between μu_g1+(1−μ)u_g2 and u_{μg1+(1−μ)g2} for μ[0,1]. This allowed us to prove the existence and uniqueness results to a family of optimal control problems over g for each heat transfer coefficient h>0, associated with the Newton law, and of another optimal control problem associated with a Dirichlet boundary condition. We prove also, when h→+∞, the strong convergence of the optimal controls and states associated with this family of optimal control problems with the Newton law to that of the optimal control problem associated with a Dirichlet boundary condition.