Optimal Control Of The Obstacle For A Parabolic Variational Inequality

. An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L 2 \Gamma 0; T ; H 2(\Omega\Gamma " H 1 0(\Omega\Gamma \Delta with / t 2 L 2 (Q) is taken as the control, and the solution to the obstacle problem taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established. Key Words: Parabolic Obstacle Problems, Optimal Control AMS Classification: 49A29, 35K85 2 1. Introduction An optimal control problem for a parabolic variational inequality is considered on a domain Q =\Omega \Theta (0; T ), with\Omega ae R n , a bounded domain with C 1 boundary @ Let z 2 L 2 (Q) be a given target profile. Our control set is U = \Phi / 2 L 2 \Gamma 0; T ; H 2 (\Omega\Gamma " H 1 0(\Omega\Gamma \Delta j/ t 2 L 2 (Q); /(x; 0) = 0 \Psi : For any / 2 U..