Optimal Control of Variational Inequalities

a.e. in Q approximating assume assumptions Ay(t boundary value problem conclude convex function defined denoted dxdt elliptic exists ǝh(u follows Green's formula H x H Hence Hilbert space infer infimum integrate on 0,t L²(n L²(Q Lemma Let y*,u letting & tend Lipschitzian locally Lipschitz lower semicontinuous maximal monotone graph maximum principle measurable subset Minimize nonlinear norm obstacle problem operator optimal control optimal control problem optimal pair pair in problem partial differential equations proof of Theorem Proposition satisfies condition scalar product Section sequence Stefan problem strongly in H subsequence subset of H tend to zero Theorem 5.1 unique solution variational inequality w¹² weak star weakly compact weakly in H weakly in L²(0,T;H yields yo(x ε ε ду