Controllability for Variational Inequalities of Parabolic Type with Nonlinear Perturbation

Let H and V be two complex Hilbert spaces. Assume that V is a dense subspace in H and the injection of V into H is continuous. If H is identified with its dual space, we may write V ⊂ H ⊂ V ∗ densely and the corresponding injections are continuous. The norm on V , H, and V ∗ will be denoted by ‖ · ‖, | · |, and ‖ · ‖∗, respectively. The duality pairing between the element v1 of V ∗ and the element v2 of V is denoted by v1, v2 , which is the ordinary inner product in H if v1, v2 ∈ H. For l ∈ V ∗, we denote l, v by the value l v of l at v ∈ V . We assume that V has a stronger topology than H and, for the brevity, we may regard that