An Inverse Problem for Evolution Inclusions

with D(φ(·,u)) ⊂ j for all u ∈ = ∏N j=1 j . The existence of an open loop of (1.1), (1.2) with φ independent of the control u, has been established in Ton [7]. With optimal shape design and with inverse problems inmind, we will consider the case when φ depends on the control u as it appears in the top order term of the partial differential operators involved in the problems. Optimal design of domains has been investigated by Barbu and Friedman [1], Canadas et al. [2], Gunzburger and Kim [3], Pironneau [6], and others. Inverse