Control problems governed by Sobolev partial differential equations

Let G be a bounded subset of Rⁿ with a smooth boundary and Q = G × (0, T]. We consider a control problem governed by the Sobolev initial-value problem My_t(u) + Ly(u) = u in L²(Q), y(·, 0; u) = 0 in L₂(G), where M = M(x) and L = L(x) are symmetric uniformly strongly elliptic operators of orders 2m and 2l, respectively. The problem is to find the control u₀ of L²(Q)-norm at most b that steers to within a prescribed tolerance ? of a given function Z in L²(G) and that minimizes a certain energy functional. Our main results establish regularity properties of u₀. We also give results concerning the existence and uniqueness of the optimal control, the controllability of Sobolev initial-value problems, and properties of the Lagrange multipliers associated with the problem constraints.