Optimal control in coefficients for weak variational problems in Hilbert space

In this paper we give sufficient conditions for the existence of solutions of a problem of parametric optimization. We use continuity with respect to a functional parameter of weak solutions of a variational problem in a Hilbert space.We consider a problem of optimization with the control in coefficients of linear parabolic equation as an example. Using results of Spagnolo we characterize the closure of the reachable set. Finally, we construct an example of an optimization problem with the control in coefficients of a parabolic equation which does not have an optimal solution.