  This paper deals with a non-standard infinite dimensional linear-quadratic
control problem arising in the physics of non-stationary states (see e.g. [6]):
finding the minimum energy to drive a fixed stationary state x = 0 into an
arbitrary non-stationary state x. The Riccati Equation (RE) associated to this
problem is not standard since the sign of the linear part is opposite to the
usual one, thus preventing the use of the known theory. Here we consider the
finite horizon case. We prove that the linear selfadjoint operator P(t),
associated to the value function, solves the above mentioned RE (Theorem 4.12).
Uniqueness does not hold in general but we are able to prove a partial
uniqueness result in the class of invertible operators (Theorem 4.13). In the
special case where the involved operators commute, a more detailed analysis of
the set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples of
applications are given.
