  We study a non standard infinite horizon, infinite dimensional
linear-quadratic control problem arising in the physics of non-stationary
states (see e.g. \cite{BDGJL4,BertiniGabrielliLebowitz05}): finding the minimum
energy to drive a given stationary state $\bar x=0$ (at time $t=-\infty$) into
an arbitrary non-stationary state $x$ (at time $t=0$). This is the opposite to
what is commonly studied in the literature on null controllability (where one
drives a generic state $x$ into the equilibrium state $\bar x=0$).
Consequently, the Algebraic Riccati Equation (ARE) associated to this problem
is non-standard since the sign of the linear part is opposite to the usual one
and since it is intrinsically unbounded. Hence the standard theory of AREs does
not apply. The analogous finite horizon problem has been studied in the
companion paper \cite{AcquistapaceGozzi17}. Here, similarly to such paper, we
prove that the linear selfadjoint operator associated to the value function is
a solution of the above mentioned ARE. Moreover, differently to
\cite{AcquistapaceGozzi17}, we prove that such solution is the maximal one. The
first main result (Theorem \ref{th:maximalARE}) is proved by approximating the
problem with suitable auxiliary finite horizon problems (which are different
from the one studied in \cite{AcquistapaceGozzi17}). Finally in the special
case where the involved operators commute we characterize all solutions of the
ARE (Theorem \ref{th:sol=proj}) and we apply this to the Landau-Ginzburg model.
