  In this paper, we develop several necessary conditions of turnpike property
for generalizaid linear-quadratic (LQ) optimal control problem in infinite
dimensional setting. The term 'generalized' here means that both quadratic and
linear terms are considered in the running cost. The turnpike property reflects
the fact that over a sufficiently large time horizon, the optimal trajectories
and optimal controls stay for most of the time close to a steady state of the
system. We show that the turnpike property is strongly connected to certain
system theoretical properties of the control system. We provide suitable
conditions to characterize the turnpike property in terms of the detectability
and stabilizability of the system. Subsequently, we show the equivalence
between the exponential turnpike property for generalized LQ and LQ optimal
control problems.
