  We analyze the consequences that the so-called turnpike property has on the
long-time behavior of the value function corresponding to a finite-dimensional
linear-quadratic optimal control problem with general terminal cost and
constrained controls.
  We prove that, when the time horizon $T$ tends to infinity, the value
function asymptotically behaves as $W(x) + c\, T + \lambda$, and we provide a
control interpretation of each of these three terms, making clear the link with
the turnpike property.
  As a by-product, we obtain the long-time behavior of the solution to the
associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is
not coercive in the momentum variable. As a result of independent interest, we
showed that linear-quadratic optimal control problems with constrained control
enjoy a turnpike property, also particularly when the steady optimum may
saturate the control constraints.
