  Considering a general nonlinear dissipative finite dimensional optimal
control problem in fixed time horizon T , we establish a two-term asymptotic
expansion of the value function as $T\rightarrow+\infty$. The dominating term
is T times the optimal value obtained from the optimal static problem within
the classical turnpike theory. The second term, of order unity, is interpreted
as the sum of two values associated with optimal stabilization problems related
to the turnpike.
