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Considering a general nonlinear dissipative finite dimensional optimalcontrol problem in fixed time horizon T , we establish a two-term asymptoticexpansion of the value function as $T\rightarrow+\infty$. The dominating termis T times the optimal value obtained from the optimal static problem withinthe classical turnpike theory. The second term, of order unity, is interpretedas the sum of two values associated with optimal stabilization problems relatedto the turnpike.

Two-term large-time asymptotic expansion of the value function for dissipative nonlinear optimal control problems