On the smoothness of optimal paths: some local turnpike results
Résumé
We study the local turnpike property for two classes of infinite-horizon discrete-time detreministic maximazition problems which common applications, e.g. optimal growth control. We follow a functional-analytic approach and rely on an implicit function theorem for the space of the sequences which converge to zero. We shall assume the existence of an optimal path which is not necessarily a steady-state. Relying on material developped in Blot and Crettez "On the smoothness of optimal paths" (Decis. Econo Finance 27, 1-34, 2004), we provide conditions under which a variation in the initial condition (i.e. capital stock and discount rate) yields an optimal solution which converges toward a reference solution when time becomes infinite. We also provide new results on bounded solutions of difference equations.