Asymptotic Stability and the Turnpike Property in Some Simple Control Problems
This chapter is devoted to the presentation of the fundamental ideas concerning the asymptotic stability of optimal trajectories and the related turnpike theorems, as they first appeared in the economic literature. In order to avoid the use of too much economic jargon, only two simple examples will be considered. The one in section 3.3 concerns the optimal exploitation of a fish population: on any fixed, sufficiently large horizon with given initial and terminal states, the optimal trajectory will first reach an “optimal steady state”, stay on this arc as long as possible and leave it only when the terminal state will have to be reached. This is an exact “turnpike” property. The example of section 3.4 is one of the simplest optimal economic growth models. Using a phase diagram technique it is possible to show that an optimal trajectory over an infinite time horizon will have to converge toward an asymptotic equilibrium and that the costate variable also has to converge.
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