Turnpike Properties for Autonomous Problems

Abstract

In this chapter we study the structure of approximate solutions of an autonomous discrete-time control system with a compact metric space of states X which is a subset of a finite-dimensional Euclidean space. This control system is described by a nonempty closed set Ω ⊂ X × X which determines a class of admissible trajectories (programs) and by a bounded upper semicontinuous function v : Ω → R ¹ which determines an optimality criterion. We are interested in turnpike properties of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals. Usually in economic dynamics turnpike properties were studied for systems such that all their good programs converge to a turnpike which was an interior point of Ω. In this chapter we prove turnpike results for a large class of control systems for which the turnpike is not necessarily an interior point of Ω. The Robinson–Solow–Srinivasan model is a particular case of the general model studied in the chapter.