A Turnpike Theorem for Continuous-time Control Systems When the Optimal Stationary Point Is Not Unique

Here, x0 ∈Ω is an assigned initial point. The multivalued mapping a : Ω→ Πc(R) has compact images and is continuous in the Hausdorff metric. We also assume that at every point x ∈Ω the set a(x) is uniformly locally connected (see [2]). The function u : Ω→ R1 is a given continuous function. In this paper, we study the turnpike property for problem (1.1) and (1.2). The term of turnpike property was first coined by Samuelson (see [17]) where it is shown that an efficient expanding economy would spend most of the time in the vicinity of a balanced equilibrium path. This property was further investigated by Radner [14], McKenzie [12], Makarov and Rubinov [7], and others for optimal trajectories of a von Neuman-Gale model with discrete time. In all these studies, the turnpike property was established under some convexity assumptions.