A turnpike theorem for continuous-time control systems when the optimal stationary point is not unique

Abstract

We study the turnpike property for the nonconvex optimal control problems described by the differential inclusion $\dot{x}\in a\left(x\right)$. We study the infinite horizon problem of maximizing the functional ${\int }_0^{T}u\left(x\left(t\right)\right)dt$ as $T$ grows to infinity. The turnpike theorem is proved for the case when a turnpike set consists of several optimal stationary points.