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

We study the turnpike property for the nonconvex optimal control problems described by the differential inclusion ẋ ∈ a(x). We study the infinite horizon problem of maximizing the functional ∫0T u(x(t))dt as T grows to infinity. The turnpike theorem is proved for the case when a turnpike set consists of several optimal stationary points.