In this paper the turnpike property is established for convex optimal control problems, involving undiscounted utility function and differential inclusions defined by multi-valued mapping having convex graph. Utility function is concave but not necessarily strictly concave. The turnpike theorem is proved under the main assumption that over any given line segment, either multi-valued mapping is strictly convex or utility function is strictly concave. In this way, the strictly convexity/concavity assumption is distributed between the mapping and utility function.
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