This paper extends the existence results obtained in Ref. [2] to the case of infinite dimensional control problems with distributed and boundary control. By appealing to the boundary control theory of Fattorini [3] and Barbu [1] we transform the boundary control system to a linear evolution equation on a Hilbert space. This allows us to exploit the results of Ref. [2] and give sufficient conditions for the "turnpike" property to hold as well as conditions for the existence of overtaking solutions.