  In this paper, we prove a theorem on the rate of convergence for the optimal
cost computed using PS methods. It is a first proved convergence rate in the
literature of PS optimal control. In addition to the high-order convergence
rate, two theorems are proved for the existence and convergence of the
approximate solutions. This paper contains several essential differences from
existing papers on PS optimal control as well as some other direct
computational methods. The proofs do not use necessary conditions of optimal
control. Furthermore, we do not make coercivity type of assumptions. As a
result, the theory does not require the local uniqueness of optimal solutions.
In addition, a restrictive assumption on the cluster points of discrete
solutions made in existing convergence theorems are removed.
