  For unconstrained control problems, a local convergence rate is established
for an $hp$-method based on collocation at the Radau quadrature points in each
mesh interval of the discretization. If the continuous problem has a
sufficiently smooth solution and the Hamiltonian satisfies a strong convexity
condition, then the discrete problem possesses a local minimizer in a
neighborhood of the continuous solution, and as either the number of
collocation points or the number of mesh intervals increase, the discrete
solution convergences to the continuous solution in the sup-norm. The
convergence is exponentially fast with respect to the degree of the polynomials
on each mesh interval, while the error is bounded by a polynomial in the mesh
spacing. An advantage of the $hp$-scheme over global polynomials is that there
is a convergence guarantee when the mesh is sufficiently small, while the
convergence result for global polynomials requires that a norm of the
linearized dynamics is sufficiently small. Numerical examples explore the
convergence theory.
