  This paper addresses the problem of control synthesis for nonlinear optimal
control problems in the presence of state and input constraints. The presented
approach relies upon transforming the given problem into an
infinite-dimensional linear program over the space of measures. To generate
approximations to this infinite-dimensional program, a sequence of
Semi-Definite Programs (SDP)s is formulated in the instance of polynomial cost
and dynamics with semi-algebraic state and bounded input constraints. A method
to extract a polynomial control function from each SDP is also given. This
paper proves that the controller synthesized from each of these SDPs generates
a sequence of values that converge from below to the value of the optimal
control of the original optimal control problem. In contrast to existing
approaches, the presented method does not assume that the optimal control is
continuous while still proving that the sequence of approximations is optimal.
Moreover, the sequence of controllers that are synthesized using the presented
approach are proven to converge to the true optimal control. The performance of
the presented method is demonstrated on three examples.
