  This paper concerns a first-order algorithmic technique for a class of
optimal control problems defined on switched-mode hybrid systems. The salient
feature of the algorithm is that it avoids the computation of Fr\'echet or
G\^ateaux derivatives of the cost functional, which can be time consuming, but
rather moves in a projected-gradient direction that is easily computable (for a
class of problems) and does not require any explicit derivatives. The algorithm
is applicable to a class of problems where a pointwise minimizer of the
Hamiltonian is computable by a simple formula, and this includes many problems
that arise in theory and applications. The natural setting for the algorithm is
the space of continuous-time relaxed controls, whose special structure renders
the analysis simpler than the setting of ordinary controls. While the space of
relaxed controls has theoretical advantages, its elements are abstract entities
that may not be amenable to computation. Therefore, a key feature of the
algorithm is that it computes adequate approximations to relaxed controls
without loosing its theoretical convergence properties. Simulation results,
including cpu times, support the theoretical developments.
