  We present a framework wherein the trajectory optimization problem (or a
problem involving calculus of variations) is formulated as a search problem in
a discrete space. A distinctive feature of our work is the treatment of
discretization of the optimization problem wherein we discretize not only
independent variables (such as time) but also dependent variables. Our
discretization scheme enables a reduction in computational cost through
selection of coarse-grained states. It further facilitates the solution of the
trajectory optimization problem via classical discrete search algorithms
including deterministic and stochastic methods for obtaining a global optimum.
This framework also allows us to efficiently use quantum computational
algorithms for global trajectory optimization. We demonstrate that the discrete
search problem can be solved by a variety of techniques including a
deterministic exhaustive search in the physical space or the coefficient space,
a randomized search algorithm, a quantum search algorithm or by employing a
combination of randomized and quantum search algorithms depending on the nature
of the problem. We illustrate our methods by solving some canonical problems in
trajectory optimization. We also present a comparative study of the
performances of different methods in solving our example problems. Finally, we
make a case for using quantum search algorithms as they offer a quadratic
speed-up in comparison to the traditional non-quantum algorithms.
