The objective of this research is to use Pontryagin's Minimum Principle to solve the constrained nonlinear minimum-fuel problem of a small craft performing a "hop" maneuver on Mars. The solution then should return a continuous optimal control or control law which is then applied to sub-optimal trajectories that are known solutions. The augmented trajectories are then evaluated through a cost function to determine the optimal trajectory to be followed to reach its destination with minimum fuel expenditure. Since the final state of the craft is known, but the time it takes to reach its destination is variable, there is no analytical solution. Therefore, numerical iterations of MatLab's bvp4c solver are required with varying guesses until an optimal solution is found. Results show that, although many solutions can be found, no sufficient solution has been determined. Further research towards a solution will likely require different numerical solvers, as solutions generated by bvp4c are highly dependent on the accuracy of the initial solution guess and the control being continuous.