The application of the exact penalty method to the problem of finding a cost-optimal tra¬jectory is considered. To minimize the exact penalty function, the apparatus of constructive nonsmooth analysis is employed. It is shown that the application of the exact penalty method in the original functional formulation leads to the generation of successive approximations in the form of non-elementary functions. Although this fact is theoretically expected, this work is the first to investigate its practical consequences: it is demonstrated that as early as the second iteration, the computational complexity increases to such an extent that the analytical implementation of the algorithm becomes infeasible (a slowdown of more than 10. 000 times compared to the Ritz method). To overcome these difficulties, a grid-based adaptation of the method is proposed, which allows obtaining a solution in a reasonable time. The obtained results may be useful for researchers planning to apply the exact penalty method to problems in the fields of calculus of variations and control theory, as well as for solving trajectory optimization and route planning problems.