The paper considers the problem of optimal control of an object described by a system with a continuously differentiable right-hand side and a nondifferentiable (but only quasidifferentiable) quality functional. We consider a problem in the form of Mayer with both a free and a fixed right end. Admissible controls are piecewise continuous and bounded vector functions, which belong to a certain polyhedron at each moment of time. Standard discretization of the initial system and control parameterization is performed, and theorems on the convergence of the solution of the obtained discrete system to the desired solution of the continuous problem are presented. Further, for the obtained discrete system, the necessary and, in some cases, sufficient minimum conditions are written in terms of quasidifferential. The quasidifferential descent method is applied to this problem. The developed algorithm is demonstrated by examples.