Algorithms for solving local and global boundary value problems for nonlinear and quasi-linear nonstationary control systems in the class of piecewise constant controls are developed. Constructive sufficient conditions are found that guarantee the existence of solutions to these problems. In addition, a Kalman-type criterion for local and global controllability of nonlinear and quasi-linear stationary systems, respectively, is obtained. The performance of the algorithms is illustrated by numerical simulation of the solution to the problem of controlling the motion of a robotic manipulator.