The article considers the problem of optimal control of an object described by a linear nonstationary system and with a piecewise affine quality functional. The problem is examined in Mayer's form with both free and partially fixed right endpoints. Piecewise continuous and bounded controls that lie in some parallelepiped at each moment of time are admissible. The standard discretization of the original system and the control parametrization are used, some convergence theorems of the discrete problem solution to the continuous problem solution are presented. Further, for the obtained discrete system, the necessary and sufficient minimum conditions are written out in terms of the codifferential, the method of the modified codifferential descent is applied to it, which guarantees to find the global minimum of this problem in a finite number of steps. The proposed algorithm is illustrated with examples.