The most important class of applied optimal control problems are time-optimal control problems. The linear speed limit problem for dynamic control objects with scalar input is discussed. By means of the Pontryagin maximum principle, the structure of extreme controls is determined and then the targeting method is used. The procedure for searching for target parameters is formalized as a nonlinear programming problem, for which a genetic algorithm is used. The proposed numerical solution of the linear problem of optimal performance has a low computational complexity. As an example, the problem of optimal performance of a 3rd-order object by the proposed method in the Matlab environment using the Control System Toolbox.