On a fixed time interval, a terminal control problem generating a phase trajectory is considered. Three points are selected on the segment: two end points and one intermediate point, they correspond to the values of the trajectory. The left end of the trajectory is fixed. Finite-dimensional linear programming problems are associated with the intermediate and last moments of time, and the corresponding values of the phase trajectory should at the same time be optimal solutions of these problems. It is required to draw a phase trajectory by choosing a control so that, starting from the left end, the trajectory passes through an intermediate point and reaches the right end of the time interval. To solve the problem, a new approach is proposed based on duality theory and Lagrangian formalism. An iterative computational process of the saddle-point type is investigated. The convergence of the process in all components of the solution is proved. It is emphasized that only evidence-based computing technologies transform mathematical models into a tool for making guaranteed solutions.