The brachistochrone problem with constraints on the curvature of the trajectory⁎⁎The paper was prepared under financial support of the Ministry of Science and Higher Education of the Russian Federation within the Program of the Center ”Supersonic” (agreement 075-15-2020-923).

The paper describes various brachistochrone problem statements. In the beginning an approach to solving the brachistochrone problem with a nonzero initial velocity using optimal control methods is presented. The substantiation of the transition to zero initial velocity is also provided. The second type of problem statement takes into account explicit restrictions on the curvature of the trajectory. The solution is obtained by using the Pontryagin’s Maximum Principle. The paper demonstrates that the optimal solution includes both regular and singular trajectory parts.