Abstract
The brachistochrone problem under the action of an accelerating force is investigated. The normal component of the support reaction is used as control. The standard problem of minimizing the time of going from an initial point to a terminal point by choosing an appropriate trajectory is solved using the Okhotsimskii-Pontryagin method of analyzing the differential of the objective function. Optimality conditions are found and investigated, and a formula for the optimal control is derived. A system of differential equations subject to initial conditions is obtained that does not involve adjoint variables; the solution of this system makes it possible to obtain extremal curves under the action of an accelerating force and the dry and viscous friction. In the absence of friction, various cases of quasi-constant accelerating force are analyzed; and the shapes of trajectories and other characteristics of motion are found.
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