Brachistochrone with dry and arbitrary viscous friction

The brachistochrone problem in the case of dry (Coulomb) and viscous friction with the coefficient that arbitrarily depends on speed is solved. According to the principle of constraint release, the normal component of the supporting curve is used as control. The standard problem of the fastest descent from a given initial point to a given terminal point assuming that the initial velocity is zero is formulated. The Okhotsimskii-Pontryagin method is used to analyze the differential of the objective function. Necessary optimality conditions are found, and a formula for the optimal control that does not involve adjoint variables is derived from them. Differential equations that allow one to obtain extremals by solving a Cauchy problem are set up. Properties of these equations are investigated. A class of simple brachistochrones is distinguished, for which singular points constituting the terminal curve and the reachability domain in the vertical plane are found. Conditions for the existence of zero controls are obtained. For some friction laws, numerical results demonstrating the shape of the determined brachistochrones and optimal time are presented.