Brachistochrone problem and two-dimensional Goddard problem

Cherkasov, O.Y.; Malykh, E.V.; Smirnova, N.V.

Авторы: Cherkasov O.Yu, Malykh E.V., Smirnova N.V.
Журнал: Nonlinear Dynamics
Год издания: 2022
Издательство: Springer Nature
Местоположение издательства: Switzerland
DOI: 10.1007/s11071-022-07857-x
Аннотация: Abstract The motion of a point mass in a verticalplane under the action of gravity, linear viscousfriction, support reaction of the curve, and the thrust isconsidered. The angle of inclination and the thrust areconsidered as control variables. The amount of fuel isindicated. The goal of the control is to maximize thehorizontal coordinate of a point in a given time. Thenovelty of the paper lies in the fact that the structure ofthe optimal thrust is determined and the optimalsynthesis is constructed in the three-dimensional space‘‘slope angle-velocity-mass’’. For the case of a motionwithout friction, it is shown that the optimal thrustcontrol takes boundary values, and the trajectoryconsists of two arcs, at the beginning with maximumthrust, and ending with zero thrust. The optimalsynthesis in the three-dimensional space ‘‘slope anglevelocity-mass’’ is constructed for a specific area of thevariables. For the case of linear viscous friction, an arcwith an intermediate thrust can be included in anextreme trajectory. Assuming that the intermediate(singular) thrust satisfies the constraints, it is shownthat the optimal thrust program consists of two arcs,maximum thrust at the beginning and zero thrust at theend, or three arcs: maximum thrust at the beginning,then intermediate thrust and zero thrust at the end. Thefollowing combination of arcs is also possible: zerothrust at the beginning, then an intermediate thrust andagain zero thrust at the end. The logic of thrust controlis similar to the well-known solution of the Goddardproblem. The results of numerical simulation illustrating the theoretical conclusions are presented. Theresults are also valid for the Brachistochrone problem,which is interrelated to the range maximizationproblem. The results of numerical simulation illustrating the theoretical conclusions are presented.Keywords Brachistochrone problem Goddardproblem Singular control Optimal thrust Optimalsynthesis
Добавил в систему: Черкасов Олег Юрьевич

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Brachistochrone problem and two-dimensional Goddard problemстатья

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