Optimal thrust control for brachistochrone with viscous friction
The problem of optimization of controlled descent of an aircraft in a homogeneous field of gravity in the presence of a resisting medium and an accelerating force is considered. Movement occurs in a vertical plane. As a control, lifting force and thrust force are applied. The model under consideration also describes the motion of a point on a curve, where the control variables are the normal component of the reaction force and the accelerating force. The goal of control is to maximize the horizontal distance ( terminal term) and the minimization of energy costs (integral term) over a fixed time interval. Range maximization problem is interrelated with Brachistochrone problem — the problem of choosing the shape of the trajectory connecting two given points in the vertical plane, the time along which will be minimal. To investigate the problem, the Pontryagin maximum principle and the methods of qualitative investigation of dynamical systems are applied. It is established that extreme trajectories correspond to motion with singular control for normal reaction of the reference curve and with regular control for thrust. Extreme control is constructed in the form of feedback on the phase variables of the initial system. The characteristic properties of the trajectories, which allowed to substantiate the results obtained by other authors by means of numerical simulation, or formulated as hypotheses. It is shown that at large time intervals extreme trajectory consists of three parts, which are output in the neighborhood of asymptotic lines, the movement in this neighborhood and out to meet the final conditions. The results obtained can be used for design of the quasi-optimal solutions and as effective initial approximations for the numerical solution of the trajectory optimization problems described by models of a higher order.
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