Об одной задаче преследования при наличии сопротивления среды

Abstract

Рассматривается задача преследования двух движущихся материальных объектов – перехватчика (преследователя) и цели (убегающего). Объекты движутся в одной плоскости под действием управляемых сил, направленных всегда перпендикулярно их скоростям. Законы изменения величин управляемых сил перехватчика и цели определяются контроллерами первого порядка. Кроме того, на объекты действуют силы сопротивления среды, пропорциональные квадратам скоростей. В рассматриваемой задаче построено управление, гарантирующее встречу. This paper considers a game pursuit problem, in which the interceptor (pursuer) and target (evader) move in the same plane under the influence of controlled forces directed always perpendicularly to their velocities. The laws of value variation of controlled forces of interceptor and target are determined by first-order controllers. Besides that, each object is influenced by the force of resistance of a medium which is proportional to the squared velocity of the object and is directed to the side which is opposite to its velocity. It is assumed that during the motion of objects, directions of their velocities are little different from an axis passing through their initial positions. It allows linearizing equations of motion of the pursuer and target. As a result of linearization it turns out that the projections of the position of objects on the axis change by a known law. When there is a coincidence of these projections, the time point prescribes the moment of the end of prosecution process. It is expected that the capture has occurred, if at this time point the module of difference of vector projections of object position on a perpendicular axis does not exceed a predetermined number. Eventually, a linear differential game of pursuit-evasion with fixed end time is obtained. Full information on the state of objects at each time point is available for players. With the help of a linear change of variables, the game comes down to a homogeneous onedimensional differential game, in which the possible values of control belong to the segments which depend on the time. As a result of the research the set of initial conditions is found, under which the capture of target is possible when in any of its allowable motion, and the control of pursuer that will ensure the capture is built.