Properties of control switching points for non-linear fourth order system

The problem of development of controlled object from the preset initial position to origin of coordinates with the preset value of course angle in a final point for the minimum time is reviewed in the paper. Object movement is described by a nonlinear system of ordinary differential equations of the fourth degree. It should be noted that similar systems were repeatedly analyzed in the past. A game approach, so-called “homicidal shauffeur” game was most commonly used. In the proposed paper an object movement research is carried out using a “classical” method of optimum control theory–a maximum principle. Quite detailed analysis of difference of the considered problem statement apart from studied and solved in the past is carried out. As a number of switching points of optimum control for initial task is unknown, some properties of acceptable trajectories of movement of controlled object, meeting necessary optimality condition-a maximum principle, are studied in the paper. It is assumed that optimum control and, therefore, an optimum trajectory of movement exists, and as that for transfer of object to the origin of coordinates with the preset value of course angle from any initial point not less than two switching of course angle control are required. A maximum quantity of switching points for speed control of object is further on a consistency open interval of this control. All reasoning is performed on the assumption that the control of course angle and speed control aren’t equal to zero. Results of the conducted analysis are presented as the respective theorem. Bibliogr. 10.