4. A theorem on the increments of controlled object coordinates. Optimal control

Annotation: Traditionally high requirements for the ideal weight and payload capabilities of spacecraft originate the need for optimal design solutions and, further, algorithms to enable the control of spacecraft maneuvering and attitude with the optimal speed and power consumption. Applying popular methods of optimal control determination encounters insurmountable complications in most practically critical situations, especially considering the persistently widening range of functions that describe controlling and disturbing forces. In this paper, the authors prove a new theorem in mechanics, which enables the solving of numerous practically critical problems related to optimal control. The core of the theorem about the increments of controlled objects’ coordinates comes down to a simple relationship between coordinate increments, the areas of acting forces, and the coordinates of these areas’ centers of gravity on the time axis. Calculating the mentioned areas and the coordinates of their centers of gravity is chiefl y an elementary problem. Therefore, the presented theorem provides an analytical solution for the algorithms of optimal control. To exhibit an example of the theorem’s practicality, the authors apply it in solving a problem concerning spacecraft attitude correction with the optimal speed and power consumption in the presence of disturbing forces. The paper also provides an analytical solution for a problem of control at a limited rate of controlling force variation. Additionally, the authors present a solution for a classical problem of spacecraft soft landing on celestial bodies. This solution is expressed in the form of a precise and simple condition for determining the ignition time for deceleration thrusters. The paper addresses the issues of optimal control for the programmed motion of actual objects, whose masses, moments of inertia, propulsion thrust, and speeds of variation of the controlling forces diff er from the ideal ones used in the computations.