Optimal trajectories of air and space vehicles

The author developed a theory of optimal trajectories for air vehicles with variable wing area and conventional wings. He applied a new theory of singular optimal solutions and obtained the optimal flight in many cases. At first glance, the results may seem strange however, this is correct and this paper will show how this new theory may be used. The main idea of the research is in using the vehicle's kinetic energy for increasing the range of missiles and projectiles. The author shows that the range of a ballistic warhead can be increased 3‐4 times if an optimal wing is added to the ballistic warhead, especially a wing with variable area. If increased range is not needed, the warhead mass can be increased. The range of big gun shells can also be increased 3‐9 times. The range of aircraft may be improved 3‐15 percent and more. The results can be used for the design of aircraft, missiles, flying bombs and shells of big guns.