A Discussion on the Optimal Control of the Shortest Time and the Minimum Energy for the Double Integrators System

Abstract

In a lecture of the optimal control about the shortest time and the minimum energy for the double integrator system, I required my students to solve a very general question, which is to calculate the shortest time and the minimum energy for the given initial position, but the answer cannot be obtained directly by the equations given in the teaching materials due to the chosen initial state with the terminal moment. In fact, one should find that the equations given in the present teaching material were not sufficient to solve the optimal shortest time and the minimum energy for the double integrators system. The general expression is given in this paper such that the optimal control of the shortest time and the minimum energy for double integrators system is solved, including the shortest time and the first switching moment and the second switching moment in the optimal control of the minimum energy under the given terminal moment. Furthermore, a discussion on the calculating of terminal moment, the first switching moment and the second switching moment is given. The new moments is defined to illustrate the existence of the first switching moment and the second switching moment for the known terminal moment under the switching function value having the opposite symbol with related to the given position.