On the variational process in optimal control theory

Abstract

The variational process is established and applied to the development of the second variation for the free-final-time optimal control problem. First, it is shown that, given a change in the control (the independent variable), the change in the state (the dependent variable) consists of all orders of the change in the control. Hence, the change in the state is a total change. This implies that variations of dependent variations exist. Next, the variational relationship between time-constant and time-free variations is developed, and the formula for taking the variation of an integral is presented. The results are used to derive the second variation following three different approaches: taking the variation of the first variation after performing the integration by parts; taking the variation of the first variation before performing the integration by parts; and using the Taylor series approach. The ability to get the same result requires the existence of the total change in the state or of the variation of the state variation. Finally, if the nominal path is not an extremal, this process gives extra terms in the second variation.