Necessary Conditions for the Impulsive Optimal Control of Multibody Mechanical Systems

In this work, necessary conditions for the impulsive optimal control of multibody mechanical systems are stated. The conditions are obtained by the application subdifferential calculus techniques to extended-valued lower semi-continuous generalized Bolza functional that is evaluated on multiple intervals. Contrary to the approach in literature so far, the instant of possibly impulsive transition is considered as a Lebesgue negligible instant. This approach is in comparison to other impulsive necessary conditions consistent with mainstream hybrid system modeling methods in which transitions happen instantaneously. The necessary conditions provide necessary criteria for the determination of optimal transition times and locations. The consideration of certain type of variations at the boundaries give birth to the concepts of internal boundary variations and discontinuous transversality conditions. The concepts are developed by the author and are presented and discussed in [20] and [22] with applications to optimal control. In this work, a characterization of these concepts in terms of upper and lower subderivatives to the extended-valued lower-semicontinuous value functional under several regularity assumptions is given. The properties of the transition sets are discussed.