Constraint reactions in optimal control of mechanical systems

Abstract

This paper is dedicated to the establishment of a general procedure of forming the optimal control problem of variable-mass nonholonomic rheonomic mechanical systems, where reactions of constraints are present in differential equations of motion. Dimensions and structure of a configuration space depend on the number of reactions of constraints that are the subject of our interest, i.e. only the reactions whose magnitudes are subjected to limitations are considered. In this paper, the procedure enables the direct application of Pontryagin’s maximum principle for the systems with limited phase state. Attention is particularly focused on discussing various modes of realizing the control by combining the active control forces and subsequent imposition of ideal holonomic mechanical constraints. Brachistochronic motions play an important role in this type of problems, because in them the control of motion can be realized exclusively with ideal constraints. The paper provides three examples of this method application, which are related to the realization of the brachistochronic motion of mechanical systems.