ABSTRACT
One of the main invariants of an extremal in a regular variational problem is its Morse index. If the extremal is optimal then its Morse index is zero. In the general case the Morse Index could be interpreted as the minimal number of additional relations that have to be satisfied by the admissible variations of the given trajectory in order to make it optimal. This chapter explores the method and compute the index for the problem with constraints on the control parameters. It considers in some detail bang-bang controls and then briefly present a universal formula valid for bang-bang as well as singular parts of the optimal trajectory. The chapter aims to the study of optimal control problems for smooth systems. Necessary conditions for optimality are formulated for controls that satisfy Pontryagin's Maximum Principle. The chapter also aims to optimal control, and consider the time-optimal problem for the system.
