ABSTRACT
This chapter discusses on several recent papers dealing with linear system with indefinite quadratic cost, brings into focus various aspects of the optimization theory. Although much of this theory is a natural extension of the classical ideas inherited from the calculus of variations, the theory also explains new and interesting phenomena not encountered in the classical studies. The chapter addresses the following questions: The necessary and sufficient conditions for optimality, Optimality properties of extremals, Fixed point boundary value problems versus the variable problem, The singular problem and its optimal synthesis, and Passage to the infinite time interval. It concentrates on the conceptual aspects of the theory, leaving the more elaborate arguments to the original sources. The singular nature of the problem demands a systematic use of Lie brackets of higher order, whose proper understanding is rooted in a generalization of the Maximum Principle.
