High-order Maximal Principles

The High-Order Maximal Principle (HMP) [15] is one of the first among A. Krener’s many major contributions to modern control theory. While some precursors had appeared in the literature in earlier years, compare the discussions in [15] and in the abbreviated announcement [14], it was [15] that became the starting point for much research in the next decades, continuing to this day – typical for many of A. Kreners’s path-breaking contributions. Originally formulated in the early 1970s [14], the HMP underwent a laborious and lengthy birth process (due to many technical intricacies) until it finally appeared as a journal article in 1977 [15]. This long delay is in large part due to the very delicate nature of precise technical conditions that must be imposed on needle variations in order to assure that higher-order approximating cones have the convexity properties that are needed for meaningful tests for optimality. It was only in the last year [6], almost 30 years after the HMP was first announced, that a counter example was constructed that authoritatively demonstrated that these highly technical conditions on the families of control variations cannot be eliminated, even for some of the most benign systems. The purpose of this note is to survey some of these recent results and put them into perspective, demonstrating that even three decades after Krener’s original article major discoveries are still made in this research area. In particular, we highlight how these recent results demarcate a definite boundary beyond which any argumentation along the lines of the Maximum Principle of optimal control cannot possibly work.