On regularity conditions in higher-order state-constrained control problems

Abstract

This work is aimed at the study of optimal control problems with state constraints of higher order. Such problems are due to multi-level dynamical control systems of triangle-type which are often encountered in various engineering applications, notably in Robotics. In the framework of the proposed investigation, a notion of higher-order regularity with respect to the state constraints (termed as k-regularity) is formulated. Under such a condition, a strengthened variant of Pontryagin’s maximum principle is derived in which the non-triviality condition is pointwise and thereby allows for the informative maximum condition. An example of a 3-order control problem is considered, which demonstrates the essentiality of k-regularity for pointwise non-triviality and also shows how the presented necessary optimality conditions can be applied.