Fuller Phenomenon in multiple input control systems

Abstract

Many optimal control problems exhibit a peculiar behavior that is not completely understood, the Fuller Phenomenon. In a naive way, this phenomenon can be described as the accumulation of discontinuities in the control function. In this paper extensions to multiple input control systems of classic results on the detection of this behavior are given. It is also given a necessary condition to an arc be singular. This condition gives a potentially new direction of $p^\bot$ which is used to extend the First Pontryagin Cone, improving the geometric comprehension of the problem. These techniques are applied to control systems derived from Hamiltonian systems, and sufficient conditions for existence of the Fuller Phenomenon in a subfamily of these systems are given.