Necessary and Sufficient Conditions for Optimality of Singular Controls in a Certain Class of Multi-Input Systems

Necessary and sufficient conditions for singular control problems to be optimal are presented for the case where the system has n states and n-2 control variables. For the second and third order control systems, one of the authors has proposed the so-colled selection rules based on the geometrical property of the sufficient conditions. The purposes of this paper are to extend the author's methods to multi-input control systems and to show that, including the case of selection rules, the sufficient conditions derived here are at the same time necessary ones. First, the given system is transformed, so that mathematically one of the state variables has a role as the independent variable, while the n-2 derivatives act as control variables. The necessary conditions for the existence of the singular controls are derived from the first variation to this new system. Next, the second variation is calculated. By definition of a new perturbation the variational system can be transformed into a homogeneous one, whose solution vanishes for all times, while the second variation is reduced to take a quadratic form with only control perturbations. Thus the necessary and sufficient conditions can be obtained. Finally, the relation between these conditions and the selection rules are discussed. To derive the selection rules, a pair of necessary and sufficient conditions which are equivalent to each other must be obtained. And it becomes clear that it is realizable only in the case of n=3.