On the structure of optimal control for certain systems linear in control

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Abstract
This paper considers the problem of optimal control for systems which are linear in control but non-linear in state. It is shown that there is a large class of control systems in which optimal singular trajectories are allowed to lie only on a certain hypersurface when a condition similar to the normality condition for linear time-optimal control systems is fulfilled. Furthermore, under the same condition, it is proved that the optimal control becomes totally bang-bang or totally singular. A necessary and sufficient condition for the optimality of these singular solutions is established. It should be noted that such a hypersurface cannot be defined when the system is linear. Finally an example is given to show that the results obtained can no longer be valid for a slightly wider class of optimal control systems.