A Criterion for an Optimal Control System to have Finite Switchings

A simple criterion is suggested which determines whether ornot an optimal control system has a finite number of switches.The results are developed using as an example the simple system with performance criterion , but are shown to hold for the more general system where f(0, 0) = 0 (i.e. the uncontrolled system has the originas an equilibrium point) subject to a performance criterion where g(0, 0) 0. The cases where g(0, 0)and are zero for all are also discussed. In particular one obtains a very simpleproof of the result that all two dimensional time optimal controlsystems with a single linear control, whose objective set consistsof the (assumed) unique critical point of the uncontrolled system,have a finite number of switches.