ИСТИНА

We address the problem of linearizing a single-input time-invariant nonlinear control system by invertible state and input transformations and invertible time scaling. We prove a necessary and sufficient condition for linearizability of single-input control systems in the above-mentioned class of transformations. We show that the problem can be solved by constructing the derived flag of the Pfaffian system I, which is obtained by eliminating the time differential from the Pfaffian system naturally associated with the control system. We prove that, for regular points of the derived flag of I, a single-input nonlinear control system is linearizable in the above-mentioned class of transformations if and only if I is diffemorphic to the Goursat normal form and an additional condition holds for the (n-2)th derived system of I where n is the dimension of the state manifold. We prove that if I is diffeomorphic to the Goursat normal form but the additional condition is not met, the control system can be linearized in the above-mentioned class of transformations after a prolongation.