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Necessary conditions involving Lie brackets for impulsive optimal control problems; the commutative case

In this article we study control problems with systems that are governed by ordinary differential equations whose vector fields depend linearly in the time derivatives of some components of the control. The remaining components are considered as classical controls. This kind of system is called `impulsive system'. We assume that the vector fields multiplying the derivatives of each component of the control are commutative. We derive new necessary conditions in terms of the adjoint state and the Lie brackets of the data functions.