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In this article we study control problems with systems that are governed byordinary differential equations whose vector fields depend linearly in the timederivatives of some components of the control. The remaining components areconsidered as classical controls. This kind of system is called `impulsivesystem'. We assume that the vector fields multiplying the derivatives of eachcomponent of the control are commutative. We derive new necessary conditions interms of the adjoint state and the Lie brackets of the data functions.

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