Abstract: Differential positivity of a dynamical system refers to the property that its linearization along trajectories is positive, that is, infinitesimally contracts a smooth cone field defined in the tangent bundle. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. Differential positivity induces a conal order which places significant constraints on the asymptotic behavior of solutions. This paper studies differentially positive systems defined on Lie groups, which constitute an important and basic class of manifolds with the structure of a homogeneous space. The geometry of a Lie group allows for the generation of invariant cone fields over the tangent bundle given a single cone in the Lie algebra. We outline the mathematical framework for studying differential positivity of a nonlinear flow on a Lie group with respect to an invariant cone field and motivate the use of this analysis framework in nonlinear control, and, in particular in nonlinear consensus theory.
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