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We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampére-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.
We compare a general controlled diffusion process with a deterministic system
where a second controller drives the disturbance against the first
controller. We show that the two models are equivalent with
respect to two properties: the viability (or controlled
invariance, or weak invariance) of closed smooth sets, and the
existence of a smooth control Lyapunov function ensuring the
stabilizability of the system at an equilibrium.
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