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We show a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under volume and convexity constraints.We prove existence of minimizers, and we describe their shape as the volume tends to zero or to infinity.
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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