On the Control of Moving Sets: Positive and Negative Confinement Results

External Ellipsoidal Approximations for Set Evolution Equations

“…Secondly, the inclusion property is preserved, i.e., whenever K (0) is contained in an ellipsoid E(0), then each of them evolves independently of the other in such a way that K (t) ⊂ E(t) holds for every t. In the following, the well-posedness of the set evolution problem and the inclusion property of any two solutions are handled even for nonlinear differential inclusions and compact (not necessarily convex) sets. This part extends various results in, e.g., [3,4,[17][18][19]37,41,44,49,53,54,62]. Then, in favor of fast numerical methods, the external approximation by ellipsoids is restricted to differential inclusions x ∈ A(•, K ) x + B(•, K ) U (i.e., linear in x and η ∈ U ).…”

Interaction Network, State Space, and Control in Social Dynamics

“…In [34], Fornasier, Piccoli and Rossi solve this problem by using a mixed granular-diffuse description of the crowd and prove convergence of the solution of the finite-dimensional problem when the number of followers tends to infinity to the solution of this new system. Similar approaches involving the coupling of microscopic dynamics for the leaders and macroscopic dynamics for the followers were adopted by Albi, Bongini, Cristiani and Kalise in [1] and by Colombo and Pogodaev in [19]. Here we present the approach of [34].…”

Estimates of the domain of dependence for scalar conservation laws

“…solves our initial confinement problem. Such ξ may be constructed by applying, for example, the technique developed in [4,5]. In particular, one may prove the following result.…”