Estimation of the growth of the degree of nonconvexity of reachable sets in terms of $\alpha$-sets

We study the properties of $\alpha$-sets, which are a generalization of convex sets. The relationship between $\alpha$-sets and weakly convex sets is established in the sense of Vial and Efimov–Stechkin. An estimate of the growth over time of the nonconvexity measure $\alpha$ of reachable sets is obtained for one class of control systems in a two-dimensional state space.