Alpha-sets in finite-dimensional Euclidean spaces and their applications in control theory

In this paper, a technique for investigating nonconvex sets that occur when describing the evolution of wave fronts, in the construction of generalized solutions of boundary value problems for equations of Hamilton–Jacobi type, in the formation of resolving structures in the problems of dynamic control is developed. An estimate is obtained for the Hausdorff distance between such sets and their convex hulls. The estimate is based on the concept of a measure of nonconvexity $\alpha$. It is shown that for small $\alpha$, nonconvex $\alpha$-sets are close to convex. An example of a solution of the optimal control problem on the basis of $\alpha$-sets is give.