Geodesic
Alpha-sets in finite-dimensional Euclidean spaces
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 3, pp. 261-272

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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.
Mots-clés : $\alpha$-set, Hausdorff distance
Keywords: convex hull, control, performance, Hamilton–Jacobi equation. 
@article{VSPUI_2018_14_3_a6,
     author = {V. N. Ushakov and A. A. Uspenskii and A. A. Ershov},
     title = {Alpha-sets in finite-dimensional {Euclidean} spaces},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {261--272},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2018_14_3_a6/}
}
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V. N. Ushakov; A. A. Uspenskii; A. A. Ershov. Alpha-sets in finite-dimensional Euclidean spaces. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 14 (2018) no. 3, pp. 261-272. http://geodesic.mathdoc.fr/item/VSPUI_2018_14_3_a6/