Abstract

We study a problem concerning the estimation of the Hausdorff deviation of convex polygons in $\mathbb R^2$ from their geometric difference with circles of sufficiently small radius. Problems with such a subject, in which not only convex polygons but also convex compacts in the Euclidean space $\mathbb R^n$ are considered, arise in various fields of mathematics and, in particular, in the theory of differential games, control theory, convex analysis. Estimates of Hausdorff deviations of convex compact sets in $\mathbb R^n$ in their geometric difference with closed balls in $\mathbb R^n$ are presented in the works of L.S. Pontryagin, his staff and colleagues. These estimates are very important in deriving an estimate for the mismatch of the alternating Pontryagin’s integral in linear differential games of pursuit and alternating sums. Similar estimates turn out to be useful in deriving an estimate for the mismatch of the attainability sets of nonlinear control systems in $\mathbb R^n$ and the sets approximating them.
The paper considers a specific convex heptagon in $\mathbb R^2$. To study the geometry of this heptagon, we introduce the concept of a wedge in $\mathbb R^2$. On the basis of this notion, we obtain an upper bound for the Hausdorff deviation of a heptagon from its geometric difference with the disc in $\mathbb R^2$ of sufficiently small radius.

Citation

Ushakov V.N., Pershakov M.V. On estimation of Hausdorff deviation of convex polygons in $\mathbb{R}^2$ from their differences with disks, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 585603.

References

 Pontryagin L.S. Linear differential games of pursuit, Mathematics of the USSRSbornik, 1981, vol. 40, no. 3, pp. 285303. https://doi.org/10.1070/SM1981v040n03ABEH001815
 Pontryagin L.S. Izbrannye nauchnye trudy. Tom 2 (Selected scientific works. Vol. 2), Moscow: Nauka, 1988.
 Nikol'skii M.S. On the alternating integral of Pontryagin, Mathematics of the USSRSbornik, 1981, vol. 44, no. 1, pp. 125132. http://doi.org/10.1070/SM1983v044n01ABEH000956
 Nikol'skii M.S. Approximate computation of the least guaranteed estimate in linear differential games with a fixed duration, Journal of Applied Mathematics and Mechanics, 1982, vol. 46, no. 4, pp. 550552. https://doi.org/10.1016/00218928(82)900442
 Polovinkin E.S. Stability of the terminal set and optimality of the pursuit time in differential games, Differ. Uravn., 1984, vol. 20, no. 3, pp. 433446 (in Russian). http://mi.mathnet.ru/de5120
 Ponomarev A.P., Rozov N.Kh. Stability and convergence of alternated Pontryagin sums, Bulletin of Moscow University. Series 15: Computational Mathematics and Cybernetics, 1978, no. 1, pp. 8290.
 Azamov A.A. Semistability and duality in the theory of the Pontryagin alternating integral, Soviet Mathematics. Doklady, 1988, vol. 37, no. 2, pp. 355359. https://zbmath.org/?q=an:0683.90108
 Polovinkin E.S., Ivanov G.E., Balashov M.V., Konstantinov R.V., Khorev A.V. An algorithm for the numerical solution of linear differential games, Sbornik: Mathematics, 2001, vol. 192, no. 10, pp. 15151542. http://doi.org/10.1070/SM2001v192n10ABEH000604
 Ushakov V.N., Pershakov M.V. On twosided approximations of reachable sets of control systems with geometric constraints on the controls, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 1, pp. 239255 (in Russian). https://doi.org/10.21538/013448892020261239255
 Polovinkin E.S., Balashov M.V. Elementy vypuklogo i sil'no vypuklogo analiza (Elements of convex and strongly convex analysis), Moscow: Fizmatlit, 2007.
