Abstract:
We consider planar zero-sum differential games with simple motions, fixed terminal time, and polygonal terminal set. The geometric constraint on the control of each player is a convex polygonal set or a straight line segment. In the case of a convex terminal set, an explicit formula is known for the solvability set (the level set of the value function, maximal u-stable bridge, viability set). The algorithm corresponding to this formula is based on the set operations of algebraic sum and geometric difference (the Minkowski difference). We propose an algorithm for the exact construction of the solvability set in the case of a nonconvex polygonal terminal set. The algorithm does not involve the additional partition of the time interval and the recovery of intermediate solvability sets at additional instants. A list of half-spaces in the three-dimensional space of time and state coordinates is formed and processed by a finite recursion. The list is based on the polygonal terminal set with the use of normals of the polygonal constraints on the controls of the players.
Keywords:
differential games with simple motions in the plane, solvability set, backward procedure.
Citation:
L. V. Kamneva, V. S. Patsko, “Construction of the solvability set in differential games with simple motions and nonconvex terminal set”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 1, 2017, 143–157; Proc. Steklov Inst. Math. (Suppl.), 301, suppl. 1 (2018), 57–71
\Bibitem{KamPat17}
\by L.~V.~Kamneva, V.~S.~Patsko
\paper Construction of the solvability set in differential games with simple motions and nonconvex terminal set
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2017
\vol 23
\issue 1
\pages 143--157
\mathnet{http://mi.mathnet.ru/timm1390}
\crossref{https://doi.org/10.21538/0134-4889-2017-23-1-143-157}
\elib{https://elibrary.ru/item.asp?id=28409374}
\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2018
\vol 301
\issue , suppl. 1
\pages 57--71
\crossref{https://doi.org/10.1134/S008154381805005X}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000453520500013}
Linking options:
https://www.mathnet.ru/eng/timm1390
https://www.mathnet.ru/eng/timm/v23/i1/p143
This publication is cited in the following 6 articles:
I. V. Izmestyev, V. I. Ukhobotov, K. N. Kudryavtsev, “Numerical solution of a control problem for a parabolic system with disturbances”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 34:1 (2024), 33–47
Vladimir N. Ushakov, Aleksandr M. Tarasev, Andrei V. Ushakov, “Minimaksnaya differentsialnaya igra s fiksirovannym momentom okonchaniya”, MTIP, 16:3 (2024), 77–112
V. N. Ushakov, A. M. Tarasyev, A. V. Ushakov, “Minimax Differential Game with a Fixed End Moment”, Dokl. Math., 110:S2 (2024), S495
V. N. Ushakov, A. A. Ershov, A. V. Ushakov, A. R. Matviichuk, “Nekotorye zadachi sblizheniya nelineinykh upravlyaemykh sistem v fiksirovannyi moment vremeni”, Izv. IMI UdGU, 62 (2023), 125–155
Pavel D. Lebedev, Alexander A. Uspenskii, “On the structure of the singularity of the solution to the time-optimal control problem in the case of a discontinuity in the curvature of the boundary of the target set”, IFAC-PapersOnLine, 56:2 (2023), 7486
P. D. Lebedev, A. A. Uspenskii, “Postroenie rasseivayuschikh krivykh v odnom klasse zadach bystrodeistviya pri skachkakh krivizny granitsy tselevogo mnozhestva”, Izv. IMI UdGU, 55 (2020), 93–112
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