Open-Loop Solvability Operator in Differential Games with Simple Motions in the Plane

  • Liudmila Kamneva
  • Valerii PatskoEmail author
Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 13)


The paper deals with an open-loop solvability operator in two-person zero-sum differential games with simple motions. This operator takes a given terminal set to the set defined at the initial instant whence the first player can bring the control system to the terminal set if the player is informed about the open-loop control of the second player. It is known that the operator possesses the semigroup property in the case of a convex terminal set. In the paper, sufficient conditions ensuring the semigroup property in the non-convex case are formulated and proved for problems in the plane. Examples are constructed to illustrate the relevance of the formulated conditions. The connection with the Hopf formula is analysed.


Planar differential games Semigroup property Simple motions Open-loop solvability operator 



We are grateful to the unknown reviewer for the valuable remarks. This research was carried out in the framework of the Program by Presidium of RAS “Fundamental problems of non-linear dynamics in mathematical and physical sciences” with financial support of Ural Branch of RAS (project No. 12-\(\Pi \)-1-1012) and was also supported by RFBR, grant No. 12-01-00537.


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© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Institute of Mathematics and MechanicsEkaterinburgRussia

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