ISSN 0005-1179, Automation and Remote Control, 2018, Vol. 79, No. 3, pp. 479–491.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
D.R. Kuvshinov, S.I. Osipov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 3, pp. 111–126.
INTELLECTUAL CONTROL SYSTEMS, DATA ANALYSIS
Numerical Construction of Stackelberg Solutions
in a Linear Positional Diﬀerential Game
Based on the Method of Polyhedra
D. R. Kuvshinov
and S. I. Osipov
Krasovsky Institute of Mathematics and Mechanics, Yekaterinburg, Russia
Yeltsin Ural Federal University, Yekaterinburg, Russia
Received February 5, 2016
Abstract—We consider the problem of constructing approximate Stackelberg solutions in a lin-
ear non-zero-sum positional diﬀerential game of two players with terminal payoﬀs and player
controls chosen on convex polyhedra. A formalization of player strategies and motions gen-
erated by them is based on the formalization and results of the theory of zero-sum positional
diﬀerential games developed by N.N. Krasovskii and his scientiﬁc school. The problem of ﬁnding
a Stackelberg solution reduces to solving nonstandard optimal control problems. We propose
an approach based on operations with convex polyhedra.
Keywords: non-zero-sum positional diﬀerential game, Stackelberg solution, convex polyhedron,
A Stackelberg solution  in a non-zero-sum game of two players is based on introducing a
hierarchy of decision making: one player is considered to be the “leader”; the other, the “follower.”
The leader tells his strategy to the follower before the game, and the follower chooses a rational
response. In this way we can pass from a game-theoretic problem to an optimal control problem.
Stackelberg solutions have been the subject of numerous works. Settings in the form of a
diﬀerential game have been considered as well. Most works in this ﬁeld can be assigned to one
(or both) of two groups: studies of solutions in the class of open-loop strategies and studies in the
class of linear–quadratic games. A survey of classical settings and approaches to solving problems
in both these groups has been given in . Usually, such problems are reduced to mathematical
programming problems which can be then solved numerically. Necessary and suﬃcient optimality
conditions for Stackelberg trajectories in positional diﬀerential games from a suﬃciently wide class
have been obtained in . As an example of recent works that consider the numerical construction
of Stackelberg solutions we can list, e.g., [4–6].
In this work we consider the problem of approximate construction of Stackelberg solutions in a
positional diﬀerential game of two players with terminal payoﬀs, linear dynamics, and constraints
on player controls deﬁned as convex polyhedra. Thus, this work does not fall into any one of these
Formalization of player strategies and motions generated by them is based on the formalization
and results of the theory of zero-sum positional diﬀerential games developed by N.N. Krasovskii
and his scientiﬁc school [7, 8]. The problem of ﬁnding solutions in non-zero-sum games reduces to
solving nonstandard optimal control problems [9, 10], which lets us pass from the original problem