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Non-stationary Differential-Difference Games of Neutral Type

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Abstract

We consider the pursuit problem for 2-person conflict-controlled process with single pursuer and single evader. The problem is given by the system of the linear functional-differential equations of neutral type. The players pursue their own goals and choose controls in the form of functions of a certain kind. The goal of the pursuer is to catch the evader in the shortest possible time. The goal of the evader is to avoid the meeting of the players’ trajectories on a whole semiinfinite interval of time or if it is impossible to maximally postpone the moment of meeting. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain moment of time for any counteractions of the evader.
Dynamic Games and Applications (2019) 9:771–779
https://doi.org/10.1007/s13235-019-00298-z
Non-stationary Differential-Difference Games of Neutral Type
Ievgen Liubarshchuk1·Yaroslav Bihun1·Igor Cherevko1
Published online: 30 January 2019
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
We consider the pursuit problem for 2-person conflict-controlled process with single pursuer
and single evader. The problem is given by the system of the linear functional-differential
equations of neutral type. The players pursue their own goals and choose controls in the form
of functions of a certain kind. The goal of the pursuer is to catch the evader in the shortest
possible time. The goal of the evader is to avoid the meeting of the players’ trajectories on a
whole semiinfinite interval of time or if it is impossible to maximally postpone the moment
of meeting. For such a conflict-controlled process, we derive conditions on its parameters
and initial state, which are sufficient for the trajectories of the players to meet at a certain
moment of time for any counteractions of the evader.
Keywords Differential-difference games ·Dynamic games ·Pursuit problem ·The Method
of Resolving Functions
1 Introduction
A variety of interesting examples stimulated the development of the Dynamic Games Theory
[1,17]. Fundamental results in Differential Games Theory were obtained by Isaacs [15],
Pontryagin et al. [22], and Krasovskii [16]. The basis for R. Isaacs’ investigations was the
Method of Dynamic Programming for Isaacs–Bellman equation. The classical Isaacs’ scheme
was later intensified by Pontryagin [21]. Pontryagin’s First Direct Method is the simplest
and the most efficient method for solving specific pursuit problems. This method affords
conveniently checkable sufficient conditions for pursuit termination. Due to its versatility,
Pontryagin’s First Direct Method gave rise to a number of extensions [18,19,23].
Further development of Pontryagin’s ideas resulted in the Method of Resolving Functions,
one of the most powerful methods of dynamic game theory, which justifies, in particular, the
BIevgen Liubarshchuk
finvara@gmail.com
Yaroslav Bihun
yaroslav.bihun@gmail.com
Igor Cherevko
i.cherevko@chnu.edu.ua
1Department of Mathematics and Informatics, Yuriy Fedkovych Chernivtsi National University,
Chernivtsi 58012, Ukraine
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Article
In this paper, we consider a two-person zero-sum differential game with incomplete information on the initial state, namely, the first player has a private information while the second player knows only a probability distribution on the initial state, and obtain the existence of its value functions. The main novelty lies in the method for which we apply a generalized weak contraction for dynamic systems and terminal cost functions instead of the Lipschitz contractive condition on a partially ordered metric space.
Article
Full-text available
If a pursuit game with many persons can be formalized in the framework of zero-sum differential games, then general methods can be applied to solve it. But difficulties arise connected with very high dimension of the phase vector when there are too many objects. Just due to this problem, special formulations and methods have been elaborated for conflict interaction of groups of objects. This paper is a survey of publications and results on group pursuit games.
Book
Game theory is a rich and active area of research of which this new volume of the Annals of the International Society of Dynamic Games is yet fresh evidence. Since the second half of the 20th century, the area of dynamic games has man­ aged to attract outstanding mathematicians, who found exciting open questions requiring tools from a wide variety of mathematical disciplines; economists, so­ cial and political scientists, who used game theory to model and study competition and cooperative behavior; and engineers, who used games in computer sciences, telecommunications, and other areas. The contents of this volume are primarily based on selected presentation made at the 8th International Symposium of Dynamic Games and Applications, held in Chateau Vaalsbroek, Maastricht, the Netherlands, July 5-8, 1998; this conference took place under the auspices of the International Society of Dynamic Games (ISDG), established in 1990. The conference has been cosponsored by the Control Systems Society of the IEEE, IFAC (International Federation of Automatic Con­ trol), INRIA (Institute National de Recherche en Informatique et Automatique), and the University of Maastricht. One ofthe activities of the ISDG is the publica­ tion of the Annals. Every paper that appears in this volume has passed through a stringent reviewing process, as is the case with publications for archival journals.
Article
"An elegantly written, introductory overview of the field, with a near perfect choice of what to include and what not, enlivened in places by historical tidbits and made eminently readable throughout by crisp language. It has succeeded in doing the near-impossible—it has made a subject which is generally inhospitable to nonspecialists because of its ‘family jargon’ appear nonintimidating even to a beginning graduate student." —The Journal of the Indian Institute of Science "The book under review gives a comprehensive treatment of basically everything in mathematics that can be named multivalued/set-valued analysis. It includes…results with many historical comments giving the reader a sound perspective to look at the subject…The book is highly recommended for mathematicians and graduate students who will find here a very comprehensive treatment of set-valued analysis." —Mathematical Reviews "I recommend this book as one to dig into with considerable pleasure when one already knows the subject...'Set-Valued Analysis’ goes a long way toward providing a much needed basic resource on the subject." —Bulletin of the American Mathematical Society "This book provides a thorough introduction to multivalued or set-valued analysis...Examples in many branches of mathematics, given in the introduction, prevail [upon] the reader the indispensability [of dealing] with sequences of sets and set-valued maps…The style is lively and vigorous, the relevant historical comments and suggestive overviews increase the interest for this work...Graduate students and mathematicians of every persuasion will welcome this unparalleled guide to set-valued analysis." —Zentralblatt Math
Article
A pursuit method in the class of counter controls with switching that is based on the use of gauge functions and their inverses is proposed. Guaranteed times for different schemes are compared. It is shown that this method yields, in particular, a complete substantiation of the parallel-pursuit rule which is well known in practice.
Book
Preface. Introduction. 1. Auxiliary Mathematical Results. 2. The Method of Resolving Functions. 3. Group Pursuit. 4. Complete Conflict Controllability. 5. Successive Pursuit. 6. Interaction of Group of Controlled Objects. Bibliographic Commentary. References. Index.
Chapter
This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.