Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game

2020 ◽  
pp. 196-244
Author(s):  
Aleksandr G. Chentsov
Keyword(s):  
Fixed Point ◽  
Differential Game ◽  
Game Problem ◽  
Time Interval ◽  
Finite Time Interval ◽  
Alternative Theorem ◽  
Phase Constraints ◽  
Target Set ◽  
Program Iteration ◽  
Solvability Set

Differential game (DG) of guidance-evasion for a finite time interval is considered;as parameters, the target set (TS) and the set defining phase constraints (PC) are used.Player I interested in realization of guidance with TS under validity PC uses set-valuedquasistrategies (nonanticipating strategies) and Player II having opposite target uses strategieswith nonanticipating choice of correction instants and finite numbers of such instants.On informative level, the setting corresponds to alternative theorem of N. N. Krasovskii andA. I. Subbotin. For position not belonging to solvability set of Player I, determination ofthe least size of neighborhoods for set-parameters under that Player I guarantees guidance(under weakened conditions) is interested. In article, this scheme is supplemented by priorityelements in questions of TS attainment and PC validity; this is realized by special parameterdefining relation for sizes of corresponding neighborhoods. Under these conditions, a functionof the least size of TS neighborhood is defined by procedure used program iteration methodfor two variants. The above-mentioned function is fixed point for one of two used “program”operators. Special type of the quality functional for which values of the above-mentionedfunction coincide with values of the minimax-maximin games is established.

scholarly journals Some questions of differential game theory with phase constraints

2020 ◽  
Vol 56 ◽  
pp. 138-184
Author(s):  
A.G. Chentsov
Keyword(s):  
Differential Game ◽  
Position Space ◽  
Natural Topology ◽  
Differential Game Theory ◽  
Phase Constraints ◽  
Target Set ◽  
Program Iteration ◽  
Guidance Problem ◽  
Relaxed Problem

Differential game (DG) of guidance-evasion is considered; moreover, its relaxations constructed with due account for priority considerations in the implementation of target set (TS) guidance and phase constraints (PC) validity are considered. We suppose that TS is closed in a natural topology of position space. With respect to the set that defines PC, it is postulated that the sections corresponding to time fixing are closed. For this setting, with the use of program iteration method (PIM), a variant of alternative for some natural (asymmetric) classes of strategies is established. A scheme of relaxation for the game guidance problem with nonclosed (in general case) set defining PC is considered. Under relaxation construction, reasons connected with priority in the implementation of guidance to TS and PC validity are taken into account (the case of asymmetric weakening of conditions of game ending is investigated). A position function is introduced, values of which (with priority correction) play the role of an analogue of least size for neighborhoods of TS and set defining PC under which it is possible to get a guaranteed solution of a relaxed problem of a player interested in approaching with TS while observing PC. It is demonstrated that the value of given function (when fixing the position of the game) is a price of DG for minimax-maximin quality functional which characterizes both the “degree” of approaching with TS and the “degree” of observance of initial PC.

On a representation of the solvability set in the retention problem

2020 ◽  
pp. 290-298
Author(s):  
Dmitriy A. Serkov
Keyword(s):  
Iterative Method ◽  
Dynamic System ◽  
Iterative Procedure ◽  
Game Problem ◽  
Resolving Set ◽  
Phase Constraints ◽  
Individual Program ◽  
Solvability Set

The paper provides another iterative method for constructing a resolving set in the game problem of retaining the movements of an abstract dynamic system in given phase constraints. In the iterative procedure, instead of the program absorption operator, it is proposed to use a family of absorption operators for individual program disturbances. Such an approach is based on theorems on the existence and representation of common fix-points of a family of mappings.

scholarly journals Constrained controllability of nonlinear stochastic impulsive systems

2011 ◽  
Vol 21 (2) ◽  
pp. 307-316 ◽  
Author(s):  
Shanmugasundaram Karthikeyan ◽  
Krishnan Balachandran
Keyword(s):  
Fixed Point ◽  
Finite Time ◽  
Stochastic Systems ◽  
Time Interval ◽  
Complete Controllability ◽  
Finite Time Interval ◽  
Impulsive Systems ◽  
Nonlinear Stochastic Systems ◽  
Constrained Controllability ◽  
Point Argument

Constrained controllability of nonlinear stochastic impulsive systemsThis paper is concerned with complete controllability of a class of nonlinear stochastic systems involving impulsive effects in a finite time interval by means of controls whose initial and final values can be assigned in advance. The result is achieved by using a fixed-point argument.

scholarly journals Linear Discrete Pursuit Game with Phase Constraints

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Asqar Raxmanov ◽  
Gafurjan Ibragimov
Keyword(s):  
Sufficient Conditions ◽  
Discrete Systems ◽  
Geometric Constraints ◽  
Time Interval ◽  
Finite Time Interval ◽  
Pursuit Game ◽  
Terminal Set ◽  
Phase Constraints ◽  
The Right ◽  
Linear Discrete Systems

We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Position of the evader satisfies phase constraints:y∈G, whereGis a subset ofRn. We considered two cases: (1) controls of the players satisfy geometric constraints, and (2) controls of the players satisfy total constraints. Terminal setMis a subset ofRnand it is assumed to have a nonempty interior. Game is said to be completed ifyk-x(k)∈Mat some stepk; thus, the evader has not the right to leave setG. To construct the control of the pursuer, at each stepi, we use the value of the control parameter of the evader at the stepi. We obtain sufficient conditions of completion of pursuit from certain initial positions of the players in finite time interval and construct a control for the pursuer in explicit form.

scholarly journals Stability defect estimation for sets in a game approach problem in a fixed moment

2019 ◽  
Vol 489 (2) ◽  
pp. 136-141
Author(s):  
V. N. Ushakov ◽  
A. G. Malev
Keyword(s):  
Fixed Point ◽  
Control System ◽  
Differential Inclusions ◽  
Game Problem ◽  
Position Space ◽  
Finite Set ◽  
Target Set ◽  
The Stability ◽  
Approach Problem ◽  
Fixed Moment

We study the game problem of approaching a control system with a target set at a fixed point in time. The question of estimating from below the stability defect of a set in the position space weakly invariant with respect to a finite set of unification differential inclusions is discussed.

scholarly journals On properties of one functional used in software constructions for solving differential games

2021 ◽  
Vol 31 (4) ◽  
pp. 668-696
Author(s):  
A.G. Chentsov
Keyword(s):  
Value Function ◽  
Sufficient Conditions ◽  
Game Problem ◽  
Limit Function ◽  
Phase Constraints ◽  
Nonlinear Differential Game ◽  
Target Set ◽  
Iteration Number ◽  
Game Position ◽  
The Value Function

Nonlinear differential game (DG) is investigated; relaxations of the game problem of guidance are investigated also. The variant of the program iterations method realized in the space of position functions and delivering in limit the value function of the minimax-maximin DG for special functionals of a trajectory is considered. For every game position, this limit function realizes the least size of the target set neighborhood for which, under proportional weakening of phase constraints, the player interested in a guidance yet guarantees its realization. Properties of above-mentioned functionals and limit function are investigated. In particular, sufficient conditions for realization of values of given function under fulfilment of finite iteration number are obtained.

scholarly journals CONTROL SYSTEM DEPENDING ON A PARAMETER

2021 ◽  
Vol 7 (1) ◽  
pp. 120
Author(s):  
Vladimir N. Ushakov ◽  
Aleksandr A. Ershov ◽  
Andrey V. Ushakov ◽  
Oleg A. Kuvshinov
Keyword(s):  
Control System ◽  
Phase Space ◽  
Specific Problem ◽  
Dimensional Euclidean Space ◽  
Time Interval ◽  
Two Dimensional ◽  
Finite Time Interval ◽  
Finite Dimensional ◽  
Dimensional Phase Space ◽  
Target Set

A nonlinear control system depending on a parameter is considered in a finite-dimensional Euclidean space and on a finite time interval. The dependence on the parameter of the reachable sets and integral funnels of the corresponding differential inclusion system is studied. Under certain conditions on the control system, the degree of this dependence on the parameter is estimated. Problems of targeting integral funnels to a target set in the presence of an obstacle in strict and soft settings are considered. An algorithm for the numerical solution of this problem in the soft setting has been developed. An estimate of the error of the developed algorithm is obtained. An example of solving a specific problem for a control system in a two-dimensional phase space is given.

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