scholarly journals An Approach for Solving Discrete Game Problems with Total Constraints on Controls

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Asqar Raxmonov ◽  
Gafurjan I. Ibragimov
Keyword(s):  
Explicit Form ◽  
Control Parameter ◽  
Sufficient Conditions ◽  
Discrete Systems ◽  
Nonempty Interior ◽  
Pursuit Game ◽  
Terminal Set ◽  
Control Forces ◽  
Linear Discrete Systems ◽  
Discrete Game

We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Controls of the players satisfy total constraints. Terminal setMis a subset ofℝnand it is assumed to have nonempty interior. Game is said to be completed ifyk-xk∈Mat some stepk. To construct the control of the pursuer, at each stepi, we use positions of the players from step 1 to stepiand the value of the control parameter of the evader at the stepi. We give sufficient conditions of completion of pursuit and construct the control for the pursuer in explicit form. This control forces the evader to expend some amount of his resources on a period consisting of finite steps. As a result, after several such periods the evader exhausted his energy and then pursuit will be completed.

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.

DIFFERENTIAL GAMES OF FRACTIONAL ORDER WITH DISTRIBUTED PARAMETERS

2021 ◽  
Vol 4 ◽  
pp. 38-47
Author(s):  
Mashrabzhan Mamatov ◽  
◽  
Jalolkon Nuritdinov ◽  
Egamberdi Esonov ◽  
◽  
...  
Keyword(s):  
Differential Games ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Initial Position ◽  
Difference Approximation ◽  
Pursuit Problem ◽  
Spatial Variables ◽  
Terminal Set ◽  
Distributed Parameters ◽  
Discrete Game

The article deals with the problem of pursuit in differential games of fractional order with distributed parameters. Partial fractional derivatives with respect to time and space variables are understood in the sense of Riemann - Liouville, and the Grunwald-Letnikov formula is used in the approximation. The problem of getting into some positive neighborhood of the terminal set is considered. To solve this problem, the finite difference method is used. The fractional Riemann-Liouville derivatives with respect to spatial variables on a segment are approximated using the Grunwald-Letnikov formula. Using a sufficient criterion for the existence of a fractional derivative, a difference approximation of the fractional-order derivative with respect to time is obtained. By approximating a differential game to an explicit difference game, a discrete game is obtained. The corresponding pursuit problem for a discrete game is formulated, which is obtained using the approximation of a continuous game. The concept of the possibility of completing the pursuit, a discrete game in the sense of an exact capture, is defined. Sufficient conditions are obtained for the possibility of completing the pursuit. It is shown that the order of approximation in time is equal to one, and in spatial variables is equal to two. It is proved that if in a discrete game from a given initial position it is possible to complete the pursuit in the sense of exact capture, then in a continuous game from the corresponding initial position it is possible to complete the pursuit in the sense of hitting a certain neighborhood. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. The methods used for this problem can be used to study differential games described by more general equations of fractional order.

scholarly journals Linear discrete systems controllability from a set of initial states to a terminal set

2018 ◽  
Vol 23 (1(31)) ◽  
pp. 81-87
Author(s):  
В. В. Пічкур ◽  
В. В. Собчук ◽  
М. С. Таірова ◽  
О. М. Башняков
Keyword(s):  
Discrete Systems ◽  
Terminal Set ◽  
Initial States ◽  
Linear Discrete Systems ◽  
Set Of Initial States

scholarly journals Discrete (h,k)-Dichotomy and Remarks on the Boundedness of the Projections

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Mihai-Gabriel Babuţia ◽  
Monteola Ilona Kovács ◽  
Mărioara Lăpădat ◽  
Mihail Megan
Keyword(s):  
Sufficient Conditions ◽  
Discrete Systems ◽  
Linear Discrete Systems

The present paper treats a concept of (h,k)-dichotomy for linear discrete systems. Sufficient conditions for the k-boundedness of the projection sequences that give the dichotomy are presented and an illustrative example shows the connection between the growth of the system and the bound of the sequence of projections. Thus the growth of the system that is assumed in the theorems is essential.

scholarly journals On the Output Controllability of Positive Discrete Linear Delay Systems

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Mouhcine Naim ◽  
Fouad Lahmidi ◽  
Abdelwahed Namir ◽  
Mostafa Rachik
Keyword(s):  
Sufficient Conditions ◽  
Discrete Systems ◽  
Necessary And Sufficient Conditions ◽  
Delay Systems ◽  
Input And Output ◽  
Linear Delay ◽  
Output Controllability ◽  
Necessary And Sufficient ◽  
Linear Discrete Systems

Necessary and sufficient conditions for output reachability and null output controllability of positive linear discrete systems with delays in state, input, and output are established. It is also shown that output reachability and null output controllability together imply output controllability.

Stability and Stabilization for Discrete System With Probabilistic Nonlinearities

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Xia Zhao ◽  
Engang Tian
Keyword(s):  
Sufficient Conditions ◽  
Discrete Systems ◽  
System Model ◽  
Linear Matrix ◽  
Time Varying Delay ◽  
Mean Square Stability ◽  
New Type ◽  
Stability And Stabilization ◽  
The Mean ◽  
Varying Delay

This paper investigates stability and stabilization of discrete systems with probabilistic nonlinearities and time-varying delay. New characters of the nonlinearities, the probability of the nonlinearities happening between different bounds, are used to build new type of system model, which can help us make a full use of the inner variation information of the nonlinearities. With the help of the new characters, new system model is proposed. Then, sufficient conditions for the mean square stability of the system can be obtained by using the Lyapunov functional approach and linear matrix inequalities technique. An example is proposed to illustrate the efficiency of the proposed method.

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