Fractional order differential pursuit games with nonlinear controls

2020 ◽  
pp. 401-409
Author(s):  
Mashrabjon Sh. Mamatov ◽  
Khakim N. Alimov
Keyword(s):  
Differential Games ◽  
Fractional Order ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Matrix Functions ◽  
Pursuit Problem ◽  
Industrial Facilities ◽  
Linear Differential Games ◽  
Generalized Matrix ◽  
Linear Differential

The article is devoted to the problems of extending the results and methods of the theory of differential games and optimal control to systems of fractional order. The research is motivated by numerous applications of fractional calculus in control problems of industrial facilities, chemical and biochemical plants, etc. The article considers the problem of pursuit in games represented by nonlinear differential equations of arbitrary fractional order in the sense of Caputo. To study this pursuit problem, we use an approach similar to the method of L. S. Pontryagin, developed for linear differential games of integer orders. In this paper, new sufficient conditions are obtained for solving the pursuit problem in the class of games under study. It has been proven that if these conditions are met, the game can be completed within a certain limited period of time. When solving the pursuit problem, we also used the representation of the solution to a differential equation in terms of generalized matrix functions.

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 Mathematical Modeling and Functional Order Pursuits with Separated Dynamics

2021 ◽  
Vol 2068 (1) ◽  
pp. 012002
Author(s):  
Mashrabjon Mamatov ◽  
Xakimjon Alimov
Keyword(s):  
Mathematical Modeling ◽  
Dynamic Systems ◽  
Sufficient Conditions ◽  
Order Equation ◽  
Matrix Functions ◽  
Pursuit Problem ◽  
Controlled Systems ◽  
Generalized Matrix ◽  
Fractional Differential ◽  
Qualitative Dynamics

Abstract This work is devoted to the study of the pursuit problem in controlled systems described by a fractional-order equation with divided dynamics. For fixed player controls, representations of solutions are established in the form of analogs of the Cauchy formula using generalized matrix functions. Sufficient conditions are obtained for the possibility of completing the pursuit. Specific types of fractional differential equations and models of fractional dynamical systems are considered. The qualitative dynamics, issues of stability and controllability of such systems are discussed. Considered, try which, the motion of the equation is described with irrational orders. Problems of the type under study are encountered in modeling the processes of economic growth and in problems of stabilizing dynamic systems.

Linear differential games

Cybernetics ◽  
1972 ◽  
Vol 4 (1) ◽  
pp. 37-41 ◽  
Author(s):  
B. N. Pshenichnyi
Keyword(s):  
Differential Games ◽  
Linear Differential Games ◽  
Linear Differential
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