scholarly journals Linear Pursuit Differential Game under Phase Constraint on the State of Evader

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Askar Rakhmanov ◽  
Gafurjan Ibragimov ◽  
Massimiliano Ferrara
Keyword(s):  
Differential Game ◽  
The State ◽  
Geometric Constraints ◽  
Phase Constraint ◽  
Phase Vector ◽  
Terminal Set

We consider a linear pursuit differential game of one pursuer and one evader. Controls of the pursuer and evader are subjected to integral and geometric constraints, respectively. In addition, phase constraint is imposed on the state of evader, whereas pursuer moves throughout the space. We say that pursuit is completed, if inclusiony(t1)-x(t1)∈Mis satisfied at somet1>0, wherex(t)andy(t)are states of pursuer and evader, respectively, andMis terminal set. Conditions of completion of pursuit in the game from all initial points of players are obtained. Strategy of the pursuer is constructed so that the phase vector of the pursuer first is brought to a given set, and then pursuit is completed.

scholarly journals Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1467
Author(s):  
Muminjon Tukhtasinov ◽  
Gafurjan Ibragimov ◽  
Sarvinoz Kuchkarova ◽  
Risman Mat Hasim
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Differential Games ◽  
Differential Game ◽  
Infinite System ◽  
The State ◽  
Geometric Constraints ◽  
Control Parameters ◽  
Systems Of Differential Equations ◽  
Pursuit And Evasion

A pursuit differential game described by an infinite system of 2-systems is studied in Hilbert space l2. Geometric constraints are imposed on control parameters of pursuer and evader. The purpose of pursuer is to bring the state of the system to the origin of the Hilbert space l2 and the evader tries to prevent this. Differential game is completed if the state of the system reaches the origin of l2. The problem is to find a guaranteed pursuit and evasion times. We give an equation for the guaranteed pursuit time and propose an explicit strategy for the pursuer. Additionally, a guaranteed evasion time is found.

On two pursuit differential game problems with state and geometric constraints in a Hilbert space

2021 ◽  
Vol 65 (3) ◽  
pp. 5-16
Author(s):  
Abbas Ja’afaru Badakaya ◽  
Keyword(s):  
Differential Game ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Nonempty Closed Convex Subset ◽  
Geometric Constraints ◽  
First Order ◽  
Control Functions ◽  
Closed Convex Set ◽  
The Given ◽  
First Order Differential Equations

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

Observation of the State of a Differential Game

1985 ◽  
Vol 18 (2) ◽  
pp. 248-249
Author(s):  
Auloge Jean-Yves
Keyword(s):  
Differential Game ◽  
The State

THE VALUE FUNCTION OF A DIFFERENTIAL GAME WITH SIMPLE MOTIONS AND AN INTEGRO-TERMINAL COST

2018 ◽  
pp. 877-890
Author(s):  
Lyubov Gennad’evna Shagalova
Keyword(s):  
Differential Equation ◽  
State Space ◽  
Differential Game ◽  
Value Function ◽  
Piecewise Linear ◽  
The State ◽  
The Value Function

An antagonistic positional differential game of two persons is considered. The dynamics of the system is described by a differential equation with simple motions, and the payoff functional is integro-terminal. For the case when the terminal function and the Hamiltonian are piecewise linear, and the dimension of the state space is two, a finite algorithm for the exact construction of the value function is proposed.

SOLUTION OF SIMPLE PURSUIT-EVASION PROBLEM WHEN EVADER MOVES ON A GIVEN CURVE

2010 ◽  
Vol 12 (03) ◽  
pp. 223-238 ◽  
Author(s):  
ATAMURAT SH. KUCHKAROV
Keyword(s):  
Differential Game ◽  
Sufficient Conditions ◽  
Initial Position ◽  
Necessary And Sufficient Conditions ◽  
Phase Constraint ◽  
Simple Motion ◽  
Pursuit Evasion ◽  
Evasion Problem ◽  
Necessary And Sufficient

We investigate a simple motion pursuit-evasion differential game of one Pursuer and one Evader. Maximal speeds of the players are equal. The Evader moves along a given curve without self-intersection. There is no phase constraint for the Pursuer. Necessary and sufficient conditions to complete pursuit from both fixed initial position and all initial positions are obtained.

Multistage Suboptimal Model Predictive Control With Improved Computational Efficiency

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Xiaotao Liu ◽  
Daniela Constantinescu ◽  
Yang Shi
Keyword(s):  
Model Predictive Control ◽  
Predictive Control ◽  
The State ◽  
Admissible Set ◽  
Admissible Sets ◽  
Prediction Horizon ◽  
Terminal Set ◽  
Minimum Number ◽  
System States ◽  
The Stability

This paper proposes a multistage suboptimal model predictive control (MPC) strategy which can reduce the prediction horizon without compromising the stability property. The proposed multistage MPC requires a precomputed sequence of j-step admissible sets, where the j-step admissible set is the set of system states that can be steered to the maximum positively invariant set in j control steps. Given the precomputed admissible sets, multistage MPC first determines the minimum number of steps M required to drive the state to the terminal set. Then, it steers the state to the (M – N)-step admissible set if M > N, or to the terminal set otherwise. The paper presents the offline computation of the admissible sets, and shows the feasibility and stability of multistage MPC for systems with and without disturbances. A numerical example illustrates that multistage MPC with N = 5 can be used to stabilize a system which requires MPC with N ≥ 14 in the absence of disturbances, and requires MPC with N ≥ 22 when affected by disturbances.

scholarly journals Discrete game problem with ring-shaped terminal set

2020 ◽  
Vol 30 (1) ◽  
pp. 18-30
Author(s):  
I.V. Izmest'ev
Keyword(s):  
Computer Simulation ◽  
Original Problem ◽  
Finite Dimension ◽  
Fixed Time ◽  
Game Problem ◽  
Optimal Controls ◽  
Fixed Duration ◽  
Phase Vector ◽  
Terminal Set ◽  
Discrete Game

In a normed space of finite dimension a discrete game problem with fixed duration is considered. The terminal set is determined by the condition that the norm of the phase vector belongs to a segment with positive ends. In this paper, a set defined by this condition is called a ring. The aim of the first player is to lead a phase vector to the terminal set at fixed time. The aim of the second player is the opposite. In this paper, optimal controls of the players are constructed. Computer simulation of the game process is performed. A modification of the original problem, in which at an unknown time there is a change in the dynamics of the first player, is considered.

scholarly journals One Game Problem for Oscillatory Systems

2021 ◽  
pp. 5-15
Author(s):  
Greta Chikrii
Keyword(s):  
Differential Game ◽  
Direct Method ◽  
Game Problem ◽  
Oscillatory Systems ◽  
Terminal Set ◽  
Damped Oscillations ◽  
Pontryagin’S Condition ◽  
Linear Differential ◽  
Second Order Systems ◽  
Time Stretching

The paper concerns the linear differential game of approaching a cylindrical terminal set. We study the case when classic Pontryagin’s condition does not hold. Instead, the modified considerably weaker condition, dealing with the function of time stretching, is used. The latter allows expanding the range of problems susceptible to analytical solution by the way of passing to the game with delayed information. Investigation is carried out in the frames of Pontryagin’s First Direct method that provides hitting the terminal set by a trajectory of the conflict-controlled process at finite instant of time. In so doing, the pursuer’s control, realizing the game goal, is constructed on the basis of the Filippov-Castaing theorem on measurable choice. The outlined scheme is applied to solving the problem of pursuit for two different second-order systems, describing damped oscillations. For this game, we constructed the function of time stretching and deduced conditions on the game parameters, ensuring termination of the game at a finite instant of time, starting from arbitrary initial states and under all admissible controls of the evader. Keywords: differential game, time-variable information delay, Pontryagin’s condition, Aumann’s integral, principle of time stretching, Minkowski’ difference, damped oscillations.

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