DETERMINISTIC DIFFERENTIAL GAMES UNDER PROBABILITY KNOWLEDGE OF INITIAL CONDITION

We study a zero-sum differential game where the players have only an unperfect information on the state of the system. In the beginning of the game only a random distribution on the initial state is available. The main result of the paper is the existence of the value obtained through an uniqueness result for Hamilton-Jacobi-Isaacs equations stated on the space of measure in ℝn. This result is the first step for future work on differential games with lack of information.

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Application of Differential Games in Mechatronic Control System

International Journal of Applied Mechanics and Engineering ◽

10.1515/ijame-2016-0051 ◽

2016 ◽

Vol 21 (4) ◽

pp. 867-878 ◽

Cited By ~ 2

Author(s):

Z. Hendzel ◽

P. Penar

Keyword(s):

Game Theory ◽

Differential Games ◽

Differential Game ◽

Approximate Solutions ◽

Control Problems ◽

Optimum Control ◽

Object A ◽

Differential Game Theory ◽

Single Link ◽

Zero Sum

Abstract Differential games are a combination of game theory and optimum control methods. Their solutions are based on Bellman's principle of optimality. In this paper, the zero-sum differential game theory has been used for the purposes of controlling a mechatronic object: a single-link manipulator. In this case, analytical solutions are unavailable, thus approximate solutions were used. Two approximation methods were compared with the use of numerical simulations and selected quality indicators. The results confirm previous assumptions and the connection between the differential game theory and H∞ control problems.

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Disturbance attenuation problem using a differential game approach for feedback linear quadratic descriptor systems

Archives of Control Sciences ◽

10.1515/acsc-2015-0029 ◽

2015 ◽

Vol 25 (4) ◽

pp. 445-462 ◽

Cited By ~ 1

Author(s):

Muhammad Wakhid Musthofa

Keyword(s):

Differential Games ◽

Differential Game ◽

Information Structure ◽

Disturbance Attenuation ◽

Descriptor Systems ◽

Linear Quadratic ◽

Feedback Information ◽

Zero Sum

Abstract This paper studies the H∞ disturbance attenuation problem for index one descriptor systems using the theory of differential games. To solve this disturbance attenuation problem the problem is converted into a reduced ordinary zero-sum game. Within a linear quadratic setting the problem is solved for feedback information structure.

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An Analytic and Numerical Investigation of a Differential Game

Axioms ◽

10.3390/axioms10020066 ◽

2021 ◽

Vol 10 (2) ◽

pp. 66

Author(s):

Aviv Gibali ◽

Oleg Kelis

Keyword(s):

Differential Game ◽

Equation Approach ◽

Linear Quadratic ◽

Control Cost ◽

Dual Representation ◽

Cost Functional ◽

Variational Derivatives ◽

Zero Sum ◽

The Cost ◽

Derivatives Of

In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation.

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Differential Games for an Infinite 2-Systems of Differential Equations

Mathematics ◽

10.3390/math9131467 ◽

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.

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Zero-sum differential games-based fast adaptive robust optimal sliding mode control design for uncertain missile autopilot with constrained-input

International Journal of Control ◽

10.1080/00207179.2021.1872802 ◽

2021 ◽

pp. 1-26

Author(s):

Yuxin Gao ◽

Chunsheng Liu ◽

Sen Jiang ◽

Shaojie Zhang

Keyword(s):

Sliding Mode Control ◽

Differential Games ◽

Sliding Mode ◽

Control Design ◽

Mode Control ◽

Zero Sum ◽

Constrained Input

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Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game

Axioms ◽

10.3390/axioms10030132 ◽

2021 ◽

Vol 10 (3) ◽

pp. 132

Author(s):

Valery Y. Glizer

Keyword(s):

Nash Equilibrium ◽

Differential Game ◽

Quadratic Functional ◽

Linear Quadratic ◽

Feedback Controls ◽

Cheap Control ◽

Optimal Values ◽

Definition Of ◽

Zero Sum ◽

Equilibrium Sequence

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.

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Local modal participation analysis of nonlinear systems using Poincaré linearization

Nonlinear Dynamics ◽

10.1007/s11071-019-05363-1 ◽

2019 ◽

Vol 99 (1) ◽

pp. 803-811 ◽

Cited By ~ 2

Author(s):

Boumediene Hamzi ◽

Eyad H. Abed

Keyword(s):

Nonlinear Systems ◽

Autonomous Systems ◽

Analytical Formula ◽

Linear Case ◽

Local Mode ◽

Initial Condition ◽

Initial State ◽

State Participation ◽

Symmetry Assumption ◽

Definition Of

AbstractThe paper studies an extension to nonlinear systems of a recently proposed approach to the definition of modal participation factors. A definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. While the definition is general, the resulting measures depend on the assumed uncertainty law governing the system initial condition, as in the linear case. The work follows Hashlamoun et al. (IEEE Trans Autom Control 54(7):1439–1449 2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure over an uncertain set of system initial state. Poincaré linearization is used to replace the nonlinear system with a locally equivalent linear system. It is found that under a symmetry assumption on the distribution of the initial state, the tractable calculation and analytical formula for mode-in-state participation factors found for the linear case persists to the nonlinear setting. This paper is dedicated to the memory of Professor Ali H. Nayfeh.

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Research on the Use of Mobile Devices and Headphones on Pedestrian Crossings—Pilot Case Study from Slovakia

Safety ◽

10.3390/safety7010017 ◽

2021 ◽

Vol 7 (1) ◽

pp. 17

Author(s):

Miroslava Mikusova ◽

Joanna Wachnicka ◽

Joanna Zukowska

Keyword(s):

Mobile Phone ◽

Mobile Devices ◽

Mobile Phones ◽

Slovak Republic ◽

Pedestrian Behavior ◽

The Road ◽

Phone Calls ◽

Lack Of Information ◽

Interaction Factor ◽

Future Work

The topic of the use of mobile devices and headphones on pedestrian crossings is much less explored in comparison to the use of the mobile phone while driving. Recent years have seen many discussions on this issue, especially in foreign countries. The Slovak Republic, however, has not been giving it enough attention (and it is not mentioned in the National Road Safety Plan for the Slovak Republic from 2011 to 2020). This paper aims to draw attention to this issue. It presents basic outputs of a pilot study on pedestrian safety, with a focus on the use of mobile devices and headphones at selected non-signalized pedestrian crossings in three Slovak cities. Overall, 9% of pedestrians used headphones or mobile devices at observed pedestrian crossings (4% of them used headphones, 1% used headphones and at same time used their mobile phone, 2% made phone calls and 2% used their mobile phones). While these numbers can be considered relatively low, the study proved that during weekdays every 2 min someone was using the crossing without fully focusing on crossing the road safely. Another main finding was that although the safety risk at pedestrian crossings is increased by factors such as rush hour traffic or reduced visibility, pedestrian behavior related to the use of mobile phones and headphones does not change. A safety assessment was also carried out at the crossings. The results show that pedestrian behavior is not affected by the level of safety of the crossing (e.g., visibility of the crossing for drivers). The results of the presented analysis suggest that action is needed to change that. Due to the lack of information about accidents involving pedestrians using mobile phones and headsets when crossing the road, no relevant statistical data could be analyzed. The dataset collected can be used as a basis for further investigation or comparisons with other countries of the relevant indicators. In future work, we would like to include a pedestrian–driver interaction factor focusing on driver speed behavior in relation to pedestrians (who are on or are about to step onto a pedestrian crossing) and identify critical situations caused by improper behavior of drivers and/or pedestrians. This will help to understand speed adjustment problems related to pedestrian crossings.

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