Principle of Time Dilation in Game Problems of Dynamics

• Published in: Recent Developments in Automatic Control Systems pp. 113 - 129
• Main Author: Chikrii, G. Ts
• Format: Book Chapter
• Language: English
• Published: River Publishers 2022
• Edition: 1
• Subjects: Control Systems; Electrical & Power Engineering; Industrial Safety; Power Generation and Transmission; Process Design, Control & Automation; Safety & Industrial Hygiene; conflict-controlled process; Cyber Physical Systems; General Problem Statements; mathematical pendulum; Variational Inequality; Measurable Choice; Theory Differential Games; Autonomous Mobile Robot; Passive Intervals; Terminal Set; Automatic Control Systems; Time Dilation; set-valued mapping; modified Pontryagin's condition; Time Delay Systems; Group Pursuit; Variational Inequality Problem; Fuzzy Systems Design; Fractional Derivatives; Game Problem; Admissible Controls; Geometric Coordinates; Fundamental Matrix; Finite Dimensional Space; Differential game; function of time dilation; Hold; Initial Interval; Measurable Selection
• ISBN: 8770226741; 9788770226745
• DOI: 10.1201/9781003339229-6

In this chapter, we consider the linear differential game of approaching a cylindrical terminal set. The case when classic Pontryagin's condition does not hold is studied. Instead, a considerably weaker condition, incorporating the function of time dilation, is introduced. This makes it possible to expand the range of problems that can be solved analytically, by using information about the enemy's control in the past. The study is carried out in the framework of the Method of Resolving Functions. The gist of the method consists in constructing certain scalar function associated with the parameters of the conflict-controlled process and characterizing the gain of the first player at each moment. If the total gain achieves the predetermined value then this means that the game terminates in a certain guaranteed time. The pursuer's control, realizing the game goal, is constructed based on the Filippov-Castaing theorem on measurable choice. To illustrate the developed scheme, we analyze in detail the problem of approaching two controlled oscillation systems in geometric coordinates.