Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems: Regularization Approach (Static & Dynamic Game Theory: Foundations & Applications)

✍ Scribed by Valery Y. Glizer, Oleg Kelis

• Publisher: Birkhäuser
• Year: 2022
• Tongue: English
• Leaves: 211
• Category: Library

✦ Synopsis

This monograph is devoted to the analysis and solution of singular differential games and singular $H_{\inf}$ control problems in both finite- and infinite-horizon settings. Expanding on the authors’ previous work in this area, this novel text is the first to study the aforementioned singular problems using the regularization approach.
After a brief introduction, solvability conditions are presented for the regular differential games and $H_{\inf}$ control problems. In the following chapter, the authors solve the singular finite-horizon linear-quadratic differential game using the regularization method. Next, they apply this method to the solution of an infinite-horizon type. The last two chapters are dedicated to the solution of singular finite-horizon and infinite-horizon linear-quadratic $H_{\inf}$ control problems. The authors use theoretical and real-world examples to illustrate the results and their applicability throughout the text,and have carefully organized the content to be as self-contained as possible, making it possible to study each chapter independently or in succession. Each chapter includes its own introduction, list of notations, a brief literature review on the topic, and a corresponding bibliography. For easier readability, detailed proofs are presented in separate subsections.
Singular Linear-Quadratic Zero-Sum Differential Games and $H_{\inf}$ Control Problems will be of interest to researchers and engineers working in the areas of applied mathematics, dynamic games, control engineering, mechanical and aerospace engineering, electrical engineering, and biology. This book can also serve as a useful reference for graduate students in these area

✦ Table of Contents

Contents
1 Introduction
References
2 Examples of Singular Extremal Problems and Some Basic Notions
2.1 Introduction
2.2 Academic Examples
2.2.1 Minimization of a Function
2.2.2 Optimal Control Problem
2.2.3 Saddle Point of a Function of Two Variables
2.2.4 Zero-Sum Differential Game
2.3 Mathematical Models of Real-Life Problems
2.3.1 Planar Pursuit-Evasion Engagement: Zero-Order Dynamics of Participants
2.3.2 Planar Pursuit-Evasion Engagement: First-Order Dynamics of Participants
2.3.3 Three-Dimensional Pursuit-Evasion Engagement: Zero-Order Dynamics of Participants
2.3.4 Infinite-Horizon Robust Vibration Isolation
2.4 Concluding Remarks and Literature Review
References
3 Preliminaries
3.1 Introduction
3.2 Solvability Conditions of Regular Problems
3.2.1 Finite-Horizon Differential Game
3.2.2 Infinite-Horizon Differential Game
3.2.3 Finite-Horizon upper H Subscript normal infinityHinfty Problem
3.2.4 Infinite-Horizon upper H Subscript normal infinityHinfty Problem
3.3 State Transformation in Linear System and Quadratic Functional
3.4 Concluding Remarks and Literature Review
References
4 Singular Finite-Horizon Zero-Sum Differential Game
4.1 Introduction
4.2 Initial Game Formulation
4.3 Transformation of the Initially Formulated Game
4.4 Regularization of the Singular Finite-Horizon Game
4.4.1 Partial Cheap Control Finite-Horizon Game
4.4.2 Saddle-Point Solution of the Game (4.8), (4.12)
4.5 Asymptotic Analysis of the Game (4.8), (4.12)
4.5.1 Transformation of the Problem (4.14)
4.5.2 Asymptotic Solution of the Terminal-Value Problem (4.23)–(4.25)
4.5.3 Asymptotic Representation of the Value of the Game (4.8), (4.12)
4.6 Reduced Finite-Horizon Differential Game
4.7 Saddle-Point Sequence of the SFHG
4.7.1 Main Assertions
4.7.2 Proof of Lemma 4.5
4.7.3 Proof of Lemma 4.6
4.8 Examples
4.8.1 Example 1
4.8.2 Example 2: Solution of Planar Pursuit-Evasion Game with Zero-Order Dynamics of Players
4.8.3 Example 3: Solution of Three-Dimensional Pursuit-Evasion Game with Zero-Order Dynamics of Players
4.9 Concluding Remarks and Literature Review
References
5 Singular Infinite-Horizon Zero-Sum Differential Game
5.1 Introduction
5.2 Initial Game Formulation
5.3 Transformation of the Initially Formulated Game
5.4 Auxiliary Lemma
5.5 Regularization of the Singular Infinite-Horizon Game
5.5.1 Partial Cheap Control Infinite-Horizon Game
5.5.2 Saddle-Point Solution of the Game (5.9), (5.29)
5.6 Asymptotic Analysis of the Solution to the Game (5.9), (5.29)
5.6.1 Transformation of the Eq. (5.33)
5.6.2 Asymptotic Solution of the Set (5.44)–(5.46)
5.6.3 Asymptotic Representation of the Value of the Game (5.9), (5.29)
5.7 Reduced Infinite-Horizon Differential Game
5.8 Saddle-Point Sequence of the SIHG
5.8.1 Main Assertions
5.8.2 Proof of Lemma 5.6
5.8.3 Proof of Lemma 5.7
5.9 Examples
5.9.1 Example 1
5.9.2 Example 2: Solution of Infinite-Horizon Vibration Isolation Game
5.10 Concluding Remarks and Literature Review
References
6 Singular Finite-Horizon upper H Subscript normal infinityHinfty Problem
6.1 Introduction
6.2 Initial Problem Formulation
6.3 Transformation of the Initially Formulated upper H Subscript normal infinityHinfty Problem
6.4 Regularization of the Singular Finite-Horizon upper H Subscript normal infinityHinfty Problem
6.4.1 Partial Cheap Control Finite-Horizon upper H Subscript normal infinityHinfty Problem
6.4.2 Solution of the upper H Subscript normal infinityHinfty Problem (6.9), (6.11)
6.5 Asymptotic Analysis of the upper H Subscript normal infinityHinfty Problem (6.9), (6.11)
6.5.1 Transformation of the Problem (6.13)
6.5.2 Asymptotic Solution of the Problem (6.20)–(6.22)
6.6 Reduced Finite-Horizon upper H Subscript normal infinityHinfty Problem
6.7 Controller for the SFHP
6.7.1 Formal Design of the Controller
6.7.2 Properties of the Simplified Controller (6.39)
6.7.3 Proof of Theorem 6.1
6.7.4 Proof of Theorem 6.2
6.8 Example
6.9 Concluding Remarks and Literature Review
References
7 Singular Infinite-Horizon upper H Subscript normal infinityHinfty Problem
7.1 Introduction
7.2 Initial Problem Formulation
7.3 Transformation of the Initially Formulated upper H Subscript normal infinityHinfty Problem
7.4 Regularization of the Singular Infinite-Horizon upper H Subscript normal infinityHinfty Problem
7.4.1 Partial Cheap Control Infinite-Horizon upper H Subscript normal infinityHinfty Problem
7.4.2 Solution of the upper H Subscript normal infinityHinfty Problem (7.9), (7.15)
7.5 Asymptotic Analysis of the upper H Subscript normal infinityHinfty Problem (7.9), (7.15)
7.5.1 Transformation of Eq. (7.17)
7.5.2 Asymptotic Solution of the Set (7.24)
7.6 Reduced Infinite-Horizon upper H Subscript normal infinityHinfty Problem
7.7 Controller for the SIHP
7.7.1 Formal Design of the Controller
7.7.2 Properties of the Simplified Controller (7.45)
7.7.3 Proof of Theorem 7.1
7.7.4 Proof of Theorem 7.2
7.8 Examples
7.8.1 Example 1
7.8.2 Example 2: upper H Subscript normal infinityHinfty Control in Infinite-Horizon Vibration Isolation Problem
7.9 Concluding Remarks and Literature Review
References
Index