Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems
Regularization Approach
- 2022
- Buch
- Verfasst von
- Valery Y. Glizer
- Oleg Kelis
- Verlag
- Springer International Publishing
Über dieses Buch
Diese Monografie widmet sich der Analyse und Lösung von singulären Differentialspielen und singulären $H _ {inf} $-Kontrollproblemen sowohl in endlichen als auch in unendlichen Horizonten. Aufbauend auf den früheren Arbeiten der Autoren auf diesem Gebiet ist dieser neuartige Text der erste, der die oben erwähnten singulären Probleme mithilfe des Regularisierungsansatzes untersucht. Nach einer kurzen Einführung werden die Lösbarkeitsbedingungen für die regulären Differentialspiele und $H _ {inf} $-Kontrollprobleme vorgestellt. Im folgenden Kapitel lösen die Autoren das singuläre linear-quadratische Differentialspiel Finite-Horizont mit Hilfe der Regularisierungsmethode. Als nächstes wenden sie diese Methode auf die Lösung eines Typs mit unendlichem Horizont an. Die letzten beiden Kapitel widmen sich der Lösung der linearen-quadratischen $H _ {inf} $-Kontrollprobleme. Die Autoren verwenden theoretische und reale Beispiele, um die Ergebnisse und ihre Anwendbarkeit im gesamten Text zu veranschaulichen, und haben den Inhalt sorgfältig so in sich geschlossen wie möglich organisiert, so dass es möglich ist, jedes Kapitel einzeln oder nacheinander zu studieren. Jedes Kapitel enthält eine eigene Einleitung, eine Liste von Notizen, eine kurze Literaturübersicht zum Thema und eine entsprechende Bibliographie. Zur besseren Lesbarkeit werden detaillierte Beweise in separaten Unterabschnitten präsentiert. Einzelne linear-quadratische Nullsummenspiele und $H _ {inf} $Kontrollprobleme werden für Forscher und Ingenieure interessant sein, die in den Bereichen angewandte Mathematik, dynamisches Spiel, Regelungstechnik, Maschinenbau und Luft- und Raumfahrt, Elektrotechnik und Biologie arbeiten. Dieses Buch kann auch als nützliche Referenz für Doktoranden in diesen Bereichen dienen.
Mit KI übersetzt
Über dieses Buch
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
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Inhaltsverzeichnis
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Frontmatter
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1. Introduction
Valery Y. Glizer, Oleg KelisAbstractA singular differential game is the game which cannot be solved by the application of the first-order solvability conditions. -
2. Examples of Singular Extremal Problems and Some Basic Notions
Valery Y. Glizer, Oleg KelisAbstractIn this chapter, we present several examples of singular extremal problems. Some of these examples are academic ones. -
3. Preliminaries
Valery Y. Glizer, Oleg KelisAbstractIn this chapter, several problems, allowing first-order solvability conditions, are considered. Namely, we consider zero-sum linear-quadratic differential games in finite-horizon and infinite-horizon settings, as well as \(H_{\infty }\) control problems in finite-horizon and infinite-horizon settings. -
4. Singular Finite-Horizon Zero-Sum Differential Game
Valery Y. Glizer, Oleg KelisAbstractIn this chapter, a finite-horizon zero-sum linear-quadratic differential game, which cannot be solved by application of the first-order solvability conditions, is considered. Thus, this game is singular. Its singularity is due to the singularity of the weight matrix in the control cost of a minimizing player in the game’s functional. For this game, novel definitions of a saddle-point equilibrium (a saddle-point equilibrium sequence) and a game value are introduced. Regularization method is proposed for obtaining these saddle-point equilibrium sequence and game value. This method consists in an approximate replacement of the original singular game with an auxiliary regular finite-horizon zero-sum linear-quadratic differential game depending on a small positive parameter. Thus, the first-order solvability conditions are applicable for this new game. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix differential equation, arising in these conditions, yields the solution (the saddle-point equilibrium sequence and the game value) to the original singular game. -
5. Singular Infinite-Horizon Zero-Sum Differential Game
Valery Y. Glizer, Oleg KelisAbstractIn this chapter, an infinite-horizon zero-sum linear-quadratic differential game, which cannot be solved by application of the first-order solvability conditions, is considered. This game is singular, and its singularity is due to the singularity of the weight matrix in the control cost of a minimizing player in the game’s functional. For this game, novel definitions of a saddle-point equilibrium (a saddle-point equilibrium sequence) and a game value are introduced. Regularization method is proposed for obtaining these saddle-point equilibrium sequence and game value. This method consists in an approximate replacement of the original singular game with an auxiliary regular infinite-horizon zero-sum linear-quadratic differential game depending on a small positive parameter. Thus, the first-order solvability conditions are applicable for this new game. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix algebraic equation, arising in these conditions, yields the solution (the saddle-point equilibrium sequence and the game value) to the original singular game. -
6. Singular Finite-Horizon Problem
Valery Glizer, Oleg KelisAbstractIn this chapter, we consider a system consisting of an uncertain controlled linear time-dependent differential equation and a linear time-dependent output algebraic equation. For this system, a finite-horizon \(H_{\infty }\) problem is studied in the case where the rank of the coefficients’ matrix for the control in the output equation is smaller than the Euclidean dimension of this control. In this case, the solvability conditions, based on the game-theoretic matrix Riccati differential equation, are not applicable to the solution of the considered \(H_{\infty }\) problem meaning its singularity. To solve this \(H_{\infty }\) problem, a regularization method is proposed. Namely, the original problem is replaced approximately with a regular finite-horizon \(H_{\infty }\) problem depending on a small positive parameter. Thus, the first-order solvability conditions are applicable to this new problem. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix differential equation, arising in these conditions, yields a controller solving the original singular \(H_{\infty }\) problem. Properties of this controller are studied. -
7. Singular Infinite-Horizon Problem
Valery Y. Glizer, Oleg KelisAbstractIn this chapter, we consider a system consisting of an uncertain controlled linear time-invariant differential equation and a linear time-invariant output algebraic equation. For this system, an infinite-horizon \(H_{\infty }\) problem is studied in the case where the rank of the coefficients’ matrix for the control in the output equation is smaller than the Euclidean dimension of this control. In this case, the solvability conditions, based on the game-theoretic matrix Riccati algebraic equation, are not applicable to the solution of the considered \(H_{\infty }\) problem meaning its singularity. To solve this \(H_{\infty }\) problem, a regularization method is proposed. Namely, the original problem is replaced approximately with a regular infinite-horizon \(H_{\infty }\) problem depending on a small positive parameter. Thus, the first-order solvability conditions are applicable to this new problem. Asymptotic analysis (with respect to the small parameter) of the Riccati matrix algebraic equation, arising in these conditions, yields a controller solving the original singular \(H_{\infty }\) problem. Properties of this controller are studied. -
Backmatter
- Titel
- Singular Linear-Quadratic Zero-Sum Differential Games and H∞ Control Problems
- Verfasst von
-
Valery Y. Glizer
Oleg Kelis
- Copyright-Jahr
- 2022
- Electronic ISBN
- 978-3-031-07051-8
- Print ISBN
- 978-3-031-07050-1
- DOI
- https://doi.org/10.1007/978-3-031-07051-8
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