Qualitative theory of hybrid dynamical systems

목차 (Table of Contents)

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• 1 Introduction.- 1.1 Hybrid Dynamical Systems.- 1.2 Two Contrasting Examples of Discretely Controlled Continuous Variable Systems.- 1.3 The Main Goal of This Book.- 1.4 Organization of the Book.- 1.5 List of Notations.- 2 Qualitative Analysis of Some Simple Hybrid Dynamical Systems.- 2.1 Introduction.- 2.2 Differential Automata and Their Trajectories.- 2.3 Cyclic Linear Differential Automata.- 2.4 Qualitative Analysis of Cyclic Linear Differential Automata.- 2.5 Switched Server Systems with a Cyclic Switching Policy.- 2.6 Switched Server Systems with Several Limit Cycles.- 2.7 Qualitative Analysis of Closed Switched Server Systems.- 2.8 Essentially Non-Periodic Dynamics of Switched Arrival Systems.- 3 General Theory of Multivalued Differential Automata.- 3.1 Introduction.- 3.2 Multivalued Differential Automata.- 3.2.1 Basic assumptions and definitions.- 3.2.2 Illustrative examples.- 3.2.3 Invariant sets.- 3.2.4 A partial classification of points in the phase space.- 3.2.5 Deterministic and well-posed systems.- 3.2.6 The skeleton and the backstepping mapping.- 3.2.7 Asymptotically stable limit cycles.- 3.3 Decomposition of Well-Posed Differential Automata.- 3.4 Existence of Periodic Trajectories.- 3.5 Proofs of the Theorems and Lemmas from Section 3.2.- 3.6 Proof of Theorem 3.2.26.- 3.7 Proofs of the Theorems from Sections 3.3 and 3.4.- 3.7.1 Proof of Theorem 3.4.3.- 4 Two-Dimensional Hybrid Dynamical Systems.- 4.1 Introduction.- 4.2 An Analog of the Poincare-Bendixon Theorem.- 4.2.1 Basic assumptions.- 4.2.2 A simple periodic dynamics.- 4.2.3 A criterion for a simple periodic dynamics.- 4.3 A Switched Arrival System with Three Buffers.- 4.4 A Switched Server System with Three Buffers.- 4.5 Proofs of the Statements from Section 4.2.- 4.5.1 Proofs of the lemmas from Section 4.2.- 4.5.2 Proof of Theorem 4.2.10 and the remarks following it.- 5 Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: General Theory.- 5.1 Introduction.- 5.2 Basic Assumptions and Definitions.- 5.2.1 Multivalued differential automata with constant derivatives.- 5.2.2 Key assumptions.- 5.3 Criteria for Existence and Stability of Limit Cycles.- 5.3.1 A complement concerning deterministic systems.- 5.4 Proofs of the Lemmas from Section 5.2.- 5.5 Proofs of the Theorems and Lemmas from Section 5.3..- 5.6 Proofs of the Theorem and Lemmas from Subsection 5.3.1.- 6 Limit Cycles in Hybrid Dynamical Systems with Constant Derivatives: Examples.- 6.1 Introduction.- 6.2 Qualitative Analysis of a Switched Server System.- 6.2.1 Description of a switched server system.- 6.2.2 A cyclic control policy.- 6.2.3 The Clear-the-Largest-Buffer-Level Policy.- 6.2.4 Structural stability of a switched server system.- 6.3 A Switched Arrival System with Three Buffers.- 6.4 Qualitative Analysis of Switched Single Server Flow Networks.- 6.4.1 Single server flow networks.- 6.4.2 A cyclic control policy.- 6.4.3 A composed cyclic control policy.- 6.4.4 A combined control policy.- 7 Globally Periodic Behavior of Switched Single Server Flow Networks.- 7.1 Introduction.- 7.2 Description of Switched Single Server Flow Networks.- 7.3 Analysis of Switched Single Server Flow Networks.- 8 Regularizability of Switched Multiple Server Flow Networks.- 8.1 Introduction 315 8.2 Description of Switched Multiple Server Flow Networks.- 8.3 Regularizable Switched Multiple Server Flow Networks.- 8.4 Illustrative Example.- 9 Open Problems.- 9.1 Introduction.- 9.2 Switched Server Systems.- 9.3 Essentially Nonperiodic Multidimensional Switched Arrival Systems.- 9.4 Switched Server/Arrival Systems with Several Servers.- 9.5 A Generalized Processor Sharing Model.- 9.6 Stabilizability of Switched Multiple Server Flow Networks.- 9.7 Chaotic Switched Flow Networks.- 9.8 Existence and Global Stability of Limit Cycles in Nonlinear Differential Automata.- References.