Exact Boundary Controllability of Nodal Profile for Quasilinear Hyperbolic Systems (SpringerBriefs in Mathematics)

This book provides a comprehensive overview of the exact boundary controllability of nodal profile, a new kind of exact boundary controllability stimulated by some practical applications. This kind of controllability is useful in practice as it does not require any precisely given final state to be attained at a suitable time t=T by means of boundary controls, instead it requires the state to exactly fit any given demand (profile) on one or more nodes after a suitable time t=T by means of boundary controls. In this book we present a general discussion of this kind of controllability for general 1-D first order quasilinear hyperbolic systems and for general 1-D quasilinear wave equations on an interval as well as on a tree-like network using a modular-structure construtive method, suggested in LI Tatsien's monograph "Controllability and Observability for Quasilinear Hyperbolic Systems"(2010), and we establish a complete theory on the local exact boundary controllability of nodal profile for 1-D quasilinear hyperbolic systems. Preface 5 Contents 7 1 First Order Quasilinear Hyperbolic Systems 10 1.1 1-D First Order Quasilinear Hyperbolic Systems 10 1.2 Characteristic Form of Hyperbolic System 12 1.3 Reducible Quasilinear Hyperbolic System. Riemann Invariants 13 1.4 Saint-Venant System for Unsteady Flows on a Single Open Canal 14 1.5 Semi-global C1 Solutions to the Mixed Initial-Boundary Value Problem 16 1.6 Exchanging the Role of t and x 19 1.7 Uniqueness of C1 Solution to the One-Sided Mixed Initial-Boundary Value Problem 20 2 Quasilinear Wave Equations 23 2.1 1-D Quasilinear Wave Equations 23 2.2 Semi-global C2 Solutions to the Mixed Initial-Boundary Value Problem 24 2.3 Uniqueness of C2 Solution to the One-Sided Mixed Initial-Boundary Value Problem 27 3 Semi-global Piecewise Classical Solutions on a Tree-Like Network 31 3.1 Introduction 31 3.2 Semi-global Piecewise C1 Solutions to 1-D First Order Quasilinear Hyperbolic Systems on a Tree-Like Network 31 3.3 Semi-global Piecewise C2 Solutions to 1-D Quasilinear Wave Equations on a Tree-Like Network 35 4 Exact Boundary Controllability of Nodal Profile for 1-D First Order Quasilinear Hyperbolic Systems 38 4.1 Introduction 38 4.2 Definitions and Main Results 39 4.3 Proof of Theorem 4.1 42 4.4 Proof of Theorem 4.2 44 4.5 Exact Boundary Controllability of Nodal Profile for Saint-Venant System ƒ 47 4.6 Remarks 53 5 Exact Boundary Controllability of Nodal Profile for 1-D First Order Quasilinear Hyperbolic Systems on a Tree-Like Network 54 5.1 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Star-Like Network of Open Canals (Case 1) 55 5.2 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Star-Like Network of Open Canals (Case 2) 60 5.3 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Star-Like Network of Open Canals (Case 3) 62 5.4 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Star-Like Network of Open Canals (Case 4) 65 5.5 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Star-Like Network of Open Canals (Case 5) 69 5.6 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Tree-Like Network of Open Canals 71 5.7 Exact Boundary Controllability of Nodal Profile for Saint-Venant System on a Tree-Like Network of Open Canals (Continued) 78 5.8 Remarks 80 6 Exact Boundary Controllability of Nodal Profile for 1-D Quasilinear Wave Equations 81 6.1 Introduction 81 6.2 Definitions and Main Results 81 6.3 Proof of Theorem 6.1 84 6.4 Proof of Theorem 6.2 86 6.5 Remarks 88 7 Exact Boundary Controllability of Nodal Profile for 1-D Quasilinear Wave Equations on a Planar Tree-Like Network of Strings 89 7.1 Introduction 89 7.2 Exact Boundary Controllability of Nodal Profile for 1-D Quasilinear Wave Equations on a Star-Like Network of Strings 89 7.3 Exact Boundary Controllability of Nodal Profile for 1-D Quasilinear Wave Equations on a Star-Like Network of Strings (Continued) 97 7.4 Exact Boundary Controllability of Nodal Profile for Quasilinear Wave Equations on a Planar Tree-Like Network of Strings 103 7.5 Exact Boundary Controllability of Nodal Profile for Quasilinear Wave Equations on a Planar Tree-Like Network (Continued) 107 7.6 Remarks 108 References 109 Index 111 Front Matter....Pages i-ix First Order Quasilinear Hyperbolic Systems....Pages 1-13 Quasilinear Wave Equations....Pages 15-22 Semi-global Piecewise Classical Solutions on a Tree-Like Network....Pages 23-29 Exact Boundary Controllability of Nodal Profile for 1-D First Order Quasilinear Hyperbolic Systems....Pages 31-46 Exact Boundary Controllability of Nodal Profile for 1-D First Order Quasilinear Hyperbolic Systems on a Tree-Like Network....Pages 47-73 Exact Boundary Controllability of Nodal Profile for 1-D Quasilinear Wave Equations....Pages 75-82 Exact Boundary Controllability of Nodal Profile for 1-D Quasilinear Wave Equations on a Planar Tree-Like Network of Strings....Pages 83-102 Back Matter....Pages 103-108