Contents Preface Athors’ biography Chapter 1 Background of Quaternion and Quaternion-valued Differential Equations 1 1.1 Background for quaternions 1 1.2 Background for QDEs 5 1.2.1 Quaternion Frenet frames in differential geometry 5 1.2.2 QDEs appears in kinematic modelling and attitude dynamics 6 1.2.3 QDE appears in fluid mechanics 8 1.2.4 QDE appears in quantum mechanics 8 1.3 History and motivation of our research 9 Chapter 2 Preliminary Concepts and Notations 12 2.1 Quaternion algebra 12 2.2 Biquaternion algebra 14 2.3 Definitions of determinants 15 2.4 Groups, rings, modules 17 2.5 Existence and uniqueness of solution to QDEs 20 Chapter 3 Basic Theory of Linear Homogeneous Quaternion-valued Differential Equations 22 3.1 Structure of general solutions for 2D QDEs 22 3.2 Structure of general solutions for any finite dimensional QDEs based on permutation 30 3.3 Fundamental matrix and solution to QDEs 40 3.4 Algorithm for computing fundamental matrix 47 3.4.1 Method 1: using expansion of exp{At} 48 3.4.2 Method 2: eigenvalue and eigenvector theory 51 Chapter 4 Algorithm for Linear Homogeneous QDEs when Linear Homogeneous System Has Multiple Eigenvalues 55 4.1 Motivations 55 4.2 Solving linear homogenous QDEs when linear homogeneous system has multiple eigenvalues 56 4.2.1 Multiple eigenvalues with enough eigenvectors 56 4.2.2 Multiple eigenvalues with fewer eigenvectors 58 Chapter 5 Floquet Theory of Quaternion-valued Differential Equations 66 5.1 Preliminary results 67 5.2 Stability of linear homogeneous QDEs with constant coefficients 70 5.3 Floquet theory for QDEs 75 5.4 Quaternion-valued Hill’s equations 84 Chapter 6 Solve Linear Nonhomogeneous Quaternion-valued Differential Equations 88 6.1 Notations 88 6.2 Main results 90 6.3 Some examples 92 Chapter 7 Linear Quaternion Dynamic Equations on Time Scale 97 7.1 Notations and preliminary results 98 7.1.1 Notations and lemmas 98 7.1.2 Calculus on time scales 100 7.2 First order linear QDETS 103 7.3 Linear systems of QDETS 109 7.4 Linear QDETS with constant coefficients 119 Chapter 8 Laplace Transform: a New Approach in Solving Linear Quaternion Differential Equations 129 8.1 Introduction 129 8.2 Biquaternion algebra 130 8.2.1 Biquaternion exponential function 131 8.2.2 Fundamental theorem of quaternion algebra and factorization theorem revisited 134 8.3 Definition and properties of the Laplace transform in biquaternion domain 134 8.4 Using QLT to solve QDEs 144 Chapter 9 Solving Quaternion Differential Equations with Two-sided Coefficients 150 9.1 Introduction 150 9.2 Notations and preliminary results 151 9.3 Solving QDEs with unilateral coefficients 152 9.4 Solving QDEs with two-sided coefficients 160 9.4.1 Homogeneous linear QDEs with two-sided coefficients 160 9.4.2 Nonhomogeneous linear QDEs with two-sided coefficients 168 Chapter 10 Controllability and Observability of Linear Quaternionvalued Systems 172 10.1 Motivations 172 10.2 Notations and preliminary results 174 10.3 Main results on the controllability and observability of linear QVS 178 10.3.1 Controllability 178 10.3.2 Observability 183 10.3.3 Duality 187 Chapter 11 Stability Analysis of Quaternion-valued Neural Networks 190 11.1 Notations and preliminary results 190 11.2 Main results 197 11.3 Examples 205 Chapter 12 Convex Function Optimization Problems with Quaternion Variables 209 12.1 Notations and preliminary results 210 12.1.1 Quaternion algebra analysis 210 12.1.2 Generalized gradient 213 12.2 Main results on the convex function optimization problems with quaternion variables 217 12.3 Examples and simulations 230 12.4 Proof of the Proposition 12.1.4 232 Chapter 13 Penalty Method for Constrained Distributed Quaternionvariable Optimization 236 13.1 Introduction 236 13.2 Preliminaries 238 13.3 Main results 245 13.4 An example 250 Bibliography 253
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