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