Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

This paper establishes a systematic frame work for the theory of linear quaternion-valued differential equations (QDEs), which can be applied to quantum mechanics, Frenet frame in differential geometry, kinematic modelling, attitude dynamics, Kalman filter design, spatial rigid body dynamics and fluid mechanics, etc. On the non-commutativity of the quaternion algebra, the algebraic structure of the solutions to the QDEs is not a linear vector space. It is actually a left- or right- module. Moreover, many concepts and properties for the ordinary differential equations (ODEs) can not be used. They should be redefined accordingly. A definition of {\em Wronskian} is introduced under the framework of quaternions which is different from standard one in the ordinary differential equations. Liouville formula for QDEs is given. Also, it is necessary to treat the eigenvalue problems with left- and right-sides, accordingly. Upon these, we studied the solutions to the linear QDEs. Furthermore, we present two algorithms to evaluate the fundamental matrix. Some concrete examples are given to show the feasibility of the obtained algorithms. Finally, a conclusion and discussion ends the paper.