Linear Quaternion Differential Equations: Basic Theory and Fundamental Results

本报告介绍四元数体上方程的基础理论和基本框架。系统性指出四元数体上微分方程与常微分方程的区别。Quaternion-valued differential equations (QDEs) are a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ordinary differential equations (ODEs) is the algebraic structure. Due to the noncommutativity of the quaternion algebra, the set of all the solutions to the linear homogenous QDEs is completely different from ODEs. It is actually a right-free module, not a linear vector space. This paper establishes a systematic frame work for the theory of linear QDEs, which can be applied to quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modeling, attitude dynamics, Kalman filter design, spatial rigid body dynamics, etc. We prove that the set of all the solutions to the linear homogenous QDEs is actually a right-free module, not a linear vector space. On the noncommutativity of the quaternion algebra, many concepts and properties for the ODEs cannot be used. They should be redefined accordingly. A definition of Wronskian is introduced under the framework of quaternions which is different from standard one in the ODEs. 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. 夏永辉，佛山大学教授。曾获省部级科技进步奖3项，其中浙江省科学技术进步一等奖1项(前三完成人),获福建青年科技奖。入选闽江学者特聘教授；2012年入选浙江省“151人才工程”第二层次。2021年，2023年科技部重点研发计划答辩会评专家组成员。多次担任科技部、教育部以及各省市基金、人才项目和科技奖励的通讯评议或者会评专家。主持国家自然科学基金3项(其中面上2项)，参与国家重点1项，主持浙江省基金重点项目1项。曾任浙江师范大学“杰出学者”特聘教授、博士生导师，与合作者一起推广了著名学者庞加莱和李雅普诺夫关于二维平面系统可积的充要条件的经典理论,将此可积理论推广到了任意有限维；建立了四元数体上微分方程理论与应用的基本框架(已经形成专著在中国科学出版社出版)；改进了经典的全局Hartman-Grobman线性化定理。