CONTROLLABILITY AND OBSERVABILITY RESULTS FOR QUATERNION-VALUED IMPULSIVE DIFFERENTIAL EQUATIONS

The purpose of this manuscript is to consider the controllability and observability of quaternion-valued impulsive differential equations (QIDEs). First, sufficient and necessary criteria for the controllability of linear QIDEs are investigated. Then, by constructing suitable control functions and employing fixed-point theorems, the controllability of nonlinear QIDEs is acquired. Next, sufficient and necessary conditions for the observability of linear QIDEs are derived. Most important of all, theoretical results obtained in the sense of complex-valued and quaternion-valued are equivalent to each other due to the adjoint matrix of quaternion matrix and the isomorphism between quaternion vector space and complex variables space. Finally, two examples are provided to indicate the feasibility of our results achieved.