Controllability of Nonlinear Quaternion-Valued Systems with Input-Delay

The utilization of quaternions in nonlinear systems with input delays is presented in this paper, aiming to investigate the controllability of nonlinear quaternion-valued systems (QVSs). In view of the non-commutativity of quaternion multiplication, the system is decomposed into four real-valued subsystems. The existence and uniqueness of solutions for QVSs with input delays are demonstrated using the contraction mapping principle and Laplace transform. To achieve system controllability amidst distinct input delays, two categories of Gram matrices along with their respective control functions are established, and their feasibility are demonstrated by Arzela-Ascoli theorem and the Schaefer fixed point theorem. Finally, numerical simulations are conducted to validate the theoretical findings.