Relative controllability of neutral delay differential equations on quaternion skew field

The research focuses on relative controllability of neutral delay differential equations on quaternion skew field (NDQDEs). First, we derive the representation of solutions for NDQDEs by quaternion determining equations and neutral delay quaternion matrix function. Then, the Gram criterion of relative controllability for NDQDEs in different cases is given with the help of the representation of solutions. By the analogy of Cayley-Hamilton theorem, the rank criterion of relative controllability is obtained. The complete characterization of quaternion control functions is obtained by a special right inverse matrix and shifted Legendre polynomials. Besides, we discuss the relative controllability of weakly nonlinear NDQDEs. Finally, we provide examples to verify the theoretical results.