Solving a constrained Sylvester-type system on the commutative quaternion ring

This paper establishes solvability conditions and general solution sets for a constrained Sylvester-type system over the commutative quaternion ring by proposing two distinct methods. As a key application, we also analyze the minimum solution of a related optimization problem when this system is solvable. Additionally, the solvability condition and the general form of the Hermitian solutions for this system over the commutative quaternion ring are established in this paper. The main results are validated through an algorithm and a numerical example.