Matrix differential equations are playing key roles in solving problems in many fields of study such as control theory. Control system theory is a branch of engineering that handles how to manipulate a dynamical system’s inputs to change the behavior or outcome of the system to something that is desired. Control systems have been around at least as far back as the float-regulated water clock developed by the Greeks in the third century BC; however, control theory as a branch of mathematics wasn’t explored until much later in the mid-19th century. Since that time, the theory has been used almost entirely by professional engineers and mathematicians. Coupled Riccati differential equations play distinguished role in the control theory  (such as optimal control problem, stochastic control problem and other control theory problems like the linear systems and dynamic games). Here, we consider a coupled Riccati differential equation and use an appropriate Green’s function to reduce it a Fredholm integral equation. Finally, we prove the existence of a periodic solution. © 2022 American Institute of Physics Inc.. All rights reserved.