Existence of a periodic solution for a coupled system of differential equations

Abstract

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 [2] (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.