It is often desired to have control over a process or a physical system, to cause it to behave optimally. Optimal control theory deals with analyzing and finding solutions for optimal control for a system that can be represented by a set of differential equations. This thesis examines such a system in the form of a set of matrix differential equations known as a continuous linear time-invariant system. Conditions on the system, such as linearity, allow one to find an explicit closed form finite solution that can be more efficiently computed compared to other known types of solutions. This is done by optimizing a quadratic cost function. The optimization leads to solving a Riccati equation. Conditions are discussed for which solutions are possible. In particular, we will obtain a solution for a stable and controllable system. Numerical examples are given for a simple system with 2x2 matrix coefficients.