Optimal control theory deals with finding the optimal control to transfer a state to another in a dynamic system such that a cost function is optimized. The Pontryagin minimum principle, PMP, is an important and useful tool in optimal control theory. It states that any optimal control and the corresponding optimal state trajectory must solve the Hamiltonian system, which is a two point boundary value problem, TPBVP, and satisfy a minimum condition for the Hamiltonian. The work in this project focuses on two main types of optimal control problems for a satellite. Minimizing the energy used when changing the satellite’s orbit and when moving the satellite forward in its orbit. These optimal control problems are solved using PMP and discretizing the TPBVP to construct a system of equations which are solved in MATLAB using fsolve. These solutions are studied and improved to find an optimal control and state trajectory which transfers the state to the desired terminal state. A key focus in this project is constructing a system that when solved will give an optimal solution which transfers the state to the desired terminal state without directly using a constraint on the terminal state.