Recent theoretical developments in dynamical systems and machine learning have allowed researchers to re-evaluate how dynamical systems are modeled and controlled. In this thesis, Koopman operator theory is used to model dynamical systems and obtain optimal control solutions for nonlinear systems using sampled system data. The Koopman operator is obtained using data generated from a real physical system or from an analytical model which describes the physical system under nominal conditions. One of the critical advantages of the Koopman operator is that the response of the nonlinear system can be obtained from an equivalent infinite dimensional linear system. This is achieved by exploiting the topological structure associated with the spectrum of the Koopman operator and the Koopman eigenfunctions. The main contributions of this thesis are threefold. First, we provide a data-driven approach for system identification, and a model-based approach for obtaining an analytic change of coordinates associated with the principle Koopman eigenfunctions for systems with hyperbolic equilibrium points. A new derivation of the Hamilton-Jacobi equations associated with the infinite time horizon nonlinear optimal control problem is obtained using the Koopman generator. Then, a learning algorithm called Koopman Policy Iteration is used to obtain the solution to the infinite horizon nonlinear optimal fixed point regulation problem without state and input constraints. Finally, the finite time nonlinear optimal control problem with state and input constraints is solved using a receding horizon optimization approach called dual mode model predictive control using Koopman eigenfunctions. Evidence supporting the convergence of these methods are provided using analytical examples
