We investigate the usefulness of deep learning when applied to both control theory and partial differential equations (PDEs). We will develop new network architectures and methodologies to approach the solving of high-dimensional problems. Specifically, we develop a network architecture called Lyapunov-Net for approximating Lyapunov functions in high-dimensions and a new methodology called Neural Control for finding solution operators for high-dimensional parabolic PDEs. The theoretical accuracy and numerical efficiency of these approaches will be investigated along with implementation details to use them in practice.