Deep neural networks (DNNs), trained with gradient-based optimization and backpropagation, are currently the primary tool in modern artificial intelligence, machine learning, and data science. In many applications, DNNs are trained offline, through supervised learning or reinforcement learning, and deployed online for inference. However, training DNNs with standard backpropagation and gradient-based optimization gives no intrinsic performance guarantees or bounds on the DNN, which is essential for applications such as controls. Additionally, many offline-training and online-inference problems, such as sim2real transfer of reinforcement learning policies, experience domain shift from the training distribution to the real-world distribution. To address these stability and transfer learning issues, we propose using techniques from control theory to update DNN parameters online. We formulate the fully-connected feedforward DNN as a continuous-time dynamical system, and we propose novel last-layer update laws that guarantee desirable error convergence under various conditions on the time derivative of the DNN input vector. We further show that training the DNN under spectral normalization controls the upper bound of the error trajectories of the online DNN predictions, which is desirable when numerically differentiated quantities or noisy state measurements are input to the DNN. The proposed online DNN adaptation laws are validated in simulation to learn the dynamics of the Van der Pol system under domain shift, where parameters are varied in inference from the training dataset. The simulations demonstrate the effectiveness of using control-theoretic techniques to derive performance improvements and guarantees in DNN-based learning systems.