  This paper presents a model predictive control (MPC) for dynamic systems
whose nonlinearity and uncertainty are modelled by deep neural networks (NNs),
under input and state constraints. Since the NN output contains a high-order
complex nonlinearity of the system state and control input, the MPC problem is
nonlinear and challenging to solve for real-time control. This paper proposes
two types of methods for solving the MPC problem: the mixed integer programming
(MIP) method which produces an exact solution to the nonlinear MPC, and linear
relaxation (LR) methods which generally give suboptimal solutions but are much
computationally cheaper. Extensive numerical simulation for an inverted
pendulum system modelled by ReLU NNs of various sizes is used to demonstrate
and compare performance of the MIP and LR methods.
