The authors describe the use of neural networks for solving optimal control problems for discrete-time linear systems with quadratic cost functions. The result is obtained by formulating the optimal control problem as a quadratic programming problem with inequality constraints and then applying a result by M. Kennedy and L. Chua (1988). The authors present numerical examples of the method, comparisions to standard Ricatti equation solutions, and extensions to Kalman filtering and other applications, including real-time, adaptive optimal control. A result that makes it possible to use a neural net to solve optimization problems is described. It is shown how to formulate the linear quadratic regulator problem as a nonlinear programming problem. It is then possible to directly apply Kennedy and Chua's result to find the optimal control solution using a neural net.