  This paper introduces a novel data-driven approach to design a linear
quadratic regulator (LQR) using a reinforcement learning (RL) algorithm that
does not require a system model. The key contribution is to perform policy
iteration (PI) by designing the policy evaluator as a two-layer quadratic
neural network (QNN). This network is trained through convex optimization. To
the best of our knowledge, this is the first time that a QNN trained through
convex optimization is employed as the Q-function approximator (QFA). The main
advantage is that the QNN's input-output mapping has an analytical expression
as a quadratic form, which can then be used to obtain an analytical expression
for policy improvement. This is in stark contrast to the available techniques
in the literature that must train a second neural network to obtain policy
improvement. The article establishes the convergence of the learning algorithm
to the optimal control, provided the system is controllable and one starts from
a stabilitzing policy. A quadrotor example demonstrates the effectiveness of
the proposed approach.
