  Stability certification and identifying a safe and stabilizing initial set
are two important concerns in ensuring operational safety, stability, and
robustness of dynamical systems. With the advent of machine-learning tools,
these issues need to be addressed for the systems with machine-learned
components in the feedback loop. To develop a general theory for stability and
stabilizability of a neural network (NN)-controlled nonlinear system subject to
bounded parametric variation, a Lyapunov-based stability certificate is
proposed and is further used to devise a maximal Lipschitz bound for the NN
controller, and also a corresponding maximal region-of-attraction (RoA) inside
a given safe operating domain. To compute such a robustly stabilizing NN
controller that also maximizes the system's long-run utility, a
stability-guaranteed training (SGT) algorithm is proposed. The effectiveness of
the proposed framework is validated through an illustrative example.
