Stability certification and identification of the stabilizable operating region of a system are two important concerns to ensure its operational safety/security and robustness. With the advent of machine-learning tools, these issues are specially important for systems with machine-learned components in the feedback loop. Here we develop a theory for stability and stabilizability of a class of neural-network controlled nonlinear systems, where the equilibria can drift when parametric changes occur. A Lyapunov based convex stability certificate is developed and is further used to devise an estimate for a local Lipschitz upper bound for a neural-network (NN) controller and a corresponding operating domain on the state space, containing an initialization set from where the closed-loop (CL) local asymptotic stability of each system in the class is guaranteed under the same controller, while the system trajectories remain confined to the operating domain. For computing such a robust stabilizing NN controller, a stability guaranteed training (SGT) algorithm is also proposed. The effectiveness of the proposed framework is demonstrated using illustrative examples.
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