Analysis on existence of compact set in neural network control for nonlinear systems

Neural network method is an effective tool for approximating the unknown function in controller design for nonlinear systems. To guarantee the validity of the approximation, state variables in approximated unknown functions need to stay in a compact set. However, in most existing results, the existence of the compact set has not been correctly proven; therefore, the proof is not actually complete in these existing works. In this paper, we analyze the existence of compact sets for two typical nonlinear systems with novel neural network-based controllers and show the strict proof for the semi-global uniform ultimate boundedness of the closed-loop system.