  In this article, two types of methods from different perspectives based on
spectral normalization are described for ensuring the stability of the system
controlled by a neural network. The first one is that the L2 gain of the
feedback system is bounded less than 1 to satisfy the stability condition
derived from the small-gain theorem. While explicitly including the stability
condition, the first method may provide an insufficient performance on the
neural network controller due to its strict stability condition. To overcome
this difficulty, the second one is proposed, which improves the performance
while ensuring the local stability with a larger region of attraction. In the
second method, the stability is ensured by solving linear matrix inequalities
after training the neural network controller. The spectral normalization
proposed in this article improves the feasibility of the a-posteriori stability
test by constructing tighter local sectors. The numerical experiments show that
the second method provides enough performance compared with the first one while
ensuring enough stability compared with the existing reinforcement learning
algorithms.
