  Reinforcement learning (RL) is promising for complicated stochastic nonlinear
control problems. Without using a mathematical model, an optimal controller can
be learned from data evaluated by certain performance criteria through
trial-and-error. However, the data-based learning approach is notorious for not
guaranteeing stability, which is the most fundamental property for any control
system. In this paper, the classic Lyapunov's method is explored to analyze the
uniformly ultimate boundedness stability (UUB) solely based on data without
using a mathematical model. It is further shown how RL with UUB guarantee can
be applied to control dynamic systems with safety constraints. Based on the
theoretical results, both off-policy and on-policy learning algorithms are
proposed respectively. As a result, optimal controllers can be learned to
guarantee UUB of the closed-loop system both at convergence and during
learning. The proposed algorithms are evaluated on a series of robotic
continuous control tasks with safety constraints. In comparison with the
existing RL algorithms, the proposed method can achieve superior performance in
terms of maintaining safety. As a qualitative evaluation of stability, our
method shows impressive resilience even in the presence of external
disturbances.
