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Online reinforcement learning is concerned with training an agent on-the-fly via dynamic interaction with the environment. Here, due to the specifics of the application, it is not generally possible to perform long pre-training, as it is commonly done in off-line, model-free approaches, which are akin to dynamic programming. Such applications may be found more frequently in industry, rather than in pure digital fields, such as cloud services, video games, database management, etc., where reinforcement learning has been demonstrating success. Online reinforcement learning, in contrast, is more akin to classical control, which utilizes some model knowledge about the environment. Stability of the closed-loop (agent plus the environment) is a major challenge for such online approaches. In this paper, we tackle this problem by a special fusion of online reinforcement learning with elements of classical control, namely, based on the Lyapunov theory of stability. The idea is to start the agent at once, without pre-training, and learn approximately optimal policy under specially designed constraints, which guarantee stability. The resulting approach was tested in an extensive experimental study with a mobile robot. A nominal parking controller was used as a baseline. It was observed that the suggested agent could always successfully park the robot, while significantly improving the cost. While many approaches may be exploited for mobile robot control, we suggest that the experiments showed the promising potential of online reinforcement learning agents based on Lyapunov-like constraints. The presented methodology may be utilized in safety-critical, industrial applications where stability is necessary.