Online learning-based control is a promising approach to control uncertain systems, where unknown components are identified during operation to improve control performance. However, resource-intensive online learning algorithms introduce non-negligible computational delays, especially when executed on systems with limited local computational resources. To mitigate this, an in-network online learning-based control structure is employed by deploying the learning-based controller on a remote computation node and connecting it via a communication channel. In this paper, control performance guarantee is first established by deriving tracking error bound for the in-network control architecture, while accounting for computational delays. The derived tracking error bound allows for diverse communication and computation strategies under a specific condition, including time-/event-triggered mechanisms. Additionally, the trade-off between communication and computation performances is shown for a given desired control performance. Furthermore, to enhance the efficiency in both communication and computation, an efficient control framework with an asynchronous event-triggered mechanism in both control and online learning is devised under the existence of computational delay. The proposed event-triggered strategy is proven to achieve the same control performance as time-triggered scenario while excluding Zeno behavior. Finally, we derive an explicit expression of the proposed event-trigger condition for exponentially stabilizable systems, and demonstrate its effectiveness through simulations.

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