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Reinforcement learning is now the mainstream paradigm for training LLMs on agentic tasks, yet standard practice keeps only the best-validation checkpoint and discards the rest. We show that this is lossy: in GRPO training of LLM agents, the best-validation checkpoint fails on problems that earlier checkpoints from the same run had already solved, and the union of problems solved across checkpoints far exceeds any single one. The discarded checkpoints thus form a teacher pool obtained at no additional training cost, which on-policy distillation (OPD) can mine to recover the lost coverage. Mining this pool raises two challenges: i) which checkpoint to distill from, as no checkpoint dominates the others and complementarity to the student is problem-specific; ii) where to trust it, as even a well-chosen teacher is only locally superior, so indiscriminate distillation overwrites behavior the student has already mastered. We propose Agentic-OPD, which treats the best-validation checkpoint as the student and all saved checkpoints as frozen teacher candidates, using the student's evolving capability gaps to guide both teacher selection and supervision filtering: the HCN policy (Highest Coverage Next) dynamically selects teachers by weighted coverage of the student's unsolved problems, while the LAD module (Locality-Aware Distillation) confines transfer to failed student rollouts on problems where the teacher performs better. Across five agentic task domains and 24 benchmarks and task subsets, Agentic-OPD improves over its initialization checkpoint by up to 7.4 points, with a 4B student surpassing the latest frontier models GPT-5.6 and DeepSeek-V4.1 on deep search. Agentic-OPD recasts a finished RL run as a reusable pool of policies and establishes post-RL capability recovery as a stage orthogonal to improvements in the RL algorithm itself.

The Control Barrier Function (CBF) method has been widely adopted in solving safety-critical control problems. However, the existence of a CBF is only a sufficient condition for system safety. The recently proposed Taylor-Lagrange Control (TLC) method addresses this limitation by providing a necessary and sufficient condition, but it is still vulnerable to the feasibility preservation problem (e.g., due to the inter-sampling effect when time is discretized). In this paper, we propose a robust TLC (rTLC) method to address the feasibility preservation problem and show how to use it to solve optimal control problems. Specifically, the rTLC method expands the safety function to an order higher than its relative degree using Taylor's expansion with Lagrange remainder, which allows the control to explicitly show up at the current time instead of a future time in the original TLC method. This implies that the safety constraint remains satisfied over the entire discretization time interval that follows. Moreover, the rTLC method naturally addresses the feasibility preservation problem with only a single hyper-parameter (the discretization time interval size during implementation), compared to significantly more parameters for its counterparts. Finally, we illustrate the effectiveness of the proposed rTLC method through an adaptive cruise control problem, and compare it with existing safety-critical control methods.