Dynamic target tracking requires a mobile robot to maintain target visibility while navigating obstacles with smooth, feasible controls. Online trajectory optimization explicitly encodes these objectives but requires repeated numerical solving, whereas waypoint-based neural policies separate geometric prediction from control execution. We introduce LECO, a framework that transfers tracking optimization into offline learning of a direct-control policy. LECO predicts control sequences and maps them to trajectories through a differentiable kinematic model, allowing trajectory-level objectives to directly guide control learning. A decaying expert-guidance curriculum initializes effective tracking behavior and progressively shifts learning toward these task objectives. At deployment, a predictive safety lock checks short-horizon obstacle clearance and invokes a classical planning and control fallback when the predicted motion is unsafe or the target prediction is invalid. Across three simulated obstacle densities, LECO reduces root-mean-square linear jerk by 14.1--31.2\% relative to the best competing method for this metric and achieves the highest tracking success rate, with per-cycle computation times of 4.25--5.63\,ms. Experiments on two robot platforms further demonstrate tracking through indoor narrow passages and outdoor forest environments. These results demonstrate that learning direct control from optimization enables smooth and robust dynamic target tracking.

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