Motion planning is central to robotics, and it is increasingly used to correct the motions proposed by learned planners and policies. In both uses, constraints must hold at every instant of the motion, not only at sampled times. To our knowledge, no algorithm provably converges to a globally optimal motion once a smoothness cost is combined with such constraints. We cast planning and steering as a single continuous-time program: robots avoid one another and static or moving obstacles, and a learned proposal is steered by keeping the motion close to it. The program is elementary to state, yet nonconvex. We propose GlobMP (Global Motion Planning), a gradient projection method over piecewise-polynomial curves, on which each constraint is one polynomial per piece that must stay nonnegative over the whole motion; it alternates cost reduction with feasibility restoration. Under mild assumptions, we show that, from a fine grid start, its cost approaches the global optimum as the number of polynomial pieces grows, with every constraint met to any chosen tolerance, or exactly on stable instances. We demonstrate its superior performance over strong baselines in real and synthetic experiments, including teams of up to $200$ robots, multi-robot planning on the Robotarium testbed and an extended benchmark, maze navigation, and steering tactile-reactive policies on a bimanual humanoid.

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