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Non-null-controllability of H^1–L^2 initial conditions in the underactuated parabolic-transport system
Regularity-gap conjecture. Not every initial condition in $H^1({\mathbb T})\times L^2({\mathbb T})$ can be steered to $0$ in time $T$ with $L^2$ controls.
Conjectureopennull-controllability parabolic-control transport-equationsarXiv posted 2 days ago

Non-null-controllability of H^1–L^2 initial conditions in the underactuated parabolic-transport system