On the small-time bilinear control of a nonlinear heat equation: Global approximate controllability and exact controllability to trajectories

In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension d. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case d = 1, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator. Dans ce travail, nous analysons les propri & eacute;t & eacute;s d'accessibilit & eacute; en temps court d'une & eacute;quation parabolique non lin & eacute;aire, & agrave; l'aide d'un contr & ocirc;le bilin & eacute;aire, pos & eacute;e sur un tore de dimension arbitraire d. Sous une hypoth & egrave;se de saturation sur les op & eacute;rateurs de contr & ocirc;le, nous montrons la contr & ocirc;labilit & eacute; approch & eacute;e en temps court entre les & eacute;tats qui ont le m & ecirc;me signe. De plus, dans le cas unidimensionnel d = 1, nous combinons cette propri & eacute;t & eacute; avec un r & eacute;sultat de contr & ocirc;labilit & eacute; locale exacte, et prouvons la contr & ocirc;labilit & eacute; exacte en temps court de tout & eacute;tat positif vers l'& eacute;tat fondamental de l'op & eacute;rateur d'& eacute;volution. (c) 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.