Gayte Delgado, Inmaculada , MARÍN GAYTE, IRENE
No
Mathematics
Article
Científica
01/02/2025
001418580100001
2-s2.0-85217651239
This work solves the exact controllability to zero in the final time for a linear parabolic problem when the control only acts in a part of the spatial domain. Specifically, it is proved, by compactness arguments, the existence of a partially distributed control. The lack of regularity in the problem prevents the use of standard techniques in this field, that is, Carleman's inequalities. Controlling a parabolic equation when the diffusion is discontinuous and only acts in a part of the domain is interesting, for example, as in the spreading of a brain tumor. The proof is based on a new maximum principle in the final time; in a linear parabolic equation, with a right-hand side that changes sign in a certain way, and an initial datum of a constant sign, the solution at the final time has the same sign as the initial datum. As a consequence of the exact control result, we prove a unique continuation theorem when the data are not regular.
exact controllability; partially distributed control; maximum strong principle; unique continuation