A New Method for the Exact Controllability of Linear Parabolic Equations

Inmaculada Gayte Delgado and Irene Marín-Gayte ()
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Inmaculada Gayte Delgado: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, 41004 Sevilla, Spain
Irene Marín-Gayte: Departamento de Métodos Cuantitativos, Universidad Loyola Andalucía, 41704 Dos Hermanas, Spain

Mathematics, 2025, vol. 13, issue 3, 1-22

Abstract: 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.

Keywords: exact controllability; partially distributed control; maximum strong principle; unique continuation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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