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A New Method for the Exact Controllability of Linear Parabolic Equations

Author

Listed:
  • Inmaculada Gayte Delgado

    (Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, 41004 Sevilla, Spain
    These authors contributed equally to this work.)

  • Irene Marín-Gayte

    (Departamento de Métodos Cuantitativos, Universidad Loyola Andalucía, 41704 Dos Hermanas, Spain
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Inmaculada Gayte Delgado & Irene Marín-Gayte, 2025. "A New Method for the Exact Controllability of Linear Parabolic Equations," Mathematics, MDPI, vol. 13(3), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:3:p:344-:d:1573291
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