Null controllability of a coupled nonlinear parabolic system

Jéssica Manghi; Juan Límaco; Mauro A. Rincon; Jéssica Manghi; Juan Límaco; Mauro A. Rincon

doi:10.3934/cam.2025040

Communications in Analysis and Mechanics

2025, Volume 17, Issue 4: 971-984. doi: 10.3934/cam.2025040

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Null controllability of a coupled nonlinear parabolic system

1.
Institute of Computing, Federal University of Rio de Janeiro, Rio de Janeiro, Rj, Brazil
2.
Institute of Mathematics, Federal University of Fluminense, UFF, Rio de Janeiro, RJ, Brazil

Received: 29 March 2025 Revised: 30 August 2025 Accepted: 29 September 2025 Published: 12 December 2025
35k59; 93B05; 35D35

This paper deals with the null controllability of a coupled nonlinear parabolic system. The coefficients of the system operators depend on the states, and the control acts through the first equation. To solve the control problem of the linearized system, we use maximum regularity results in the spaces $ L^{q}(0, T, L^{p}(\Omega)) $, and we use Liusternick's inverse function theorem for the nonlinear controllability problem. In addition, as an appendix, we prove the well-posedness of the system.
- Quasi-linear parabolic problem,
- controllability,
- strong solution to PDE's
Citation: Jéssica Manghi, Juan Límaco, Mauro A. Rincon. Null controllability of a coupled nonlinear parabolic system[J]. Communications in Analysis and Mechanics, 2025, 17(4): 971-984. doi: 10.3934/cam.2025040

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Abstract

This paper deals with the null controllability of a coupled nonlinear parabolic system. The coefficients of the system operators depend on the states, and the control acts through the first equation. To solve the control problem of the linearized system, we use maximum regularity results in the spaces $ L^{q}(0, T, L^{p}(\Omega)) $, and we use Liusternick's inverse function theorem for the nonlinear controllability problem. In addition, as an appendix, we prove the well-posedness of the system.

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