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Controllability under positive constraints for quasilinear parabolic PDEs

  • * Corresponding author: Miguel R. Nuñez-Chávez

    * Corresponding author: Miguel R. Nuñez-Chávez
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  • This paper deals with the analysis of the internal controllability with constraint of positive kind of a quasilinear parabolic PDE. We prove two results about this PDE: First, we prove a global steady state constrained controllability result. For this purpose, we employ the called "stair-case method". And second, we prove a global trajectory constrained controllability result. For this purpose, we employ the well-known "stabilization property" in $ L^2 $ norms. Furthermore, for both results an important argument is needed: the exact local controllability to trajectories. Then we prove the positivity of the minimal controllability time using arguments of comparison principle. Some additional comments and open problems concerning other systems are presented.

    Mathematics Subject Classification: Primary: 35K59, 93C20; Secondary: 35K15.


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  • Figure 1.  The stair-case method. Trajectory of state-control in blue. Path of steady states in red

    Figure 2.  Trajectory of state-control in blue. Target trajectory in red

    Figure 3.  Initial datum $ \varphi^T $ for the adjoint system

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