A global Carleman inequality and exact controllability of parabolic equations with mixed boundary conditions

2008 ◽  
Vol 24 (2) ◽  
pp. 265-280
Author(s):  
Tujin Kim ◽  
Qian-shun Chang ◽  
Jing Xu
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Mixed Boundary ◽  
Exact Controllability ◽  
Mixed Boundary Conditions ◽  
Carleman Inequality

scholarly journals Cost of null controllability for parabolic equations with vanishing diffusivity and a transport term

2021 ◽  
Author(s):  
Jon Asier Bárcena-Petisco
Keyword(s):  
Boundary Conditions ◽  
Parabolic Equations ◽  
Space Variable ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Mixed Boundary Conditions ◽  
Null Controllability ◽  
Dirichlet Boundary Conditions ◽  
The Cost ◽  
Transport Term

In this paper we consider the heat equation with Neumann, Robin and mixed boundary conditions (with coefficients on the boundary which depend on the space variable). The main results concern the behaviour of the cost of the null controllability with respect to the diffusivity when the control acts in the interior. First, we prove that if we almost have Dirichlet boundary conditions in the part of the boundary in which the flux of the transport enters, the cost of the controllability decays for a time $T$ sufficiently large. Next, we show some examples of Neumann and mixed boundary conditions in which for any time $T>0$ the cost explodes exponentially as the diffusivity vanishes. Finally, we study the cost of the problem with Neumann boundary conditions when the control is localized in the whole domain.

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