Jornada de primavera en EDPs. Viernes, 9 de abril de 2021, 10h- 11:25h

**Jornada de primavera en EDPs. ****Viernes, 9 de abril de 2021, 10h- 11:25h
**

Enlace: https://conectaha.csic.es/b/mar-e3k-scr-8wi

Esteve C., Zuazua E.,The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes, SIAM J. Math. Anal., Vol. 52, No. 6, pp. 5627–5657 (2020). https://doi.org/10.1137/20M1330130

__Title:__ Cost of null controllability for parabolic equations with vanishing viscosity

__Abstract: __The transport-diffusion equation with vanishing diffusivity describes the dynamics

of physical and biological phenomena in which the transport dynamics dominates the

diffusive dynamics. Since these systems are of parabolic nature, it is well-known that

they are null controllable. However, there are many open questions on the asymptotic

behaviour of the cost of null contrallability when the diffusion parameter vanishes.

In this talk we study the transport-diffusion equation with Neumann, Robin and mixed

boundary conditions. The main results concern the behaviour of the cost of the null

controllability when the diffusivity vanishes and 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. 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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