Shirshendu Chowdhury (IISER Kolkata, India) Slides
In the first part of the talk, we introduce the concept: Controllability of Differential Equations. Then we give some examples in finite (ODE) and infinite dimensional(PDE) contexts. We recall the controllability results of the Transport and Heat equation.
In the second part of the talk, we consider compressible Navier-Stokes equations in one dimension, linearized around a positive constant steady state . It is a Coupled system of Transport (for density) and Heat type (for velocity) equations. We study the boundary null-controllability of this linearized system in an interval when a Dirichlet control function is acting either only on the density or only on the velocity component at one end of the interval. In this setup, we state some new control results which we have obtained. We see that these controllability results are optimal/sharp concerning the regularity of initial states (in the velocity case) and time (in the density case). The proof is based on a spectral analysis and on solving a mixed parabolic-hyperbolic moments problem and a parabolic hyperbolic joint Ingham-type inequality. This is a joint work with Kuntal Bhandari, Rajib Dutta and Jiten Kumbhakar. Finally, the talk ends with some ongoing and future directions of research.