Séminaire: Uniform controllability for metastable systems

In this talk, we consider the issue of the null-controllability for some problems presenting a metastable behavior. We will in particular consider the one-dimensional Burgers equation linearized at a stationary shock, and the Allen–Cahn equation linearized at a n-node solution. We give an upper and a lowerbound on the control time required for the controllability cost to remain bounded as the viscosity term tends to 0, aswell as a rough description of an admissible control. The proof relies on complex analysis and a precise spectral analysis of the operators at stake, and adapts methods previously used to tackle similar issues with very regular terms.