CHAT

The aim of the CHAT network is to deepen and widen the cross-fertilization collaborations that recently appeared between researchers in 3 fields of mathematics: harmonic analysis, complex analysis and control theory of partial differential equations. The questions addressed are directly motivated by controllability issues. Recall that the aim is to bring a system governed by a PDE from any initial state to a fixed final state, in minimal time, with a reasonable cost, by acting on the parameters of that PDE. This can be done bu acting on a source term (inner controllability) or via the boundary conditions of the PDE (boundary control). The classical questions of the field (approximate controllability, null-controllability, exact controllability, observability, reachable space...), that are still open for many PDEs, are at the centre of the CHAT project. Collaborations between researchers of the 3 mathematical fields represented in CHAT allowed to make significant progress recently. For instance, the description of the reachable space for the heat equation (in 1D) was obtained through the collaboration of specialists of holomorphic function spaces and experts in control theory. The description of the control sets of this same equation (in any dimension) resulted from the recent quantitative version of Logvinenko-Sereda's Uncertainty Principle, a classical result in harmonic analysis. In the opposite direction, questions in control theory triggered new interest in classical questions of harmonic and complex analysis (spaces of holomorphic functions, their sets of uniqueness, interpolation or sampling, uncertainty principles...)