Ivica Nakić: Control cost of the heat equation in large domains

Abstract:

I will present new bounds on the control cost for the abstract heat equation, obtained from a spectral inequality or uncertainty relation for spectral projectors. These bounds will be applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. Here I will assume that the observability/control set obeys an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that the obtained control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.

Joint work with Matthias Täufer, Martin Tautenhahn and Ivan Veselić.