Stabilizing unstable flows by coarse mesh observables and actuators – a pavement to data assimilation
One of the main characteristics of infinite-dimensional dissipative evolution equations, such as the Navier-Stokes equations and reaction-diffusion systems, is that their long-time dynamics are determined by finitely many parameters—finite number of determining modes, nodes, volume elements and other determining interpolants. In this talk I will show how to explore this finite-dimensional feature, of the long-time behavior of infinite-dimensional dissipative systems, to design finite-dimensional feedback control for stabilizing their solutions. Moreover, based on this approach I will also present a data assimilation (downscaling) algorithm for weather and climate predictions employing discrete coarse mesh measurements. Notably, numerical implementation of this algorithm yields errors that are bounded uniformly in time; consequently it can be reliably used for long-time integration and statistics. Finally, computational demonstrations implementing this algorithm will exhibit that its performance remarkably exceeds what is suggested by the theory.