Data assimilation and local-in-time global control of inviscid systems using partially resolved measurements

Abstract

Most data assimilation algorithms have focused on dissipative systems, as the method relies on the existence of a global attractor. In this paper, we extend the framework of continuous data assimilation to inviscid models. As a first step towards this goal, we consider two inviscid systems: the passive scalar transport equation and the Euler equation. The data assimilation algorithm we employ utilizes nudging, a method based on a Newtonian relaxation scheme motivated by feedback control. We consider the two systems in an analytic space with a time-dependent analytic radius. This allows us to extract an artificial dissipative term, which is necessary for the application of data assimilation techniques. We establish exponential decay of the error between the data assimilated solution and the reference solution on a finite time interval $[0,T_{\ast })$, and estimate the error at the end of the algorithm (as $t\to T_{\ast }$).