Instantaneous sentinel for the identification of the pollution term in Navier-Stokes system

Abstract

The aim of this paper is about presenting some results of the Sentinel Theory in Connection with Control Theory of Distributed Systems. There is of course a large variety of models where the results to follow could be applied. What we have particularly in mind is the classical set of Navier-Stokes equations. We shall denote here by the velocity field and the pressure, which is a quite unusual notation in the "Turbulence" circle. The reason is simply that in all what is following that we think of as the state of our system, this state depends on control functions, these control functions are being either "artificial" or "natural". Also in this work we are(trying) working to get instantaneous information at fixed instant "T" on pollution term in Navier-Stokes system in which the initial condition is incomplete. The best method which can solve this problem is the sentinel one; it allows the estimation of the pollution term at which we look for information independently of the missing term that we do not want to identify. So, we prove the existence of such instantaneous sentinel by solving a problem of controllability with a constraint on the control.