A least-squares formulation for the approximation of controls for the Stokes system

Abstract

This work deals with the approximation of distributed null controls for the Stokes system. The goal is to compute an approximation of controls that drives the solution from a prescribed initial state at $t=0$ to zero at $t=T$. The existence of square-integrable controls has been obtained in Fursikov and Imanuvilov (Controllability of evolution equations, pp 1–163, 1996) via Carleman type estimates. We introduce and analyze a least-squares formulation of the controllability problem, and we show that it allows the construction of convergent sequences of functions toward null controls for the Stokes system. The approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time—in particular the null controllability condition at time $T$—and then finding among this class one element satisfying the Stokes system. This second step is done by minimizing a quadratic functional, among the admissible corrector functions of the Stokes system. Numerical experiments for the two-dimensional case are performed in the framework of finite element approximations and demonstrate the interest of the approach. The method described here does not make use of duality arguments and, therefore, avoids the introduction of numerical ill-posed problem, as is typical when parabolic-type equation is considered. This work extends (Munch and Pedregal in Eur J Appl Math, 2014) where the case of the heat equation is discussed.