Local null controllability of a class of non-Newtonian incompressible viscous fluids

Abstract

We investigate the null controllability property of systems that mathematically describe the dynamics of some non-Newtonian incompressible viscous flows. The principal model we study was proposed by O. A. Ladyzhenskaya, although the techniques we develop here apply to other fluids having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum Principle, we utilize a bootstrapping argument to prove that sufficiently smooth controls to the forced linearized Stokes problem exist, as long as the initial data in turn has enough regularity. From there, we extend the result to the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to compute the states and a control, which we prove to converge in an appropriate sense. We finish the work with some numerical experiments.

References in corpus (3)

• Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain
• On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations
• Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions