Analysis of the Velocity Tracking Control Problem for the 3D Evolutionary Navier--Stokes Equations

Eduardo Casas,Konstantinos Chrysafinos

doi:10.1137/140978107

Abstract

The velocity tracking problem for the evolutionary Navier--Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the control problem is possible. First and second order necessary and sufficient optimality conditions are proved. A fully discrete scheme based on a discontinuous (in time) Galerkin approach, combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, $\tau$ and $h,$ respectively, satisfy $\tau \leq Ch^2$, the $L^2(\Omega_T)$ error estimates of order $O(h)$ are proved for the difference between the locally optimal controls and their discrete approximations. Finally, combining these techniques and the approach of Casas, Herzog, and Wachsmuth [SIAM J. Optim., 22 (2012), pp. 795--820], we extend our results to the case of $L^1(\Omega_T)$ type functionals that allow sparse c...

Highlights

IntroductionWe study the following velocity tracking control problem associated to the evolutionary Navier–Stokes equations for three-dimensional (3D) flows:
In this paper, we study the following velocity tracking control problem associated to the evolutionary Navier–Stokes equations for three-dimensional (3D) flows: (1.1)yt − νΔy + (y · ∇)y + ∇p = f + u in ΩT = (0, T ) × Ω, div y = 0 in ΩT, y(0) = y0 in Ω, y = 0 on ΣT = (0, T ) × Γ.In these equations, y = (y1, y2, y3) is the velocity field of the fluid, p is the pressure, ν > 0 is the viscosity, f and u represent the body forces, and y0 denotes the initial velocity
Optim., 22 (2012), pp. 795–820], we extend our results to the case of L1(ΩT ) type functionals that allow sparse controls

Summary

Introduction

We study the following velocity tracking control problem associated to the evolutionary Navier–Stokes equations for three-dimensional (3D) flows:. Yt − νΔy + (y · ∇)y + ∇p = f + u in ΩT = (0, T ) × Ω, div y = 0 in ΩT , y(0) = y0 in Ω, y = 0 on ΣT = (0, T ) × Γ In these equations, y = (y1, y2, y3) is the velocity field of the fluid, p is the pressure, ν > 0 is the viscosity, f and u represent the body forces, and y0 denotes the initial velocity. For two-dimensional (2D) flows, Ω ⊂ R2, an existence and uniqueness theorem for a solution of (1.1) has been known for a long time. The study is more complicated for the 3D flows, Ω ⊂ R3 In this case, two different types of solutions are distinguished: weak and strong. In the 2D case, weak and strong solutions coincide, and we have existence and uniqueness of a solution

EDUARDO CASAS AND KONSTANTINOS CHRYSAFINOS

Let us consider the space

We also consider a subspace Yh of Zh defined by

Nτ tn