L. S. Hou

Boundary velocity control of incompressible flow with an application to viscous drag reduction

An optimal boundary control problem for the Navier–Stokes equations is presented. The control is the velocity on the boundary, which is constrained to lie in a closed, convex subset of

Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls

H^{{1 / 2}}

Boundary Value Problems and Optimal Boundary Control for the Navier--Stokes System: the Two-Dimensional Case

of the boundary. A necessary condition for optimality is derived. Computations are done when the control set is actually finite-dimensional, resulting in an application to viscous drag reduction.

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

We examine certain analytic and numerical aspects of optimal control problems for the stationary Navier-Stokes equations. The controls considered may be of either the distributed or Neumann type; the functionals minimized are either the viscous dissipation or the L4_distance of candidate flows to some desired flow. We show the existence of optimal solutions and justify the use of Lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced. We study the regularity of solutions of this system. Then, we consider the approximation, by finite element methods, of solutions of the optimality system and derive optimal error estimates.

Numerical Approximation of Optimal Flow Control Problems by a Penalty Method: Error Estimates and Numerical Results

We study optimal boundary control problems for the two-dimensional Navier--Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier--Stokes equations; namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier--Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.

Optimal Boundary Control for the Evolutionary Navier--Stokes System: The Three-Dimensional Case

We introduce a penalized Neumann boundary control approach for solving an optimal Dirichlet boundary control problem associated with the two- or three-dimensional steady-state Navier--Stokes equations. We prove the convergence of the solutions of the penalized Neumann control problem, the suboptimality of the limit, and the optimality of the limit under further restrictions on the data. We describe the numerical algorithm for solving the penalized Neumann control problem and report some numerical results.

Semidiscrete Finite Element Approximations of a Linear Fluid-Structure Interaction Problem

The purpose of this paper is to present numerically convenient approaches to solve optimal Dirichlet control problems governed by the steady Navier--Stokes equations. We will examine a penalized Neumann control approach for solving Dirichlet control problems from numerical and computational points of view. The control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction. The control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one. We develop sequential quadratic programming methods to solve these optimal control problems. The effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized Neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous, incompressible fluid flow in a two-dimensional channel and in a cavity geometry.

Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation

Optimal boundary control problems for the three-dimensional, evolutionary Navier--Stokes equations in the exterior of a bounded domain are studied. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with almost minimal possible regularity. The control objective is to minimize the drag functional. The existence of an optimal solution is proved. A strong form of an optimality system of equations is derived on the basis of regularity results established in this work for the adjoint Oseen equations with regular initial data which do not satisfy the compatibility conditions.

Error Estimates for Semidiscrete Finite Element Approximations of the Stokes Equations Under Minimal Regularity Assumptions

Semidiscrete finite element approximations of a linear fluid-structure interaction problem are studied. First, results concerning a divergence-free weak formulation of the interaction problem are reviewed. Next, semidiscrete finite element approximations are defined, and the existence of finite element solutions is proved with the help of an auxiliary, discretely divergence-free formulation. A discrete inf-sup condition is verified, and the existence of a finite element pressure is established. Strong a priori estimates for the finite element solutions are also derived. Then, by passing to the limit in the finite element approximations, the existence of a strong solution is demonstrated and semidiscrete error estimates are obtained.

A numerical method for exact boundary controllability problems for the wave equation

Abstract We consider finite element approximations of an optimal control problem associated with a scalar version of the Ginzburg-Landau equations of superconductivity. The control is the Neumann data on the boundary and the optimization goal is to obtain a best approximation, in the least squares sense, to some desired state. The existence of optimal solutions is proved. The use of Lagrange multipliers is justified and an optimality system of equations is derived. Then, the regularity of solutions of the optimality system is studied, and finally, finite element algorithms are defined and optimal error estimates are obtained.