Semidiscretization and error estimates for distributed control of the instationary Navier-Stokes equations

Summary.Optimal control problems governed by the two-dimensional instationary Navier–Stokes equations and their spatial discretizations with finite elements are investigated. A concept of semi–discrete solutions to the control problem is introduced which is utilized to prove existence and uniqueness of discrete controls in neighborhoods of regular continuous solutions. Furthermore, an optimal error estimate in terms of the spatial discretization parameter is given.

[1]  M. Hinze.  Optimal and instantaneous control of the instationary Navier-Stokes equations , 2002 .

[2]  K. Deimling.  Nonlinear functional analysis , 1985 .

[3]  Max D. Gunzburger,et al.  The Velocity Tracking Problem for Navier-Stokes Flows With Boundary Control , 2000, SIAM J. Control. Optim..

[4]  Michael Hinze,et al.  Error Estimates in Space and Time for Tracking-type Control of the Instationary Stokes System , 2003 .

[5]  L. S. Hou,et al.  Optimization Problems for the Navier-Stokes Equations with Regular Boundary Controls , 1993 .

[6]  L. Hou,et al.  Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls , 1991 .

[7]  Fredi Tröltzsch,et al.  Semidiscrete Ritz-Galerkin Approximation of Nonlinear Parabolic Boundary Control Problems , 1993 .

[8]  R. Temam,et al.  On some control problems in fluid mechanics , 1990 .

[9]  R. Temam.  Navier-Stokes Equations , 1977 .

[10]  L. S. Hou,et al.  Numerical Approximation of Optimal Flow Control Problems by a Penalty Method: Error Estimates and Numerical Results , 1999, SIAM J. Sci. Comput..

[11]  Max D. Gunzburger,et al.  Analysis and Approximation of the Velocity Tracking Problem for Navier-Stokes Flows with Distributed Control , 2000, SIAM J. Numer. Anal..

[12]  E. Wagner.  International Series of Numerical Mathematics , 1963 .

[13]  B. Craven.  Control and optimization , 2019, Mathematical Modelling of the Human Cardiovascular System.

[14]  Karl Kunisch,et al.  Second Order Methods for Optimal Control of Time-Dependent Fluid Flow , 2001, SIAM J. Control. Optim..

[15]  Fredi Tröltzsch,et al.  On Convergence of Semidiscrete Ritz-Galerkin Schemes Applied to the Boundary Control of Parabolic Equations with Non-Linear Boundary Condition , 1992 .