Riccati-based boundary feedback stabilization of incompressible Navier-Stokes flow

In this article a boundary feedback stabilization approach for incompressible Navier--Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790--828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038--1063]. The nested iteration resul...

[1]  Rolf Rannacher,et al.  Numerical methods for the Navier-Stokes equations , 1994 .

[2]  W. Schiesser.  The Numerical Method of Lines: Integration of Partial Differential Equations , 1991 .

[3]  Howard C. Elman,et al.  Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics , 2014 .

[4]  Peter Benner,et al.  An improved numerical method for balanced truncation for symmetric second-order systems , 2013 .

[5]  Peter Benner,et al.  Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method , 2012, Numerical Algorithms.

[6]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[7]  D. Kleinman.  On an iterative technique for Riccati equation computations , 1968 .

[8]  R. Glowinski,et al.  Numerical methods for the navier-stokes equations. Applications to the simulation of compressible and incompressible viscous flows , 1987 .

[9]  Eugene L. Wachspress,et al.  The ADI Model Problem , 2013 .

[10]  Jean-Marie Buchot,et al.  A stabilization algorithm of the Navier–Stokes equations based on algebraic Bernoulli equation , 2012, Numer. Linear Algebra Appl..

[11]  J. Oden,et al.  Finite Element Methods for Flow Problems , 2003 .

[12]  P. Benner,et al.  Solving large-scale control problems , 2004, IEEE Control Systems.

[13]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[14]  John A. Burns,et al.  Mesh Independence of Kleinman--Newton Iterations for Riccati Equations in Hilbert Space , 2008, SIAM J. Control. Optim..

[15]  Karl Kunisch,et al.  The linear regulator problem for parabolic systems , 1984 .

[16]  H. Weichelt.  Riccati-Based Feedback Stabilized Navier-Stokes Flow , 2013 .

[17]  Jean-Pierre Raymond,et al.  HINFINITY Feedback Boundary Stabilization of the Two-Dimensional Navier-Stokes Equations , 2011, SIAM J. Control. Optim..

[18]  Gene H. Golub,et al.  Matrix computations , 1983 .

[19]  Peter Benner,et al.  Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow , 2013, SIAM J. Sci. Comput..

[20]  P. Benner,et al.  Optimal Control-Based Feedback Stabilization of Multi-field Flow Problems , 2014 .

[21]  Jean-Pierre Raymond,et al.  Local Boundary Feedback Stabilization of the Navier-Stokes Equations , 2006 .

[22]  A. Spence,et al.  Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics , 1994, SIAM J. Matrix Anal. Appl..

[23]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[24]  P TaylorC.Hood,et al.  Navier-Stokes equations using mixed interpolation , 1974 .

[25]  Sabine Hein,et al.  MPC/LQG-Based Optimal Control of Nonlinear Parabolic PDEs , 2009 .

[26]  Jean-Pierre Raymond,et al.  Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions , 2007 .

[27]  V. Perrier,et al.  Submitted to: 2007 Industrial Simulation Conference SIMULATION-BASED OPTIMIZATION OF AGENT SCHEDULING IN MULTISKILL CALL CENTERS , 2007 .

[28]  Y. Saad,et al.  Numerical solution of large Lyapunov equations , 1989 .

[29]  E. Bänsch,et al.  Simulation of instationary, incompressible flows. , 1998 .

[30]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[31]  Peter Benner,et al.  Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems , 2008, Numer. Linear Algebra Appl..

[32]  William L. Briggs,et al.  1. Model Problems , 2000 .

[33]  C. Amrouche,et al.  Navier-Stokes Equations with Nonhomogeneous Boundary Conditions , 2009 .

[34]  Jean-Pierre Raymond,et al.  Feedback Boundary Stabilization of the Two-Dimensional Navier--Stokes Equations , 2006, SIAM J. Control. Optim..

[35]  R. Rannacher,et al.  Benchmark Computations of Laminar Flow Around a Cylinder , 1996 .

[36]  Jacob K. White,et al.  Low-Rank Solution of Lyapunov Equations , 2004, SIAM Rev..

[37]  J. Weickert.  Navier-Stokes equations as a differential-algebraic system , 1998 .

[38]  H. Banks,et al.  A numerical algorithm for optimal feedback gains in high dimensional linear quadratic regulator problems , 1991 .

[39]  Peter Benner,et al.  Stabilization of Incompressible Flow Problems by Riccati-based Feedback , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[40]  Eberhard Bänsch,et al.  Local mesh refinement in 2 and 3 dimensions , 1991, IMPACT Comput. Sci. Eng..

[41]  E B Ansch,et al.  Simulation of Instationary, Incompressible Flows , 1998 .

[42]  Danny C. Sorensen,et al.  Balanced Truncation Model Reduction for a Class of Descriptor Systems with Application to the Oseen Equations , 2008, SIAM J. Sci. Comput..

[43]  Enrique S. Quintana-Ortí,et al.  Efficient algorithms for generalized algebraic Bernoulli equations based on the matrix sign function , 2007, Numerical Algorithms.

[44]  R. Schneider,et al.  Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern , 2001 .

[45]  Jens Saak,et al.  Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction , 2009 .

[46]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[47]  K. Kunisch,et al.  Second order methods for boundary control of the instationary Navier‐Stokes system , 2004 .

[48]  S. Sieniutycz,et al.  Optimal Control: An Introduction , 2001 .

[49]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[50]  R. F. Boisvert,et al.  The Matrix Market Exchange Formats: Initial Design | NIST , 1996 .

[51]  P. Lancaster,et al.  The Algebraic Riccati Equation , 1995 .

[52]  R. Maccormack.  Numerical Methods for the Navier-Stokes Equations , 1985 .

[53]  Ekkehard W. Sachs,et al.  Inexact Kleinman-Newton Method for Riccati Equations , 2009, SIAM J. Matrix Anal. Appl..

[54]  Alain Bensoussan,et al.  Representation and Control of Infinite Dimensional Systems, 2nd Edition , 2007, Systems and control.

[55]  Jean-Pierre Raymond,et al.  Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations , 2007 .

[56]  Peter Benner,et al.  Efficient Solvers for Large-Scale Saddle Point Systems Arising in Feedback Stabilization of Multi-field Flow Problems , 2013, System Modelling and Optimization.

[57]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[58]  Serkan Gugercin,et al.  Model Reduction of Descriptor Systems by Interpolatory Projection Methods , 2013, SIAM J. Sci. Comput..