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OPTIMAL FEEDBACK CONTROL FOR UNDAMPED WAVE EQUATIONS BY SOLVING A HJB EQUATION
, 2014
"... An optimal finitetime horizon feedback control problem for (semilinear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary HamiltonJacobiBellman (HJB) equation. Classical discretization methods based on finit ..."
Abstract

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An optimal finitetime horizon feedback control problem for (semilinear) wave equations is presented. The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary HamiltonJacobiBellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional spaces which makes the numerical resolution by the HJB approach infeasible. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach.
unknown title
"... Abstract. 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 dif ..."
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Abstract. 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 resulting from applying this approach within the NewtonADI method to solve the LQR algebraic Riccati equation is the key to compute a feedback matrix that in turn can be applied within a closedloop simulation. Numerical examples for various parameters influencing the different levels of the nested iteration are given. Finally, the stabilizing property of the computed feedback matrix is demonstrated using the von Kármán vortex street within a finite element based flow solver.
Problem Setting
"... Derive and investigate numerical algorithms for optimal controlbased (normal and tangential) boundary feedback stabilization of multifield flow problems. Explore the potentials and limitations of feedbackbased (Riccati) stabilization techniques. Extend current methods for flow described by Navier ..."
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Derive and investigate numerical algorithms for optimal controlbased (normal and tangential) boundary feedback stabilization of multifield flow problems. Explore the potentials and limitations of feedbackbased (Riccati) stabilization techniques. Extend current methods for flow described by NavierStokes equations (NSE) to flow problems coupled with other field equations of increasing complexity. Major Challenge Numerical solution of algebraic Riccati equations associated to special
Previous Work
"... • Derive and investigate numerical algorithms for optimal controlbased boundary feedback stabilization of multifield flow problems. • Explore the potentials and limitations of feedbackbased (Riccati) stabilization techniques. • Employ recent advances in reducing complexity of Riccati solvers to ach ..."
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• Derive and investigate numerical algorithms for optimal controlbased boundary feedback stabilization of multifield flow problems. • Explore the potentials and limitations of feedbackbased (Riccati) stabilization techniques. • Employ recent advances in reducing complexity of Riccati solvers to achieve stabilization with cost proportional to the simulation of the forward problem.
A NonConforming Composite
"... Quadrilateral Finite Element Pair for Feedback Stabilization of the Stokes Equations MAX−PLANCK−INSTITUT ..."
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Quadrilateral Finite Element Pair for Feedback Stabilization of the Stokes Equations MAX−PLANCK−INSTITUT
EFFICIENT SOLUTION OF LARGESCALE SADDLE POINT SYSTEMS ARISING IN RICCATIBASED BOUNDARY FEEDBACK STABILIZATION OF INCOMPRESSIBLE STOKES FLOW
, 2012
"... Abstract. We investigate numerical methods for solving largescale saddle point systems which arise during the feedback control of flow problems. We focus on the Stokes equations that describe instationary, incompressible flows for low and moderate Reynolds numbers. After a mixed finite element disc ..."
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Abstract. We investigate numerical methods for solving largescale saddle point systems which arise during the feedback control of flow problems. We focus on the Stokes equations that describe instationary, incompressible flows for low and moderate Reynolds numbers. After a mixed finite element discretization [23] we get a differentialalgebraic system of differential index two [45]. To reduce this index, we follow the analytic ideas of Raymond [34, 35, 36] coupled with the projection idea of Heinkenschloss et al. [22]. Avoiding this explicit projection leads to solving a series of largescale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners based on the approaches of Wathen et al. [19, 42]. In addition, the main results can be extended to the nonsymmetric case of linearized NavierStokes equations. We conclude with numerical examples showcasing the performance of our preconditioned iterative saddle point solver.