Abstract not found

The optimal boundary control of Navier--Stokes flow is formulated as a constrained optimization problem and a sequential quadratic programming (SQP) approach is studied for its solution. Since SQP methods treat states and controls as independent variables and do not insist on satisfying the constraints during the iterations, care must be taken to avoid a possible incompatibility of Dirichlet boundary conditions and incompressibility constraint. In this paper, compatibility is enforced by choosing appropriate function spaces. The resulting optimization problem is analyzed. Differentiability of the constraints and surjectivity of linearized constraints are verified and adjoints are computed. An SQP method is applied to the optimization problem and compared with other approaches.

this article we use global control (the entire body is subjected to prescribed motion)compared to the approach of local control (e.g. blowing=suction as reported by Li et al. [41])

this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows. I. Introduction

this paper, these two approaches produce the same control input #. 5. Results As was mentioned in 4, the pressures on the cylinder surface are measured at # 6 # 6 # and the blowing and suction are applied at # 6 # 6 #.In 5.1, sensings and actuations are carried out all over the cylinder surface, i.e. (#, #)=(#,#)=(0, 2#). Local sensings and actuations are performed in 5.2. Finally, open-loop controls are investigated in 5.3. For all cases investigated in this study, we have used the computational time step #t = 0.015, and the control time interval #t c = 0.06. That is, the sensing and actuation are updated at every four computational time steps. We have also 136 C. Min and H. Choi 1.5 1.4 1.3 1.2 1.1 1.0 0 30 60 90 120 t J 3 (c) 1.4 0.8 1.2 1.0 0.4 0 30 60 90 120 J 2 (b) 1.1 1.0 0.9 0.8 0.7 0 30 60 90 120 J 1 (a) 0.6 Figure 4. Time histories of the cost functional with (#, #)=(#,#)=(0, 2#): , # max =0.1; --------, 0.2; ---, 0.3; --------, 0.4. (a) J 1 ;(b) J 2 ;(c) J 3 . investigated a few di#erent combinations of #t and #t c , but the results showed only a slight change compared to those obtained from #t = 0.015 and #t c = 0.06. The Reynolds numbers investigated in this study (two-dimensional computations) are 100 and 160; according to the recent result by Henderson (1997), the two-dimensional wake becomes absolutely unstable to long-wavelength spanwise perturbations and bifurcates to a three-dimensional flow at Re # 190 (mode A; see also Williamson 1988). All controls begin at t = 30 and the maximum blowing/suction value relative to the free-stream velocity, # max = max 06#<2# |#(#)|, is kept constant during the control. 5.1. Sensing and actuation all over the cylinder surface We have applied the actuation values of decreasing J 1 and J 2...

. This paper studies constrained LQR problems in distributed boundary control systems governed by the Stokes equation with point velocity observations. Although the objective function is not well-defined, we are able to use hydrostatic potential theory and a variational inequality in a Banach space setting to derive a first order optimality condition and then a characterization formula of the optimal control. Since matrix-valued singularities appear in the optimal control, a singularity decomposition formula is also established, with which the nature of the singularities is clearly exhibited. It is found that in general, the optimal control is not defined at observation points. A necessary and sufficient condition that the optimal control is defined at observation points is then proved. Key words. LQR, Stokes fluid, distributed boundary control, point observation, hydrostatic potential, BIE, singularity decomposition. AMS subject classifications. 49N10,49J20,76D07,76D10,93C20,65N38 ...

Key words and phrases: Least-squares principles, Navier-Stokes equations, boundary control, finite elements. 1.

In this paper, we first study an optimal boundary control problem governed by the Stokes system with point pressure observations on the boundary. A constrained LQR approach with some hydrostatic potential theory, boundary integral equations and a variational inequality in a Banach space setting is applied to establish a state feedback characterization of the optimal control. Since a hyper-singularity is involved in regularity/singularity analysis and the kernel is only p.v.-integrable, many Lebesgue integral related tools cannot be applied directly. We develop a method to handle such a hyper-singularity in establishing some new regularity results. Robust optimal control problems, i.e., to minimize the velocity at observation points while bringing down the pressure there, are then solved in the last section. Keywords. LQR, Stokes fluid, distributed boundary control, point pressure observation, hydrostatic potential, BIE, hyper-singularity AMS(MOS) subject classifications. 49N10,49J20,7...

Abstract. In this paper, we investigate an efficient numerical method to identify an optimal impedance factor for mitigating radiated engine noise. The engine tone-noise wavenumber is treated as a random variable. We prove the existence of the sensitivity derivative of the state variable (which is the acoustic pressure) with respect to the random wavenumber. The proposed numerical method is based on the stratified Monte Carlo algorithm whose convergence is accelerated by exploiting the sensitivity derivative information.

ABSTRACT. This paper introduces an explicit numerical criterion for the stabilization of steady state solutions of the Navier-Stokes equations (NSE) with linear feedback control. Given a linear feedback controller that stabilizes a steady state solution to the closed-loop standard Galerkin (or nonlinear Galerkin) NSE discretization, it is shown that, if the number of modes involved in the computation is large enough, this controller stabilizes a nearby steady state of the closed-loop NSE. We provide an explicit estimate, in terms of the physical parameters, for the number of modes required in order for this criterion to hold. Moreover, we provide an estimate for the distance between the stabilized numerical steady state and the actually stabilized steady state of the closed-loop Navier-Stokes equations. More accurate approximation procedures, based on the concept of postprocessing the Galerkin results, are also presented. All the criterion conditions are imposed on the computed numerical solution, and no a priori knowledge is required about the steady state solution of the full PDE. This criterion holds for a large class of unbounded linear feedback operators and can be slightly modified to include certain nonlinear parabolic systems such as reaction-diffusion systems.