stabilization of a plane periodic channel flow by noise wall normal controllers the steady state flows in a 2 d channel with periodic conditions along the axis are stabilized by a boundary feedback stochastic controller with vertical velocity observation which acts on the normal component of velocity only

a closed form feedback controller for stabilization of the linearized 2 d navier stokes poiseuille system we present a formula for a boundary control law which stabilizes the parabolic profile of an infinite channel flow which is linearly unstable for high reynolds numbers also known as the poiseuille flow this problem is frequently cited as a paradigm for transition to turbulence whose stabilization for arbitrary reynolds numbers without using discretization has so far been an open problem our result achieves exponential stability in the l2 h1 and h2 norms for the linearized navier stokes equations explicit solutions are obtained for the closed loop system this is the first time explicit formulae are produced for solutions of the linearized navier stokes equations in a channel flow with feedback in the boundary conditions used to make this possible the result is presented for the 2d case for clarity of exposition an extension to 3d is available and will be presented in a future publication

a general framework for robust control in fluid mechanics abstract the application of optimal control theory to complex problems in fluid mechanics has proven to be quite effective when complete state information from high resolution numerical simulations is available p moin t r bewley appl mech rev part 2 47 6 1994 s3 s13 t r bewley p moin r temam j fluid mech 1999 submitted for publication in this approach an iterative optimization algorithm based on the repeated computation of an adjoint field is used to optimize the controls for finite horizon nonlinear flow problems f abergel r temam theoret comput fluid dyn 1 1990 303 325 in order to extend this infinite dimensional optimization approach to control externally disturbed flows in which the controls must be determined based on limited noisy flow measurements alone it is necessary that the controls computed be insensitive to both state disturbances and measurement noise for this reason robust control theory a generalization of optimal control theory has been examined as a technique by which effective control algorithms which are insensitive to a broad class of external disturbances may be developed for a wide variety of infinite dimensional linear and nonlinear problems in fluid mechanics an aim of the present paper is to put such algorithms into a rigorous mathematical framework for it cannot be assumed at the outset that a solution to the infinite dimensional robust control problem even exists in this paper conditions on the initial data the parameters in the cost functional and the regularity of the problem are established such that existence and uniqueness of the solution to the robust control problem can be proven both linear and nonlinear problems are treated and the 2d and 3d nonlinear cases are treated separately in order to get the best possible estimates several generalizations are discussed and an appropriate numerical method is proposed

on the approximate and null controllability of the navier stokes equations this paper presents some known results on the approximate and null controllability of the navier stokes equations all of them can be viewed as partial answers to a conjecture of j l lions

existence and multiplicity of normalized solutions for the nonlinear kirchhoff type problems abstract in this paper we prove the existence and multiplicity results of solutions with prescribed l 2 norm for a class of kirchhoff type problems a b r 3 u 2 d x u u f u in r 3 where a b 0 are constants r and f c r r to obtain such solutions we look into critical points of the energy functional e b u a 2 r 3 u 2 b 4 r 3 u 2 2 r 3 f u constrained on the l 2 spheres s c u h 1 r 3 u 2 2 c here c 0 and f s 0 s f t d t under some mild assumptions on f we show that critical points of e b unbounded from below on s c exist for c 0 in addition we establish the existence of infinitely many radial critical points u n b of e b on s c provided that f is odd finally the asymptotic behavior of u n b as b 0 is analyzed these conclusions extend some known ones in previous papers

magnetohydrodynamic state estimation with boundary sensors we present a pde observer that estimates the velocity pressure electric potential and current fields in a magnetohydrodynamic mhd channel flow also known as hartmann flow this flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field the system is described by the inductionless mhd equations a combination of the navier stokes equations and a poisson equation for the electric potential under the so called inductionless mhd approximation in a low magnetic reynolds number regime we identify physical quantities measurable on the wall of the channel that are sufficient to generate convergent estimates of the velocity pressure and electric potential field away from the walls our observer consists of a copy of the linearized mhd equations combined with linear injection of output estimation error with observer gains designed using backstepping pressure skin friction and current measurements from one of the walls are used for output injection for zero magnetic field or nonconducting fluid the design reduces to an observer for the navier stokes poiseuille flow a benchmark for flow control and turbulence estimation we show that for the linearized mhd model the estimation error converges to zero in the l 2 norm despite being a subject of practical interest the problem of observer design for nondiscretized 3 d mhd or navier stokes channel flow has so far been an open problem

adaptive output feedback stabilization of non local hyperbolic pdes we address the problem of adaptive output feedback stabilization of general first order hyperbolic partial integro differential equations pide such systems are also referred to as pdes with non local in space terms we apply control at one boundary take measurements on the other boundary and allow the system s functional coefficients to be unknown to deal with the absence of both full state measurement and parameter knowledge we introduce a pre transformation which happens to be based on backstepping of the system into an observer canonical form in that form the problem of adaptive observer design becomes tractable both the parameter estimator and the control law employ only the input and output signals and their histories over one unit of time prior to presenting the adaptive design we present the non adaptive baseline controller which is novel in its own right and facilitates the understanding of the more complex adaptive system the parameter estimator is of the gradient type based on a parametric model in the form of an integral equation relating delayed values of the input and output for the closed loop system we establish boundedness of all signals pointwise in space and time and convergence of the pde state to zero pointwise in space we illustrate our result with a simulation

optimal controls of navier stokes equations this paper studies optimal control problems of the fluid flow governed by the navier stokes equations two control problems are formulated in the case of the driven cavity and flow through a channel with sudden expansion and solved successfully using a numerical optimization algorithm based on the augmented lagrangian method existence and the first order optimality condition of the optimal control are established a convergence result on the augmented lagrangian method for nonsmooth cost functional is obtained

brief control of mixing by boundary feedback in 2d channel flow we address the problem of enhancing mixing by means of boundary feedback control in 2d channel flow this is done by first designing feedback control strategies for the stabilization of the parabolic equilibrium flow then applying this feedback with the sign of the input reversed the result is enhanced instability of the parabolic equilibrium flow which leads rapidly to highly complex flow patterns simulations of the deformation of dye blobs positioned in the flow indicate qualitatively that effective mixing is obtained for small control effort as compared with the nominal uncontrolled flow a mixedness measure p e is constructed to quantify the mixing observed and is shown to be significantly enhanced by the application of the destabilizing control feedback

stabilization of a plane channel flow by wall normal controllers abstract the steady state flows in a two dimensional channel with periodic conditions along the axis can be stabilized by boundary controllers with vertical velocity observation which acts on the normal component of velocity only

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 h 1 2 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

brief paper exponential stabilization of the linearized navier stokes equation by pointwise feedback noise controllers we address the problem of exponential stabilization in probability of the linearized navier stokes equations in an equilibrium point this is done by designing a linear stochastic feedback controller with support in a point or on a discrete set of points of the domain this controller consists of a steady state impulse component with support in a finite set of points modulated by an unsteady feedback noise controller

drag reduction in transitional linearized channelow using distributed control this work focuses on feedback control of incompressible transitional newtonian channel flow described by the twodimensional linearized navier stokes equations the control objective is to use distributed feedback to achieve stabilization of the parabolic velocity profile for values of the reynolds number for which this profile is unstable and therefore to reduce the frictional drag exerted on the lower channel wall compared to the open loop values the control system uses measurements of shear stresses on the lower channel wall and the control actuation is assumed to be in the form of electromagnetic lorentz forces applied to the flow near the bottom wall galerkin s method is initially used to derive a high order discretization of the linearized flow field that captures the flow instability and accounts for the effect of control actuation on all the modes then a low order approximation of the linearized flow field is derived and used for the synthesis of a linear output feedback controller that enfor

existence of optimal controls for viscous flow problems a class of optimal control problems in viscous flow is studied main result is the existence theorem for optimal control three typical flow control problems are formulated within this general class

skin friction and pressure the footprints of turbulence abstract the problems of exact state reconstruction and approximate state estimation based on wall information in a wall bounded incompressible unsteady flow are addressed it is shown that if in an arbitrarily small neighborhood of time t precise measurements are made of the two components of wall skin friction and the wall pressure all terms in the taylor series expansions of the unsteady flow state near the wall at time t may be determined in the linear setting this determination may be made based on skin friction measurements alone combining this fact with the analyticity of solutions of the nonlinear navier stokes equation and the unique continuation theorem for analytic functions in theory complete reconstruction of a fully developed turbulent flow in a channel at any reynolds number at time t is possible given only information about the unsteady flow available at the wall in a neighborhood of time t without knowledge of the initial conditions of the flow thus skin friction and pressure measurements on the wall in a neighborhood of time t provide a unique footprint of the entire unsteady turbulent flow state no other flow can have the same footprint indeed higher order terms are shown to uniformly improve the correlation of truncated taylor series expansions with the dns of a turbulent flow near the wall however such series extrapolations amplify measurement noise as they require differentiation in both space and time of the measurements and the radius of convergence of the taylor series expansions is less than 10 wall units the so called linear stochastic estimation technique in which the polynomials forming the basis of the series expansion are replaced by well behaved functions such as pod modes on the entire flow domain also demonstrates very poor convergence in light of these limitations on direct extrapolations from measurements in the practical setting an adjoint based algorithm is presented and numerically tested for estimating the state of an entire turbulent channel flow system based on a time history of noisy measurements at the wall this algorithm effectively uses the unsteady nonlinear navier stokes equation itself as a filter to find the flow solution that is most consistent with the available measurements

dynamics and approximations of a velocity tracking problem for the navier stokes flows with piecewise distributed controls we study the dynamics of a piecewise in time distributed optimal control problem for the navier stokes equations which models velocity tracking over time we also study the dynamics of semidiscrete and fully discrete approximations of this velocity tracking problem we prove that the rates of velocity tracking are exponential some computational results are presented which reinforces the theoretical results derived

mhd channel flow control in 2d mixing enhancement by boundary feedback a nonlinear lyapunov based boundary feedback control law is proposed for mixing enhancement in a 2d magnetohydrodynamic mhd channel flow also known as hartmann flow which is electrically conducting incompressible and subject to an external transverse magnetic field the mhd model is a combination of the navier stokes pde and the magnetic induction pde which is derived from the maxwell equations pressure sensors magnetic field sensors and micro jets embedded into the walls of the flow domain are employed for mixing enhancement feedback the proposed control law designed using passivity ideas is optimal in the sense that it maximizes a measure related to mixing which incorporates stretching and folding of material elements while at the same time minimizing the control and sensing efforts a dns code is developed based on a hybrid fourier pseudospectral finite difference discretization and the fractional step technique to numerically assess the controller

h control theory of fluid dynamics robust or h control theory has been an expanding subject in recent years because of its wide applications and its close connection to operator theory on hardy spaces we develop an hcontrol theor

necessary and sufficient conditions for optimal controls in viscous flow problems a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation characterising the feedback problem the maximum principle is established by two quite different methods

the time optimal control of navier stokes equations the maximum principle for the time optimal control of the navier stokes equations in 2 d is established

a dissipative feedback control synthesis for systems arising in fluid dynamics a dissipative feedback control synthesis is constructed to regulate the systems arising in fluid dynamics the feedback law is obtained by utilizing nonlinear dynamic programming techniques the control law is designed for driving the system to a prescribed equilibrium state and enhancing the energy dissipation effects of the dynamical system two dimensional navier stokes equations and burgers equation are used for numerical experiments to illustrate the effects of the feedback synthesis and the theoretical results

exact controllability and stabilizability of the korteweg de vries equation in this paper we consider distributed control of the system described by the korteweg de vries equation i atu uaxu e3 u f on the interval 0 0 with periodic boundary conditions ii au 27r t ku o t k 0 1 2 where the distributed control f f x t is restricted so that the volume f u x t dx of the solution is conserved both exact controllability and stabilizibility questions are studied for the system in the case of open loop control if the control f is allowed to act on the whole spatial domain 0 27r it is shown that the system is globally exactly controllable i e for given t 0 and functions q x x with the same volume one can alway find a control f so that the system i ii has a solution u x t satisfying u x 0 0 x u x t x if the control f is allowed to act on only a small subset of the domain 0 27r then the same result still holds if the initial and terminal states b and 0 have small amplitude in a certain sense in the case of closed loop control the distributed control f is assumed to be generated by a linear feedback law conserving the volume while monotonically reducing f u x t 2dx the solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state as in the open loop control case a small amplitude assumption is needed if the control is allowed to act on only a small subdomain the smoothing property of the periodic linear kdv equation discovered recently by bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper

dynamic programming of the navier stokes equations a class of optimal control problems in viscous flow is studied main results are the pontryagin maximum principle and the verification theorem for the hamilton jacobi bellman equation

lp exponential stability for the equilibrium solutions of the navier stokes equations the exponential stability of the solution of the navier stokes equation in lp p 2 p 3 in bounded domain is considered in this paper under some assumptions on the external force it can be shown that the bounded solution of the navier stokes equation with initial and boundary conditions approaches the stationary solution of the system exponentially when time t goes to infinite

dynamics for controlled navier stokes systems with distributed controls the long time behavior of solutions for an optimal distributed control problem associated with the navier stokes equations is studied first a linear feedback solution for the navier stokes equations is constructed this feedback solution possesses decay in time properties then some preliminary estimates for the long time behavior of all solutions of the navier stokes equations are derived next the existence of a solution for the optimal control problem is proved finally the long time decay properties for the optimal solutions are established

empirical likelihood based semiparametric inference for the treatment effect in the two sample problem with censoring to compare two samples of censored data we propose a unified semiparametric inference for the parameter of interest when the model for one sample is parametric and that for the other is nonparametric the parameter of interest may represent for example a comparison of means or survival probabilities the confidence interval derived from the semiparametric inference which is based on the empirical likelihood principle improves its counterpart constructed from the common estimating equation the empirical likelihood ratio is shown to be asymptotically chi squared simulation experiments illustrate that the method based on the empirical likelihood substantially outperforms the method based on the estimating equation a real dataset is analysed

boundary value problems and optimal boundary control for the navier stokes system the two dimensional case 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

stability enhancement by boundary control in 2 d channel flow in this paper we stabilize the parabolic equilibrium profile in a two dimensional 2 d channel flow using actuators and sensors only at the wall the control of channel flow was previously considered by speyer and coworkers and bewley and coworkers who derived feedback laws based on linear optimal control and implemented by wall normal actuation with an objective to achieve global lyapunov stabilization we arrive at a feedback law using tangential actuation using teamed pairs of synthetic jets or rotating disks and only local measurements of wall shear stress allowing to embed the feedback in microelectromechanical systems mems hardware without need for wiring this feedback is shown to guarantee global stability in at least h sup 2 norm which by sobolev s embedding theorem implies continuity in space and time of both the flow field and the control as well as their convergence to the desired steady state the theoretical results are limited to low values of reynolds number however we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher

stability enhancement by boundary control in the kuramoto sivashinsky equation where we refer to 0 as the anti di ussion parameter note that a more general form ut 1uxxxx 2uxx 3uux 0 can always be reduced to 1 1 by appropriate rescaling of t x and u eq 1 1 was derived independently by kuramoto et al 20 22 as a model for phase turbulence in reaction di usion systems and by sivashinsky 33 as a model for plane ame propagation describing the combined in uence of di usion and thermal conduction of the gas on stability of a plane ame front so far it has been well understood that the ks equation can also serve as a mathematical model for cellular instabilities in a variety of situations the ow of thin liquid lms on inclined planes 30 in the limit of large surface tension dendritic fronts in dilute binary alloys 31 and alfven drift waves in plasmas 23 as a nonlinear saturation mechanism of the dissipative trapped ion modes

global ln solution and its decay property for the navier stokes equations in half space r n soit r n x 1 x 2 x n 1 x n x n 0 n 2 dans q r n 0 on considere les equations de navier stokes u t u u u 0 dans q divu 0 dans q u x u n 0 0 u 0 a on demontre l existence et l unicite d une l n solution globale forte pour des donnees initiales petites