analysis and approximation for linear feedback control for tracking the velocity in navier stokes flows abstract some systematic approaches to the mathematical formulation and numerical resolution of the linear feedback control problem for tracking the velocity in navier stokes flows in a bounded two dimensional domain with bounded distributed control are presented semidiscrete in time and full space time discrete approximations are also studied some computational results are presented and compared with analogous results from optimal control theory

relaxation theorems differential inclusions and filippov s theorem for relaxed controls in semilinear infinite dimensional systems abstract under suitable assumptions trajectories of semilinear control systems in banach spaces corresponding to relaxed controls can be uniformly approximated by trajectories corresponding to ordinary controls moreover relaxed trajectories can be defined nonparametrically that is by relaxed differential inclusions

existence theory and the maximum principle for relaxed infinite dimensional optimal control problems existence theorems are considered for relaxed optimal control problems described by semilinear systems in banach spaces relaxed controls are used whose values are finitely additive probability measures this class of relaxed controls does not require special assumptions such as compactness on the control set under suitable conditions relaxed trajectories coincide with those obtained from differential inclusions existence theorems for relaxed controls are obtained that apply to distributed parameter systems described by semilinear parabolic and wave equations as well as a version of pontryagin s maximum principle for relaxed optimal control problems

real hyperbolic hyperplane complements in the complex hyperbolic plane abstract this paper studies riemannian manifolds of the form m s where m 4 is a complete four dimensional riemannian manifold with finite volume whose metric is modeled on the complex hyperbolic plane c h 2 and s is a compact totally geodesic codimension two submanifold whose induced riemannian metric is modeled on the real hyperbolic plane h 2 in this paper we write the metric on c h 2 in polar coordinates about s compute formulas for the components of the curvature tensor in terms of arbitrary warping functions theorem 7 1 and prove that there exist warping functions that yield a complete finite volume riemannian metric on m s whose sectional curvature is bounded above by a negative constant theorem 1 1 1 the cases of m s modeled on h n h n 2 and c h n c h n 1 were studied by belegradek in 4 and 3 respectively one may consider this work as part 3 to this sequence of papers

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

optimal control of navier stokes equations by oseen approximation a non standard sequential quadratic programming type iterational process based on oseen s approximation is proposed and analyzed to solve an optimal control problem for the steady state navier stokes equations further numerical approximation by a finite element method and sample computational experiments are presented too

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

pinching constants for hyperbolic manifolds we show in this paper that for everyn 4 there exists a closedn dimensional manifoldv which carries a riemannian metric with negative sectional curvaturek but which admits no metric with constant curvaturek 1 we also estimate the pinching constantsh for which our manifoldsv admit metrics with 1 k h

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

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

an optimal control problem in exterior hydrodynamics in this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure an existence theorem for the speed trajectory which corresponds to the absolute minimum is provided the results are valid for arbitrary reynolds numbers

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

control of low speed airfoil aerodynamics nomenclature cd drag coefficient 2d pulc cf local skin friction coefficient 2rw pul cl lift coefficient 2l pu c cq suction coefficient i vw i uq c airfoil s chord d drag force per unit span l lift force per unit span p instantaneous hydrostatic pressure po pressure outside boundary layer p mean pressure r wall s radius of curvature 6 displacement thickness reynolds number uod v ro momentum thickness reynolds number u0de v re reynolds number based on distance from leading edge or chord and freestream velocity t instantaneous temperature t mean temperature j7 instantaneous velocity component uj mean velocity component u0 velocity outside the boundary layer t oo freestream velocity hi fluctuating velocity component u friction velocity vr p vw normal velocity of fluid injected or withdrawn through the wall x cartesian coordinates x streamwise distance from leading edge y normal distance from the wall y normal distance in wall units yu v z spanwise coordinate a angle of attack 6 boundary layer thickness 60 momentum thickness 6 displacement thickness x dynamic coefficient of viscosity v kinematic viscosity v u viscous length scale wall unit p density p uv tangential reynolds stress t w shear stress at the wall pu ajo instantaneous spanwise vorticity at the wall fijo rnean spanwise vorticity at the wall du dy q

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

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

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

recognizing constant curvature discrete groups in dimension 3 we characterize those discrete groups g which can act properly discontinuously isometrically and cocompactly on hyperbolic 3 space ihi3 in terms of the combinatorics of the action of g on its space at infinity the major ingredients in the proof are the properties of groups that are negatively curved in the large that is gromov hyperbolic the combinatorial riemann mapping theorem and the sullivan tukia theorem on groups which act uniformly quasiconformally on the 2 sphere

numerical solution of a flow control problem vorticity reduction by dynamic boundary action in order to laminarize an unsteady internal flow the vorticity field is minimized in a least squares sense using an optimal control approach the flow model is the navier stokes equation for a viscous incompressible fluid and the flow is controlled by suction and blowing on a part of the boundary a quasi newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term the navier stokes equations are approximated using a finite difference scheme in time and finite element approximations in space accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm therefore first order necessary conditions for a minimizer of the objective function are derived in the fully discrete case a memory saving device is discussed without which problems of any realistic size especially in three space dimensions would remain computationally intractable the feasibility of the optimal control approach for flow control problems is demonstrated by numerical experiments for a two dimensional flow in a rectangular cavity at a reynolds number high enough for nonlinear effects to be important

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

sutured manifold hierarchies essential laminations and dehn surgery x0 introduction a compact orientable surface f with nonnegative euler characteristic is either a sphere a disk a torus or an annulus if a 3 manifold m contains such an essential surface then it is said to be reducible reducible toroidal or annular respectively any such surface can be used to decompose the manifold further into simpler manifolds we say that m is a simple manifold if it has no such surfaces a simple manifold is expected to have a nice geometric structure if m has nonempty boundary then the geometrization theorem of thurston for haken manifolds says that m with boundary tori removed admits a nite volume hyperbolic structure with totally geodesic boundary when m has no boundaries thurston s geometrization conjecture asserts that m is either hyperbolic or is a seifert b er space with orbifold a sphere with at most 3 cone points suppose t is a torus boundary component of m we use m to denote the manifold obtained by dehn lling on t so that the slope on t bounds a disk in the dehn lling solid torus when m e k is the exterior of a knot k in s 3 denote m by k and call it the manifold obtained by surgery on the knot k it is well known that if m is simple then there are only nitely many dehn llings on t which produce non simple manifolds if m 1 and m 2 are non simple manifolds then the geometric intersection number between 1 and 2 denoted by 1 2 is proved to be at most 8 by gordon gor1 there are 10 dieren t cases according to the types of nonnegative euler characteristic surfaces in m i in many cases the upper bounds for 1 2 have been established see table 0 1

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

the lower central series and pseudo anosov dilatations the theme of this paper is that algebraic complexity implies dynamical complexity for pseudo anosov homeomorphisms of a closed surface s g of genus g penner proved that the logarithm of the minimal dilatation for a pseudo anosov homeomorphism of s g tends to zero at the rate 1 g we consider here the smallest dilatation of any pseudo anosov homeomorphism of s g acting trivially on gamma gamma k the quotient of gamma pi 1 s g by the k rm th term of its lower central series k geq 1 in contrast to penner s asymptotics we prove that this minimal dilatation is bounded above and below independently of g with bounds tending to infinity with k for example in the case of the torelli group cal i s g we prove that l cal i s g the logarithm of the minimal dilatation in cal i s g satisfies 197 l cal i s g 4 127 in contrast we find pseudo anosov mapping classes acting trivially on gamma gamma k whose asymptotic translation lengths on the complex of curves tend to 0 as g to infty