determining the optimal parameters for the mhd flow and heat transfer with variable viscosity and hall effect abstract the direct and optimal control solution of the laminar fully developed steady mhd flow of an incompressible electrically conducting fluid in a duct is considered together with the heat transfer the flow is driven by a constant pressure gradient and an external uniform magnetic field the fluid viscosity is temperature dependent varying exponentially and the hall effect viscous and joule dissipations are taken into consideration the control problem is solved by the discretize then optimize approach using mixed finite element method for the mhd and energy equations the control formulations with the hall and viscosity parameters the hartmann and brinkmann number are given to regain the desired velocity and temperature of the mhd flow

control of flow separation over a forward facing step by model reduction abstract design and implementation of reduced order optimal controller for flow separation are investigated the reduced order controller design is based on proper orthogonal decomposition pod and galerkin projection detailed finite element simulations are performed and pod is applied to the resulting data to extract the most energetic eigenmodes these global eigenmodes are then used in conjunction with the galerkin projection to obtain a reduced order low dimensional model reduced order models are not only attractive for real time control computation but also crucial for detailed stability and bifurcation analysis we investigate the design of optimal controller for flow separation in a channel with this model where actuation is performed on a small part of the boundary two different types of surface actuation are considered tangential blowing and suction through a single slot best locations for actuation and adaptive procedure for reduced order model are explored the proposed approaches are evaluated by investigating the control of flow separation over a forward facing step channel our methods are found to be efficient and fast and our methods demonstrate a significant reduction in computational time and feasibility dramatic separation delays are observed on all cases it is found that the tangential blowing is more efficient in mitigating flow separation and reducing wake spread

reduced order optimal control based on approximate inertial manifolds for nonlinear dynamical systems a reduced order method for optimal control problems in infinite dimensions based on approximate inertial manifolds is developed convergence of the cost optimal controls and optimal states of the finite dimensional reduced order optimal control problems to the original optimal control problem is analyzed special attention is given to the particular case when the dynamics are described by the navier stokes equations in dimension two

a generalization of the schur siegel smyth trace problem abstract let be a totally positive algebraic integer and define its absolute trace to be tr deg the trace of divided by the degree of elementary considerations show that the absolute trace is always at least one while it is plausible that for any 0 the absolute trace is at least 2 with only finitely many exceptions this is known as the schur siegel smyth trace problem our aim in this paper is to show that the schur siegel smyth trace problem can be considered as a special case of a more general problem

numerical model reduction of 2d steady incompressible laminar flows application on the flow over a backward facing step the numerical solution of most fluid mechanics problems usually needs such a fine mesh that the associated computational times become non negligible parts in any design process in order to couple numerical modelling schemes with inversion or control algorithms the size of such models needs to be highly reduced the identification method is a way to build low order models that fit with the original ones the laminar flow over a backward facing step is used as a test case presented solutions are found to be in good agreement with experimental and numerical results found in the literature

on singular q curvature type equations this paper is devoted to q curvature type equations with singularities mainly we give existence and regularity results of solutions to have positive solutions which will be meaningful in conformal geometry we restrict ourself to special manifolds

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

domain decomposition methods for advection dominated linear quadratic elliptic optimal control problems abstract we present an optimization level domain decomposition dd preconditioner for the solution of advection dominated elliptic linear quadratic optimal control problems which arise in many science and engineering applications the dd preconditioner is based on a decomposition of the optimality conditions for the elliptic linear quadratic optimal control problem into smaller subdomain optimality conditions with dirichlet boundary conditions for the states and the adjoints on the subdomain interfaces these subdomain optimality conditions are coupled through robin transmission conditions for the states and the adjoints the parameters in the robin transmission condition depend on the advection this decomposition leads to a schur complement system in which the unknowns are the state and adjoint variables on the subdomain interfaces the schur complement operator is the sum of subdomain schur complement operators the application of which is shown to correspond to the solution of subdomain optimal control problems which are essentially smaller copies of the original optimal control problem we show that under suitable conditions the application of the inverse of the subdomain schur complement operators requires the solution of a subdomain elliptic linear quadratic optimal control problem with robin boundary conditions for the state numerical tests for problems with distributed and with boundary control show that the dependence of the preconditioners on mesh size and subdomain size is comparable to its counterpart applied to a single advection dominated equation these tests also show that the preconditioners are insensitive to the size of the control regularization parameter

second order methods for optimal control of time dependent fluid flow second order methods for open loop optimal control problems governed by the two dimensional instationary navier stokes equations are investigated optimality systems based on a lagrangian formulation and adjoint equations are derived the newton and quasi newton methods as well as various variants of sqp methods are developed for applications to optimal flow control and their complexity in terms of system solves is discussed local convergence and rate of convergence are proved a numerical example illustrates the feasibility of solving optimal control problems for two dimensional instationary navier stokes equations by second order numerical methods in a standard workstation environment

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

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

adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition an adaptive reduced order controller design is presented for flow control using proper orthogonal decomposition pod in reduced order controller design the idea is to start with an ensemble of data obtained from numerical simulation of the underlying partial differential equations pdes pod is then used to obtain a reduced set of basis functions which is then used to derive a reduced order model of the pdes via galerkin projection this reduced order model allows us to derive a reduced order controller however it is not clear a priori what is the best way to obtain an ensemble of data that would give basis functions that represent the influence of the control action on the system in this paper we explore an adaptive procedure for reduced order controller design that improves the reduced order model by successively updating the ensemble of data during the optimization iterations we illustrate this method on a control problem in thermal flow system modeled by a thermally coupled navier stokes equations numerical results are presented for a vorticity regulation problem in fluid flows using boundary temperature as control mechanism through our numerical experiments we demonstrate the feasibility and applicability of the adaptive reduced order controllers

resolution of singularities and modular galois theory i shall sketch a brief history of the desingularization problem from riemann thru zariski to hironaka including the part i played in it and the work on galois theory which this led me to and how that caused me to search out many group theory gurus i shall also formulate several conjectures and suggest numerous thesis problems

optimal boundary feedback flow stabilization by model reduction abstract in this paper we consider the optimal boundary feedback stabilization of fluid flows governed by navier stokes equations using model reduction the model reduction is carried out using a combination of proper orthogonal decomposition pod and galerkin projection the resulting reduced order model is employed in the optimal linear quadratic regulator lqr design to derive a feedback control the feedback control is then used in the nonlinear navier stokes equation to stabilize the system sufficient conditions are derived for the linear control to stabilize the nonlinear system feasibility of the proposed boundary feedback stabilization design is numerically demonstrated on the problem of stabilizing the flow past an airfoil the control is effected through unsteady blowing on the airfoil surface the computational results show that the closed loop system amplitudes converge to the desired equilibrium state showing the stability predicted by the theory in addition they show that downstream directed blowing on the upper surface of the airfoil near the leading edge is more efficient in stabilizing the state

numerical approximation of optimal flow control problems by a penalty method error estimates and numerical results the purpose of this paper is to present numerically convenient approaches to solve optimal dirichlet control problems governed by the steady navier stokes equations we will examine a penalized neumann control approach for solving dirichlet control problems from numerical and computational points of view the control is affected by the suction or injection of fluid through the boundary or by boundary surface movements in the tangential direction the control objective is to minimize the vorticity in the flow or to drive the velocity field to a desired one we develop sequential quadratic programming methods to solve these optimal control problems the effectiveness of the optimal control techniques in flow controls and the feasibility of the proposed penalized neumann control approaches for flow control problems are demonstrated by numerical experiments for a viscous incompressible fluid flow in a two dimensional channel and in a cavity geometry

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

bivariate factorizations connecting dickson polynomials and galois theory in his ph d thesis of 1897 dickson introduced certain permutation polynomials whose galois groups are essentially the dihedral groups these are now called dickson polynomials of the first kind to distinguish them from their variations introduced by schur in 1923 which are now called dickson polynomials of the second kind in the last few decades there have been extensive investigations of both of these types which are related to the classical chebyshev polynomials we give new bivariate factorizations involving both types of dickson polynomials these factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups and lead to the construction of explicit equations with orthogonal groups as galois groups

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

analysis of optimal control problems for the 2 d stationary boussinesq equations this paper deals with optimal control problems associated with the 2 d boussinesq equations the controls considered may be of either the distributed or the neumann type these problems are first put into an appropriate mathematical formulation then the existence of optimal solutions is proved the use of lagrange multiplier techniques is justified and an optimality system of equations is derived

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

on the einstein scalar field lichnerowicz equations on compact riemannian manifolds in this paper we investigate the existence of a positive solution to the einstein scalar field lichnerowicz equation on the compact riemannian manifold we use a variational method which allows us to derive a new existence result which depends on the spectral properties of an appropriate operator a kazdan and warner type obstruction is also found in the specific case when the initial value for the scalar field is a constant in the standard unit sphere in this paper we investigate the existence of a positive solution to the einstein scalar field lichnerowicz equation on the compact riemannian manifold we use a variational method which allows us to derive a new existence result which depends on the spectral properties of an appropriate operator a kazdan and warner type obstruction is also found in the specific case when the initial value for the scalar field is a constant in the standard unit sphere

analysis of neumann boundary optimal control problems for the stationary boussinesq equations including solid media this article deals with neumann boundary optimal control problems associated with the boussinesq equations including solid media these problems are first put into an appropriate mathematical formulation then the existence of optimal solutions is proved the use of lagrange multiplier techniques is justified and an optimality system of equations is derived

further nice equations for nice groups nice sextinomial equations are given for unramified coverings of the affine line in nonzero characteristic p with p 2m q and 2m q as galois groups where m 3 is any integer and q 1 is any power of p 2

optimal control of thermally convected fluid flows we examine the optimal control of stationary thermally convected fluid flows from the theoretical and numerical point of view we use thermal convection as control mechanism that is control is effected through the temperature on part of the boundary control problems are formulated as constrained minimization problems existence of optimal control is given and a first order necessary condition of optimality from which optimal solutions can be obtained is established we develop numerical methods to solve the necessary condition of optimality and present computational results for control of cavity and channel type flows showing the feasibility of the proposed approach

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

adjoint based iterative method for robust control problems in fluid mechanics in this article we study the convergence of an adjoint based iterative method recently proposed in t r bewley r temam and m ziane phys d 138 2000 pp 360 392 for the numerical solution of a class of nonlinear robust control problems in fluid mechanics under weaker assumptions than those of t tachim medjo numer funct anal optim 23 2002 pp 849 873 we prove the convergence of the algorithm and we obtain an estimate of the convergence rate numerical solutions of a robust control problem related to data assimilation in oceanography are presented to illustrate the method

on the rayleigh benard marangoni system and a related optimal control problem abstract in this paper we study a boundary control problem associated to the stationary rayleigh benard marangoni rbm system in presence of controls for the velocity and the temperature on parts of the boundary we analyze the existence uniqueness and regularity of weak solutions for the stationary rbm system in polyhedral domains of r 3 and then we prove the existence of the optimal solution by using the theorem of lagrange multipliers we derive an optimality system we also give a second order sufficient optimality condition and we establish a result of uniqueness of the optimal solution

control of the freezing interface motion in two dimensional solidification processes using the adjoint method the aim of this work is to calculate the optimum history of boundary cooling conditions that in two dimensional conduction driven solidification processes results in a desired history of the freezing interface location motion the freezing front velocity and heat flux on the solid side of the front define the obtained solidification microstructure that can be selected such that desired macroscopic mechanical properties and soundness of the final cast product are achieved the so called two dimensional inverse stefan design problem is formulated as an infinite dimensional minimization problem the adjoint method is developed in conjunction with the conjugate gradient method for the solution of this minimization problem

reduced order control based on approximate inertial manifolds a reduced order method based on approximate inertial manifolds is applied to optimal control problems in infinite dimensional state spaces a detailed analysis of the method is given for the linear quadratic regulator problem the method can also be applied to higher order control systems with an appropriate decomposition of the state space in terms of slow and fast exponential decay

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