error estimates of stochastic optimal neumann boundary control problems we study mathematically and computationally optimal control problems for stochastic partial differential equations with neumann boundary conditions the control objective is to minimize the expectation of a cost functional and the control is of the deterministic boundary value type mathematically we prove the existence of an optimal solution and of a lagrange multiplier we represent the input data in terms of their karhunen loeve expansions and deduce the deterministic optimality system of equations computationally we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces

noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

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

the velocity tracking problem for navier stokes flows with bounded distributed controls we present some systematic approaches to the mathematical analysis and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in bounded two dimensional domains with bounded distributed controls we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and fully space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

observer based feedback boundary stabilization of the navier stokes equations abstract this paper aims at designing an observer based feedback law which locally stabilizes the solution to the two dimensional navier stokes equations with mixed boundary conditions we consider a finite number of controls acting on a portion of the boundary through robin boundary conditions and construct a linear luenberger observer based on the point observations of the linearized navier stokes equations the sensor location for the point observations is determined by the response of feedback functional gains we prove that the nonlinear system coupled with the observer through the feedback law is locally exponentially stable numerical experiments based on a taylor hood finite element method are presented to illustrate the design for different reynolds numbers

a penalized neumann control approach for solving an optimal dirichlet control problem for the navier stokes equations we introduce a penalized neumann boundary control approach for solving an optimal dirichlet boundary control problem associated with the two or three dimensional steady state navier stokes equations we prove the convergence of the solutions of the penalized neumann control problem the suboptimality of the limit and the optimality of the limit under further restrictions on the data we describe the numerical algorithm for solving the penalized neumann control problem and report some numerical results

on the second real eigenvalue of nonegative and z matrices abstract we give bounds for the second real eigenvalue of nonegative matrices and z matrices furthermore we establish upper bounds for the maximal spectral radii of principal submatrices of nonnegative matrices using these bounds we prove that our inequality for the second real eigenvalue of the adjacency matrix of a connected regular graph improves a well known bound for the second eigenvalue using cheeger s inequality

on spectral integral variations of mixed graphs in this paper we characterize the mixed graphs with exactly one laplacian eigenvalue moving up by an integer and other laplacian eigenvalues remaining invariant when an edge is added the results extend those of fan linear and multilinear algebra 50 2002 133 for general graphs and so linear and multilinear algebra 46 1999 193 for simple graphs

linear feedback control and approximation for a system governed by unsteady mhd equations abstract we are concerned with the feedback control of unsteady flow of an electrically conducting fluid confined to a bounded region of space and driven by a combination of distributed body force and externally generated currents the flow is governed by the navier stokes equations and maxwell s equations coupled via ohm s law and the lorentz force the mathematical formulation and numerical resolution of the linear feedback control problem for tracking velocity and magnetic fields of electrically conducting flows in two dimensional domain are presented semi discrete in time and full space time discrete approximations are also studied computational results showing the efficacy of the feedback control are presented and compared with analogous results using optimal control theory

real time computational algorithm for optimal control of an mhd flow system the flow of weakly electrically conducting fluids can be controlled by electromagnetic forces originating from electrodes and permanent magnets we consider the possibility of separation control for an electrically conducting bluff body flow with reduced order modeling the reduced order model is derived using the proper orthogonal decomposition pod and galerkin projection the pod allows one to extract the dominant modes in the observed physical phenomena and to define the smallest possible solution space this can be exploited in a galerkin framework to reduce break the nonlinear infinite dimensional model to a small finite dimensional model we investigate the design of a reduced order nonlinear optimal controller for flow separation in a channel we present numerical experiments with an applied spanwise magnetic field distribution and a boundary electric potential actuation on a part of the boundary our methods are found to be efficient and fast and our methods demonstrate a significant reduction in computational time

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

an energy space finite element approach for elliptic dirichlet boundary control problems in this paper we present a finite element analysis for a dirichlet boundary control problem where the dirichlet control is considered in a convex closed subspace of the energy space h 1 2 gamma h 1 2 as an equivalent norm in h 1 2 gamma h 1 2 we use the energy norm induced by the so called steklov poincare operator which realizes the dirichlet to neumann map and which can be implemented by using standard finite element methods the presented stability and error analysis of the discretization of the resulting variational inequality is based on the mapping properties of the solution operators related to the primal and adjoint boundary value problems and their finite element approximations some numerical results are given which confirm on one hand the theoretical estimates but on the other hand indicate the differences when modelling the control in l 2 gamma l 2

infimum of a pair of matrices abstract in this paper answers are provided to the following questions given a matrix e1 of order k m does there exist a matrix e2 of order m k m such that the matrix e e 1 e 2 is square idempotent and of the same rank as e1 further when is the choice of e2 unique given a subspace t virtually disjoint with the row span of e1 when does there exist a matrix g g 1 g 2 such that g is square idempotent of rank equal to rank g1 rank e1 and g1 is congruent to e1 mod t these results are then used to determine if a pair of matrices a and b has an infimum in the minus partial order further the existence of the sharp order infimum is studied

shorted matrices an extended concept and some applications the shorting of an operator hitherto considered by krein 11 and by anderson and trapp 3 only for positive operators is extended to rectangular matrices and square matrices not necessarily hermitian nonnegative definite some applications of the shorted matrix in mathematical statistics are discussed

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

on the perron roots of principal submatrices of co order one of irreducible nonnegative matrices let a be an irreducible nonnegative matrix and a be the perron root spectral radius of a denote by min a the minimum of the perron roots of all the principal submatrices of co order one it is well known that the interval min a a does not contain any eigenvalues of a consider any principal submatrix a v of co order one whose perron root is equal to min a we show that the jordan structure of min a as an eigenvalue of a is obtained from that of the perron root of a v as follows one largest jordan block disappears and the others remain the same so if only one jordan block corresponds to the perron root of the submatrix then min a is not an eigenvalue of a by schneider s theorem this holds if and only if there is a hamiltonian chain in the singular digraph of a v in the general case the jordan structure for the perron root of the submatrix a v and therefore that for the eigenvalue min a of a can be arbitrary but if the perron root a w of a principal submatrix a w of co order one is strictly greater than min a then a w is a simple eigenvalue of a w we also obtain different representations for the generalized eigenvectors corresponding to the eigenvalues of a contained in the annulus min a a

optimal control of vortices in non homogeneous navier stokes flows abstract optimal control of time dependent fluid flow governed by the incompressible navier stokes equations is considered a cost functional based on a local dynamical systems characterization of vortices is investigated the resulting functional is a non convex function of the velocity gradient tensor the optimality system based on a lagrangian formulation and adjoint equations describing first order necessary optimality conditions is provided the gradient and the second derivative of the cost functional with respect to the control are derived

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

least squares methods for optimal control optimal control and optimal shape design problems for the navier stokes equations arise in many important practical applications such as design of optimal profiles 7 drag minimization 9 11 and heating and cooling 12 among others typically optimal control problems for the navierstokes equations combine lagrange multiplier techniques to enforce the constraints and to derive an optimality system see e g 11 12 with mixed galerkin discretization for the state equations resulting methods are well studied theoretically for example an abstract framework that can be used for the analyses of such optimal control methods has been suggested in 13 however the use of lagrange multipliers and mixed galerkin discretizations is associated with some complications in the numerical computations which can reduce the overall efficiency and robustness of corresponding algorithms for example resulting discrete problems are in general indefinite similarly it is now well understood that stability of mixed discretizations does not allow one to choose independently the approximation spaces for the velocity and the pressure and that these spaces are subject to a restrictive stability condition known as the inf sup or lbb condition see 8 one possibility to remedy these difficulties is to consider optimal control methods in which the navier stokes constraint is treated by augmented lagrangian techniques see 6 nevertheless the use of mixed galerkin discretization in the method of 6 still requires approximation by finite element spaces that are subject to the inf sup condition

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

a dissipative pseudospectral method for the two dimensional navier stokes equations a pseudospectral method for solving the two dimensional navier stokes equations with periodic boundary conditions is developed the numerical method preserves the zero dissipation property of the nonlinear convective term using weak variational forms the convergence properties of the authors scheme are analyzed

first order sufficient optimality conditions for navier stokes flow dual feedback controls the navier stokes system of partial differential equations with navier boundary condition that models the evolution of viscous incompressible flows is studied as a controllable system and the optimal cost functional with respect to that control is invstigated a dual dynamic programming approach is proposed and sufficient optimality conditions in terms of verification theorem are proved the essential novelty of the paper is that all equations under consideration are treated in weak form dual feedback controls are proposed for the problem under study

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

schur complements and its applications to symmetric nonnegative and z matrices in linear algebra appl 177 1992 137 smith proved that if h is a hermitian semidefinite matrix and a is a nonsingular principal submatrix then the eigenvalues of the schur complement h a interlace those of h in this paper we refine the latter result and use it to derive eigenvalues interlacing results on an irreducible symmetric nonnegative matrix that involve perron complements for an irreducible symmetric nonnegative matrix we give lower and upper bounds for its spectral radius and also a lower bound for the maximal spectral radius of its principal submatrices of a fixed order we apply our results to an irreducible symmetric z matrix and to the adjacency matrix or the general laplacian matrix of a connected weighted graph the equality cases for the bounds for spectral radii or least eigenvalues are also examined

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

a superconvergent hybridizable discontinuous galerkin method for dirichlet boundary control of elliptic pdes we begin an investigation of hybridizable discontinuous galerkin hdg methods for approximating the solution of dirichlet boundary control problems governed by elliptic pdes these problems can involve atypical variational formulations and often have solutions with low regularity on polyhedral domains these issues can provide challenges for numerical methods and the associated numerical analysis we propose an hdg method for a dirichlet boundary control problem for the poisson equation and obtain optimal a priori error estimates for the control specifically under certain assumptions for a 2d convex polygonal domain we show the control converges at a superlinear rate we present 2d and 3d numerical experiments to demonstrate our theoretical results

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

on some characterizations of the star partial ordering for matrices and rank subtractivity abstract the main result is that drazin s star partial ordering a b holds if and only if a b and b a b a where a b is defined by a a a b and aa ba and where a b denotes rank subtractivity here a and b are m n complex matrices and the superscript denotes the moore penrose inverse several other characterizations of a b are given with particular emphasis on what extra condition must be added in order that rank subtractivity becomes the stronger star order a key tool is a new canonical form for rank subtractivity connections with simultaneous singular value decompositions schur complements and idempotent matrices are also mentioned

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

lattice properties of the order for complex matrices abstract it is shown that the set c m n of complex m n matrices forms a lower semilattice under the partial ordering a b defined by a a a b aa ba where a denotes the conjugate transpose of a as a special case of a result for division rings it is further shown that over any field f for m n 2 and any proper involution of f2 2 the corresponding intersections a b all exist

preconditioned iterative methods for navier stokes control problems pde constrained optimization problems are a class of problems which have attracted much recent attention in scientific computing and applied science in this paper we discuss preconditioned iterative methods for a class of time independent navier stokes control problems one of the main problems of this type in the field of fluid dynamics having detailed the picard type iteration we use to solve the problems and derived the structure of the matrix system to be solved at each step we utilize the theory of saddle point systems to develop efficient preconditioned iterative solution techniques for these problems we also require theory of solving convection diffusion control problems as well as a commutator argument to justify one of the components of the preconditioner

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

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

a reduced order method for simulation and control of fluid flows this article presents a reduced order modeling approach for simulation and control of viscous incompressible flows the reduced order models suitable for control and which capture the essential physics are developed using the reduced basis method the so called lagrange approach is used to define reduced bases and the basis functions in this approach are obtained from the numerical solutions the feasibility of this method for flow control is demonstrated on boundary control problems in closed cavity and in wall bounded channel flows control action is effected through boundary surface movement on a part of the solid wall our formulation of the reduced order method applied to flow control problems leads to a constrained minimization problem and is solved by applying newton like methods to the necessary conditions of optimality through our computational experiments we demonstrate the feasibility and applicability of the reduced order method for simulation and control of fluid flows

a semi smooth newton method for control constrained boundary optimal control of the navier stokes equations abstract in this paper we study optimal control of the navier stokes equations when the control acts as a pointwise constrained boundary condition of dirichlet type the problem is analyzed in the control space h 00 1 2 the optimality system and second order sufficient optimality conditions are derived for the numerical solution we apply a semi smooth newton method to a regularized version of the original problem and show convergence properties of the method and of the regularized solutions towards the original one

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

an estimate for the nonstochastic eigenvalues of doubly stochastic matrices abstract we introduce a new measure of irreducibility of a doubly stochastic matrix and find the best estimate from below for the distance of nonstochastic eigenvalues from one the estimate is used to obtain a cheeger type inequality for the second laplacian eigenvalue of a weighted graph

optimal control of two and three dimensional incompressible navier stokes flows the focus of this work is on the development of large scale numerical optimization methods for optimal control of steady incompressible navier stokes flows the control is affected by the suction or injection of fluid on portions of the boundary and the objective function represents the rate at which energy is dissipated in the fluid we develop reduced hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration both quasi newton and newton variants are developed and compared to the approach of eliminating the flow equations and variables which is effectively the generalized reduced gradient method optimal control problems are solved for two dimensional flow around a cylinder and three dimensional flow around a sphere the examples demonstrate at least an order of magnitude reduction in time taken allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation

semidiscrete approximations of optimal robin boundary control problems constrained by semilinear parabolic pde an optimal robin boundary control problem associated with semilinear parabolic partial differential equations is considered existence of an optimal solution is proved and an optimality system of equations is derived semidiscrete finite element approximations of the optimality system are defined and error estimates are obtained

the velocity tracking problem for navier stokes flows with boundary control we present some systematic approaches to the mathematical formulation and numerical approximation of the time dependent optimal control problem of tracking the velocity for navier stokes flows in a bounded two dimensional domain with boundary control we study the existence of optimal solutions and derive an optimality system from which optimal solutions may be determined we also define and analyze semidiscrete in time and full space time discrete approximations of the optimality system and a gradient method for the solution of the fully discrete system the results of some computational experiments are provided

feedback stabilization of a thermal fluid system with mixed boundary control we consider the problem of local exponential stabilization of the nonlinear boussinesq equations with control acting on portion of the boundary in particular given a steady state solution on an bounded and connected domain r 2 we show that a finite number of controls acting on a part of the boundary through neumann robin boundary conditions is sufficient to stabilize the full nonlinear equations in a neighborhood of this steady state solution dirichlet boundary conditions are imposed on the rest of the boundary we prove that a stabilizing feedback control law can be obtained by solving a linear quadratic regulator lqr problem for the linearized boussinesq equations numerical result are provided for a 2d problem to illustrate the ideas

lower bounds for the first eigenvalue of certain m matrices associated with graphs abstract we give lower bounds for the smallest eigenvalue of the laplacian of corresponding undirected connected multigraphs in terms of the expansion coefficients and norm estimates upper bounds for the convergence rate of certain nonnegative irreducible symmetric matrices and irreducible diagonally symmetrizable stochastic matrices are given

finite element approximations of stochastic optimal control problems constrained by stochastic elliptic pdes abstract in this paper we study mathematically and computationally optimal control problems for stochastic elliptic partial differential equations the control objective is to minimize the expectation of a tracking cost functional and the control is of the deterministic distributed type the main analytical tool is the wiener ito chaos or the karhunen loeve expansion mathematically we prove the existence of an optimal solution we establish the validity of the lagrange multiplier rule and obtain a stochastic optimality system of equations we represent the input data in their wiener ito chaos expansions and deduce the deterministic optimality system of equations computationally we approximate the optimality system through the discretizations of the probability space and the spatial space by the finite element method we also derive error estimates in terms of both types of discretizations

second order necessary and sufficient optimality conditions for optimization problems and applications to control theory this paper deals with a class of nonlinear optimization problems in a function space where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type second order necessary and sufficient optimality conditions are established where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces the results are applied to some optimal control problems for ordinary and partial differential equations

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

nonlinear control of incompressible fluid flow application to burgers equation and 2d channel flow abstract this paper proposes a methodology for the synthesis of nonlinear finite dimensional feedback controllers for incompressible newtonian fluid flows described by two dimensional navier stokes equations combination of galerkin s method with approximate inertial manifolds is employed for the derivation of low order ordinary differential equation ode systems that accurately describe the dominant dynamics of the flow these ode systems are subsequently used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed loop system to follow the reference input asymptotically the method is successfully used to synthesize nonlinear finite dimensional output feedback controllers for the burgers equation and the two dimensional channel flow that enhance the convergence rate to the spatially uniform steady state and the parabolic velocity profile respectively the performance of the proposed controllers is successfully tested through simulations and is shown to be superior to the one of linear controllers

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

constrained optimal control of navier stokes flow by semismooth newton methods abstract we propose and analyze a semismooth newton type method for the solution of a pointwise constrained optimal control problem governed by the time dependent incompressible navier stokes equations the method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation we analyze the flow control problem and prove q superlinear convergence of the method in the numerical implementation adjoint techniques are combined with a truncated conjugate gradient method numerical results are presented that support our theoretical results and confirm the viability of the approach

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

optimization problems for the navier stokes equations with regular boundary controls abstract we study the necessary conditions for an optimal boundary control problem associated with the stationary navier stokes equations with regular controls the control is the velocity on part or all of the boundary of the given flow domain we present a rigorous justification for the use of the lagrange multiplier rule to derive first order necessary conditions for optimality these are expressed as a system of partial differential equations we study the regularity of solutions of this system and finally we give some examples of specific functionals useful in applications

error estimates for discontinuous galerkin time stepping schemes for robin boundary control problems constrained to parabolic pdes we consider fully discrete finite element approximations of a robin optimal boundary control problem constrained by linear parabolic pdes with rough initial data conforming finite element methods for spatial discretization combined with discontinuous time stepping galerkin schemes are being used for the space time discretization error estimates are proved under weak regularity hypotheses for the state adjoint and control variables computational examples validating our expected rates of convergence are also provided

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

left star and right star partial orderings abstract two partial orderings in the set of complex matrices are introduced by combining each of the conditions a a a b and aa ba which define the star partial ordering with one of the conditions m a m b and m a m b which define the space preordering several properties of these orderings are examined with main emphasis on comparing the new orderings with the star ordering the minus ordering and other related partial orderings moreover some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived with the main emphasis on characterizations involving reflexive generalized inverses

convergence and quasi optimality of an adaptive finite element method for elliptic robin boundary control problem abstract in this paper we focus on the convergence and quasi optimality of an adaptive finite element method for elliptic robin boundary control problems we use piecewise linear finite elements to approximate the state and the adjoint state variables and the variational discretization to approximate the control variable under mild assumption on the initial mesh we prove the contraction property for the sum of the energy errors of the state and adjoint state and the scaled error estimator on two consecutive adaptive loops the resulting linear convergence yields the quasi optimal convergence rate for the afem algorithm applied to our problem additionally some numerical results are provided to support our theoretical analysis