analysis and discretization of an optimal control problem for the time periodic mhd equations we consider the mathematical formulation and analysis of an optimal control problem associated with the tracking of the velocity and the magnetic field of a viscous incompressible electrically conducting fluid in a bounded two dimensional domain through the adjustment of distributed controls existence of optimal solutions is proved and first order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls semidiscrete in time approximations are defined and their convergence to the exact optimal solutions is shown

error estimates for the discretization of the velocity tracking problem in this paper we are continuing our work casas and chrysafinos siam j numer anal 50 5 2281 2306 2012 concerning a priori error estimates for the velocity tracking of two dimensional evolutionary navier stokes flows the controls are of distributed type and subject to point wise control constraints the discretization scheme of the state and adjoint equations is based on a discontinuous time stepping scheme in time combined with conforming finite elements in space for the velocity and pressure provided that the time and space discretization parameters tau and h h respectively satisfy tau le ch 2 ch2 error estimates of order mathcal o h 2 o h2 and mathcal o h frac 3 2 frac 2 p o h32 2p with p 3 p 3 depending on the regularity of the target and the initial velocity are proved for the difference between the locally optimal controls and their discrete approximations when the controls are discretized by the variational discretization approach and by using piecewise linear functions in space respectively both results are based on new duality arguments for the evolutionary navier stokes equations

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

a monte carlo method for solving unsteady adjoint equations traditionally solving the adjoint equation for unsteady problems involves solving a large structured linear system this paper presents a variation on this technique and uses a monte carlo linear solver the monte carlo solver yields a forward time algorithm for solving unsteady adjoint equations when applied to computing the adjoint associated with burgers equation the monte carlo approach is faster for a large class of problems while preserving sufficient accuracy

birkhoff integral for multi valued functions abstract the aim of this paper is to study birkhoff integrability for multi valued maps f cwk x where is a complete finite measure space x is a banach space and cwk x is the family of all non empty convex weakly compact subsets of x it is shown that the birkhoff integral of f can be computed as the limit for the hausdorff distance in cwk x of a net of riemann sums n an f tn we link birkhoff integrability with debreu integrability a notion introduced to replace sums associated to correspondences when studying certain models in mathematical economics we show that each debreu integrable multi valued function is birkhoff integrable and that each birkhoff integrable multi valued function is pettis integrable the three previous notions coincide for finite dimensional banach spaces and they are different even for bounded multi valued functions when x is infinite dimensional and x is assumed to be separable we show that when f takes values in the family of all non empty convex norm compact sets of a separable banach space x then f is pettis integrable if and only if f is birkhoff integrable in particular these pettis integrable f s can be seen as single valued pettis integrable functions with values in some other adequate banach space incidentally to handle some of the constructions needed we prove that if x is an asplund banach space then cwk x is separable for the hausdorff distance if and only if x is finite dimensional

ramanujan s theories of elliptic functions to alternative bases in his famous paper 3 10 pp 23 39 ramanujan offers several beautiful series representations for 1 pi he first states three formulas one of which is frac 4 pi sum limits mathbf n 0 infty frac left 6 mathbf n 1 right left frac 1 2 right 3 n left mathbf n right 3 4 n where a o 1 and for each positive integer n left mathbf a right n mathbf a left mathbf a 1 right left mathbf a 2 right left mathbf a mathbf n 1 right

measurable selectors and set valued pettis integral in non separable banach spaces kuratowski and ryll nardzewski s theorem about the existence of measurable selectors for multi functions is one of the keystones for the study of set valued integration one of the drawbacks of this result is that separability is always required for the range space in this paper we study pettis integrability for multi functions and we obtain a kuratowski and ryll nardzewski s type selection theorem without the requirement of separability for the range space being more precise we show that any pettis integrable multi function f cwk x defined in a complete finite measure space with values in the family cwk x of all non empty convex weakly compact subsets of a general non necessarily separable banach space x always admits pettis integrable selectors and that moreover for each a the pettis integral afd coincides with the closure of the set of integrals over a of all pettis integrable selectors of f as a consequence we prove that if x is reflexive then every scalarly measurable multi function f cwk x admits scalarly measurable selectors the latter is also proved when x w is angelic and has density character at most 1 in each of these two situations the pettis integrability of a multi function f cwk x is equivalent to the uniform integrability of the family supx f x bx r results about norm borel measurable selectors for multi functions satisfying stronger measurability properties but without the classical requirement of the range banach space being separable are also obtained

inequalities for the generalized elliptic integrals and modular functions abstract in this paper the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions such as m a r and a r making use of these results the authors obtain some sharp inequalities for the so called ramanujan s generalized modular functions

gauge integrals and selections of weakly compact valued multifunctions abstract in the paper henstock mcshane birkhoff and variationally multivalued integrals are studied for multifunctions taking values in the hyperspace of convex and weakly compact subsets of a general banach space x in particular the existence of selections integrable in the same sense of the corresponding multifunctions has been considered

numerical approximations of exact controllability problems by optimal control problems for parabolic differential equations we consider the exact controllability problems for systems governed by linear parabolic differential equations we prove that the solution of an associated optimal control problem converges to the solution of the exact controllability problem the numerical solution of the exact controllability problem is then obtained by solving the optimal control problem numerically numerical experiments are presented

analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations this work concerns analysis and error estimates for optimal control problems related to implicit parabolic equations the minimization of the tracking functional subject to implicit parabolic equations is examined existence of an optimal solution is proved and an optimality system of equations is derived semi discrete in space error estimates for the finite element approximations of the optimality system are presented these estimates are symmetric and applicable for higher order discretizations finally fully discrete error estimates of arbitrarily high order are presented based on a discontinuous galerkin in time and conforming in space scheme two examples related to the lagrangian moving mesh galerkin formulation for the convection diffusion equation are described

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

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

class number three ramanujan type series for 1 p abstract there is a class of remarkable series for 1 of the form c 3 n 0 a nb c 3n 6n 3n n 3 where a b c are certain algebraic numbers these were first examined by ramanujan these examples arise by computing singular invariants for j and the constants involved will have degree equal to the class number of the associated imaginary quadratic field our intention is compute all such class number three examples the largest such example with discriminant 907 adds 37 additional digit accuracy per term a class number four example with discriminant 1555 gives 50 digits per term

a non conforming computational methodology for modeling coupled problems a variety of engineering applications that involve coupling different physical phenomena often require detailed finite element analysis to be carried out over complex domains often such analysis may be accomplished by dividing the global domain into several local subdomains over each of which a local model can be analyzed independently the global solution can then be constructed by suitably piecing together local solutions from these individually modeled subdomains however during the assembly it is often too cumbersome or even infeasible to coordinate the meshes over separate subdomains one must therefore employ non conforming techniques to accomplish such modeling in this paper we develop a non conforming computational methodology via the mortar finite element method to solve a coupled problem where we are interested in determining the effects of temperature variations on a given flow or the transfer of heat within the flow using this method the solution over different subdomains with different multigrid levels is efficiently computed our numerical results clearly suggest that the proposed methodology is robust and stable 2005 elsevier ltd all rights reserved

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

identification of space time distributed parameters in the gierer meinhardt reaction diffusion system we consider parameter identification for the classic gierer meinhardt reaction diffusion system the original gierer meinhardt model a gierer and h meinhardt kybernetik 12 1972 pp 30 39 was formulated with constant parameters and has been used as a prototype system for investigating pattern formation in developmental biology in our paper the parameters are extended in time and space and used as distributed control variables the methodology employs pde constrained optimization in the context of image driven spatiotemporal pattern formation we prove the existence of optimal solutions derive an optimality system and determine optimal solutions the results of numerical experiments in two dimensions are presented using the finite element method which illustrates the convergence of a variable step gradient algorithm for finding the optimal parameters of the system a practical target function is constructed for the optimal control algorithm corresponding to the actual image of a marine angel

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

eigen series solutions to terminal state tracking optimal control problems and exact controllability problems constrained by linear parabolic pdes abstract terminal state tracking optimal control problems for linear parabolic equations are studied in this paper the control objectives are to track a desired terminal state and the control is of the distributed type explicit solution formulae for the optimal control problems are derived in the form of eigen series pointwise in time l 2 norm estimates for the optimal solutions are obtained and approximate controllability results are established exact controllability is shown when the target state vanishes on the boundary of the spatial domain one dimensional computational results are presented which illustrate the terminal state tracking properties for the solutions expressed by the series formulae

series expansions for 1 πm and πm abstract in terms of the telescoping method we establish a general identity then it is used to derive several interesting series expansions for 1 m and m with m n

the controllability of systems governed by parabolic differential equations abstract the main result of this paper is a new weaker condition for the exact controllability of linear parabolic partial differential equations the result is derived through an examination of a related optimal control problem it is shown that if the terminal state belongs to the sobolev space of functions having two square integrable derivatives then the parabolic equation is exactly controllable

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

solving inverse problems involving the navier stokes equations discretized by a lagrange galerkin method abstract in this article we are investigating the numerical approximation of an inverse problem involving the evolution of a newtonian viscous incompressible fluid described by the navier stokes equations in 2d this system is discretized using a low order finite element in space coupled with a lagrange galerkin scheme for the nonlinear advection operator we introduce a full discrete linearized scheme that is used to compute the gradient of a given cost function by ensuring its consistency using gradient based optimization algorithms we are able to deal with two fluid flow inverse problems the drag reduction around a moving cylinder and the identification of a far field velocity using the knowledge of the fluid load on a rectangular bluff body for both fixed and prescribed moving configurations

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

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

conjugate gradient boundary element solution for distributed elliptic optimal control problems an optimality system of equations for the optimal control problem governed by helmholtz type equations is derived by the associated first order necessary optimality condition we obtain the conjugate gradient method cgm in the continuous case introducing the sequence of higher order fundamental solutions we propose an iterative algorithm based on the conjugate gradient boundary element method using the multiple reciprocity method cgm mrbem for solving the discrete control input this algorithm has an advantage over that of the existing literatures because the main attribute the reduced dimensionality of the boundary element method is fully utilized finally the local error estimates for this scheme are obtained and a test problem is given to illustrate the efficiency of the proposed method

on the controllability of a fractional order parabolic equation the null controllability property of a 1 d parabolic equation involving a fractional power of the laplace operator delta alpha is studied the control is a scalar time dependent function g g t acting on the system through a given space profile f f x on the interior of the domain thus the control g determines the intensity of the space control f applied to the system the latter being given a priori we show that if alpha leq 1 2 and the shape function f is say in l 2 no initial datum belonging to any sobolev space of negative order may be driven to zero in any time this is in contrast with the existing positive results for the case alpha 1 2 and in particular for the heat equation that corresponds to alpha 1 this negative result exhibits a new phenomenon that does not arise either for finite dimensional systems or in the context of the heat equation on the contrary if more regularity of the shape function f is assumed then we show that there are initial data in any sobolev space h m that may be controlled once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which when increasing the regularity of the control profile the space of controllable data decreases these results show that in order for the control properties of the heat equation to be true the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when alpha 1 2 for the fractional powers of the dirichlet laplacian in 1 d the results we present here are in nature and with respect to techniques of proof similar to those on the control of the heat equation in unbounded domains in s micu and e zuazua trans amer math soc 353 2000 pp 1635 1659 and s micu and e zuazua portugal math 58 2001 pp 1 24 we also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models

control of a simplified unsteady film cooling flow using gradient based optimization maximization of film cooling effectiveness in an idealized film cooling flow is accomplished using a gradient based control strategy the state equations are the two dimensional unsteady incompressible navier stokes equations and the temperature equation the control is a spatially and temporally varying normal velocity boundary condition subject to a constraint of zero mass and energy flux the control that maximizes the objective function is found using a conjugate gradient method in which the gradient of the objective function with respect to the control variables is obtained from solving a set of adjoint equations the effectiveness of choices for the objective function are examined along with the effect of the event horizon over which the control is obtained by the optimizer drastic improvements in the film cooling effectiveness are obtained for certain choices of the objective function and event horizon although not a specific target of the optimizer one result of the control is that the wall jet remains attached to the wall the numerical issues involved with finding the control are discussed and the features of the resulting control are analyzed with the goal of understanding the mechanisms that affect the film cooling effectiveness

sufficient second order optimality conditions for convex control constraints abstract in this article sufficient optimality conditions are established for optimal control problems with pointwise convex control constraints here the control is a function with values in r n the constraint is of the form u x u x where u is a set valued mapping that is assumed to be measurable with convex and closed images the second order condition requires coercivity of the lagrange function on a suitable subspace which excludes strongly active constraints together with first order necessary conditions it ensures local optimality of a reference function in an l neighborhood the analysis is done for a model problem namely the optimal distributed control of the instationary navier stokes equations

the pettis integral for multi valued functions via single valued ones abstract we study the pettis integral for multi functions f cwk x defined on a complete probability space with values into the family cwk x of all convex weakly compact non empty subsets of a separable banach space x from the notion of pettis integrability for such an f studied in the literature one readily infers that if we embed cwk x into l b x by means of the mapping j cwk x l b x defined by j c x sup x c then j f is integrable with respect to a norming subset of b l b x a natural question arises when is j f pettis integrable in this paper we answer this question by proving that the pettis integrability of any cwk x valued function f is equivalent to the pettis integrability of j f if and only if x has the schur property that is shown to be equivalent to the fact that cwk x is separable when endowed with the hausdorff distance we complete the paper with some sufficient conditions involving stability in talagrand s sense that ensure the pettis integrability of j f for a given pettis integrable cwk x valued function f

on a sequence arising in series for π in a recent investigation of dihedral quartic fields 6 a rational sequence a n was encountered we show that these a n are positive integers and that they satisfy surprising congruences modulo a prime p they generate unknown p adic numbers and may therefore be compared with the cubic recurrences in 1 where the corresponding p adic numbers are known completely 2 other unsolved problems are presented the growth of the a n is examined and a new algorithm for computing a n is given an appendix by d zagicr which carries the investigation further is added

an integral constrained parabolic problem with applications in thermochronology we investigate an inverse source problem with an integral constraint for a parabolic equation the constraint is motivated by an application in thermochronology a branch of geology the existence and uniqueness of weak solutions are established by means of the rothe method and an energy method respectively the elliptic problem resulting from the time discretization is solved by homogenizing the integral constraint the implicit scheme used in the proof of existence lends itself readily for numerical studies and we present the results of numerical experiments we also report on the errors and convergence rates

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

henstock kurzweil pettis integrability of compact valued multifunctions with values in an arbitrary banach space abstract the aim of this paper is to describe henstock kurzweil pettis hkp integrable compact valued multifunctions such characterizations are known in case of functions see di piazza and musial 2006 16 it is also known see di piazza and musial 2010 19 that each hkp integrable compact valued multifunction can be represented as a sum of a pettis integrable multifunction and of an hkp integrable function invoking to that decomposition we present a pure topological characterization of integrability having applied the above results we obtain two convergence theorems that generalize results known for hkp integrable functions we emphasize also the special role played in the theory by weakly sequentially complete banach spaces and by spaces possessing the schur property

a constrained least squares regularization method for nonlinear ill posed problems this paper deals with a method for solving ill posed nonlinear hilbert space operator equations f x y regularization is obtained by solving a constrained least squares regularization problem min f x y 2 quad text subject to j x leqq beta 2 beta serves as a regularization parameter and j x is a quadratic penalty functional to robustly and efficiently solve this regularization problem we apply a trust region method at each iteration the quadratic penalty constraint is retained a gauss newton approximation to the objective functional is taken and we add a quadratic trust region constraint the resulting quadratic subproblem is then reformulated as a nonlinear complementarily problem and solved using newton s method this paper applies methods to find approximate solutions to a severely ill posed nonlinear first kind integral equation arising in geophysics the method of generalized cross validation gcv is used to pick the regularization parameter when random er

a self contained automated methodology for optimal flow control this paper describes a self contained automated methodology for active flow control which couples the time dependent navier stokes system with an adjoint navier stokes system and optimality conditions from which optimal states i e unsteady flow fields and controls e g actuators may be determined the problem of boundary layer instability suppression through wave cancellation is used as the intital validation case to test the methodology here the objective of control is to match the stress vector along a portion of the boundary to a steady base flow control is effected through the injection or suction of fluid through a single orifice on the boundary the results demonstrate that instability suppression can be achieved without any a priori knowledge of the flow unsteadiness such as frequencies instability type etc the present methodology has been extended to three dimensions and may potentially be applied to separation control relaminarization and turbulence control applications using one to many sensors and actuators

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

a summation formula and ramanujan type series abstract using some properties of the general rising shifted factorial and the gamma function we derive a variant form of dougall s f 4 5 summation for the classical hypergeometric functions this variant form allows us to derive easily many ramanujan type series for 1 and ramanujan type series for some other constants

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

hypergeometric series summations and π formulas abstract by two slater s hypergeometric series identities we establish two general summation formulas with six free parameters specializing certain parameters in these formulas we obtain a list of ramanujan type series formulas for 2 and 3

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

a discontinuous galerkin time stepping scheme for the velocity tracking problem the velocity tracking problem for the evolutionary navier stokes equations in two dimensions is studied the controls are of distributed type and are submitted to bound constraints first and second order necessary and sufficient conditions are proved a fully discrete scheme based on the discontinuous in time galerkin approach combined with conforming finite element subspaces in space is proposed and analyzed provided that the time and space discretization parameters tau and h respectively satisfy tau leq ch 2 then l 2 error estimates of order o h are proved for the difference between the locally optimal controls and their discrete approximations

analysis and approximation of the velocity tracking problem for navier stokes flows with distributed control we consider the mathematical formulation analysis and the numerical solution of a time dependent optimal control problem associated with the tracking of the velocity of a navier stokes flow in a bounded two dimensional domain through the adjustment of a distributed control the existence of optimal solutions is proved and the first order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls semidiscrete in time and fully discrete space time approximations are defined and their convergence to the exact optimal solutions is shown a gradient method for the solution of the fully discrete equations is examined as are its convergence properties finally the results of some illustrative computational experiments are presented

a forward approach to numerical data assimilation variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena most notably in weather prediction and are usually solved via an ill posed optimal control problem for the initial state at the time of the first available measurements an alternative forward approach focuses on computation of the final state after this interval which is just as suitable for prediction purposes and is well posed without additional regularization specifically it is possible to compute projections of the unknown final state on all elements of an orthonormal basis which theoretically allows for the complete reconstruction of the final state in this paper an efficient numerical method for linear evolution equations of diffusive type is presented and convergence of the numerical approximation based on a discontinuous galerkin discretization is proved the key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition numerical examples for a scalar convection diffusion equation in two and three dimensions show the effectiveness of the method