the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

feasible and noninterior path following in constrained minimization with low multiplier regularity primal dual path following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed regularity properties of the path are proved the path structure allows us to define approximating models which are used for controlling the path parameter in an iterative process for computing a solution of the original problem the moreau yosida regularized subproblems of the new path following technique are solved efficiently by semismooth newton methods the overall algorithmic concept is provided and numerical tests including a comparison with primal dual path following interior point methods for state constrained optimal control problems show the efficiency of the new concept

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

improving order and efficiency composition with a modified newton s method in this paper a zero finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented the key idea in deriving this procedure is to compose a given iterative method with a modified newton s method that introduces just one evaluation of the function to carry out this procedure some classical methods with different orders of convergence are used to obtain new methods that can be generalized in banach spaces

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

on chebyshev halley methods with sixth order convergence for solving non linear equations abstract in this paper we present a family of new variants of chebyshev halley methods the new methods have sixth order convergence although they only add one evaluation of the function at the point iterated by chebyshev halley methods the numerical results presented show that the new methods work better not only in order but in efficiency

primal dual interior point methods for pde constrained optimization this paper provides a detailed analysis of a primal dual interior point method for pde constrained optimization considered are optimal control problems with control constraints in l p it is shown that the developed primal dual interior point method converges globally and locally superlinearly not only the easier l setting is analyzed but also a more involved l q analysis q is presented in l the set of feasible controls contains interior points and the frechet differentiability of the perturbed optimality system can be shown in the l q setting which is highly relevant for pde constrained optimization these nice properties are no longer available nevertheless a convergence analysis is developed using refined techniques in parti cular two norm techniques and a smoothing step are required the l q analysis with smoothing step yields global linear and local superlinear convergence whereas the l analysis without smoothing step yields only global linear convergence

dynamics of a family of chebyshev halley type methods in this paper the dynamics of the chebyshev halley family is studied on quadratic polynomials a singular set that we call cat set appears in the parameter space associated to the family this set has interesting similarities with the mandelbrot set the parameter space has allowed us to find different elements of the family which have bad convergence properties since periodic orbits and attractive strange fixed points appear in the dynamical plane of the corresponding method

constrained dirichlet boundary control in l 2 for a class of evolution equations optimal dirichlet boundary control based on the very weak solution of a parabolic state equation is analyzed this approach allows us to consider the boundary controls in l 2 which has advantages over approaches which consider control in sobolev spaces involving fractional derivatives pointwise constraints on the boundary are incorporated by the primal dual active set strategy its global and local superlinear convergences are shown a discretization based on space time finite elements is proposed and numerical examples are included

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

sixth order variants of chebyshev halley methods for solving non linear equations abstract in this paper we present a family of new variants of chebyshev halley methods with sixth order convergence compared with chebyshev halley methods the new methods require one additional evaluation of the function the numerical results presented show that the new methods compete with chebyshev halley methods

modified chebyshev halley methods with sixth order convergence abstract in this paper we present a family of modified chebyshev halley methods the new methods add one evaluation of the function at the point iterated by chebyshev halley methods analysis of convergence shows that these methods have sixth order convergence numerical results show that the new methods can be of practical interest

a composite third order newton steffensen method for solving nonlinear equations in this paper we suggest a third order method formed by the composition of newton and steffensen methods for finding simple and real roots of a nonlinear equation in single variable per iteration the formula requires two evaluations of the function and single evaluation of the derivative experiments show that the method is suitable in the cases where newton and steffensen methods fail

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

semi smooth newton methods for state constrained optimal control problems abstract a regularized optimality system for state constrained optimal control problems is introduced and semi smooth newton methods for its solution are analyzed convergence of the regularized problems is proved numerical tests confirm the theoretical results and demonstrate the efficiency of the proposed methodology

state constrained optimal control of the three dimensional stationary navier stokes equations abstract in this paper an optimal control problem for the stationary navier stokes equations in the presence of state constraints is investigated existence of optimal solutions is proved and first order necessary conditions are derived the regularity of the adjoint state and the state constraint multiplier is also studied lipschitz stability of the optimal control state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated

two sided approximation for some newton s type methods abstract we suggest and analyze a combination of a damped newton s method and a simplified version of newton s one we show that the proposed iterations give two sided approximations of the solution which can be efficiently used as posterior estimations some numerical examples illustrate the efficiency and performance of the method proposed

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

zero finder methods derived from obreshkov s techniques in this paper two families of zero finding iterative methods for solving nonlinear equations f x 0 are presented the key idea to derive them is to solve an initial value problem applying obreshkov like techniques more explicitly obreshkov s methods have been used to numerically solve an initial value problem that involves the inverse of the function f that defines the equation carrying out this procedure several methods with different orders of local convergence have been obtained an analysis of the efficiency of these methods is given finally we introduce the concept of extrapolated computational order of convergence with the aim of numerically test the given methods a procedure for the implementation of an iterative method with an adaptive multi precision arithmetic is also presented

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

causal input output representation of 2d systems in the behavioral approach in this paper the concept of causal input output representation of two dimensional 2d systems in the behavioral approach is introduced these representations provide an interesting connection between classical 2d systems theory and 2d systems theory in the behavioral approach some characterizations of such representations are presented finally a technique that allows us to obtain causal input output representations is proposed

regularized state constrained boundary optimal control of the navier stokes equations abstract the numerical solution of the dirichlet boundary optimal control problem of the navier stokes equations in presence of pointwise state constraints is investigated two different regularization techniques are considered first a moreau yosida regularization of the problem is studied optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved a source representation of the control combined with a lavrentiev type regularization strategy is also presented the analysis concerning optimality conditions and convergence of the regularized solutions is carried out in the last part of the paper numerical experiments are presented for the numerical solution of each regularized problem a semi smooth newton method is applied

an optimal reconstruction of chebyshev halley type methods for nonlinear equations having multiple zeros abstract establishment of a new optimal higher order iterative scheme for multiple zeros with known multiplicity m 1 of univariate function is one of the hard and demanding task in the area of computational mathematics and numerical analysis we write this paper with the aim to propose a new higher order reconstruction of chebyshev halley type iteration functions for multiple zeros in a simple way minimum order of convergence of our scheme is seven and maximum reaches to eight without using any extra functional evaluation with regard to computational cost each member of our scheme needs four functional evaluations at each step so the maximum efficiency index of our scheme is 1 6818 for 1 in addition we thoroughly study the convergence analysis which confirms the theoretical order of convergence moreover we check the efficiency and compare them to the existing ones on four real life problems namely fractional conversion chemical engineering continuous stirred tank reactor and eigen value problems and two well known academic problems finally we confirm on the basis of obtained results that our methods have better residual error errors between two consecutive iterations and stable computational order of convergence as compared to the other existing ones

energy corrected fem for optimal dirichlet boundary control problems in the presence of re entrant corners the solution of pde constrained optimization problems in general has singular components even when the given data are smooth consequently the use of standard finite element discretization gives rise to the so called pollution effect this means that only a reduced convergence rate compared to the best approximation error is obtained we discuss how optimal convergence rates in weighted norms can be achieved using the idea of energy corrected finite element methods applied to optimal dirichlet boundary control problem in the energy space we present optimal error estimates in weighted norms for the state variable and for the control several numerical examples illustrate the obtained theoretical results

bilateral approximations for some aitken steffensen hermite type methods of order three abstract we study the local convergence of some aitken steffensen hermite type methods of order three we obtain that under some reasonable conditions on the monotony and convexity of the nonlinear function the iterations offer bilateral approximations for the solution which can be efficiently used as a posteriori estimations

a fourth order iterative method for solving nonlinear equations the present paper illustrates an iterative numerical method to solve nonlinear equations of the form f x 0 especially those containing the partial and non partial involvement of transcendental terms comparative analysis shows that the present method is faster than newton raphson method hybrid iteration method new hybrid iteration method and others cost is also found to be minimum than these methods the beauty in our method can be seen because of the optimization in important effecting factors i e lesser number of iteration steps lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods this work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative

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

primal dual active set strategy for a general class of constrained optimal control problems an algorithm for efficient solution of optimal control problems with pointwise constraints is introduced it is based on an active set strategy using primal as well as dual variables under certain assumptions a decrease of an appropriately chosen merit function and convergence of the algorithm are proved numerical examples for a parabolic optimal control problem demonstrate the efficiency of the proposed method for control constrained problems

a two step steffensen s method under modified convergence conditions a simplification of a third order iterative method is proposed the main advantage of this method is that it does not need to evaluate neither any frechet derivative nor any bilinear operator a semilocal convergence theorem in banach spaces under modified kantorovich conditions is analyzed a local convergence analysis is also performed finally some numerical results are presented

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

an improvement of the jarratt method in this paper we present a variant of jarratt method for solving non linear equations per iteration the new method adds the evaluation of the function at another point in the procedure iterated by jarratt method as a consequence the local order of convergence is improved from four for jarratt method to six for the new method the new multistep iteration scheme based on the new method is developed and numerical tests verifying the theory are also given

a family of fifth order iterations composed of newton and third order methods abstract in this paper we present a family of fifth order methods formed by the composition of newton and third order methods for solving non linear equations per iteration the new methods require two evaluations of the function one of its first derivative and one of its second derivative analysis of efficiency shows that the new methods can compete with newton s method and the classical third order methods numerical results show that the methods have definite practical utility

some variants of chebyshev halley methods with fifth order convergence abstract in this paper we present some new variants of chebyshev halley methods the local order of convergence is increased from 3 to 5 the new methods add one evaluation of the function at the point iterated by chebyshev halley methods the numerical results show that the new methods work better not only in order but in efficiency

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

an analysis of the properties of the variants of newton s method with third order convergence for the last five years the variants of the newton s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non linear equations these methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives in this paper we investigate about the relationship between these methods which are in fact based on the approximation of the second order derivative present in the third order limited taylor expansion we also prove that they are different forms of the halley method and are all contractive iterative methods in a common neighbourhood we extend some of these variants to multivariate cases and prove their respective local cubic convergence from their corresponding linear models

some modifications of newton s method with fifth order convergence in this paper we present some new modifications of newton s method for solving non linear equations analysis of convergence shows that these methods have order of convergence five numerical tests verifying the theory are given and based on these methods a class of new multistep iterations is developed

on a steffensen hermite method of order three in this paper we study a third order steffensen type method obtained by controlling the interpolation nodes in the hermite inverse interpolation polynomial of degree 2 we study the convergence of the iterative method and we provide new convergence conditions which lead to bilateral approximations for the solution it is known that the bilateral approximations have the advantage of offering a posteriori bounds of the errors the numerical examples confirm the advantage of considering these error bounds

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

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

some new sixth order methods for solving non linear equations abstract in this paper we present some new sixth order methods based on the chebyshev halley methods per iteration the new methods require an additional evaluation of the function compared with chebyshev halley methods analysis of efficiency shows that the new methods can compete with newton s method and the classical third order methods as we show in numerical results

trajectory control and interconnection of 1d and nd systems in this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems we show that for one dimensional linear time invariant systems the ability to obtain a given subsystem by regular interconnection a prerequisite for any feedback type structure is equivalent to the ability to drive any trajectory into that subsystem however in the case of multidimensional systems the former is a stronger property than the latter trajectory controllability can however be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables this leads as a by product to the notion of controlling a system by means of latent variables

causal and stable input output structures on multidimensional behaviors in this work we study multidimensional nd linear differential behaviors with a distinguished independent variable called time we define in a natural way causality and stability of input output structures with respect to this distinguished direction we make an extension of some results in the theory of partial differential equations demonstrating that causality is equivalent to a property of the transfer matrix which is essentially hyperbolicity of the p c operator defining the behavior cal b c 0 y we also quote results which in effect characterize time autonomy for the general systems case stability is likewise characterized by a property of the transfer matrix we prove this result for the 2d case and for the case of a single equation for the general case it requires solution of an open problem concerning the geometry of a particular set in mathbb c n in order to characterize input output stability we also develop new results on inclusions of kernels freeness of variables and closure with respect to cal s cal s prime and associated spaces which are of independent interest we also discuss stability of autonomous behaviors which we believe to be governed by a corresponding condition

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

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

state space realization of 2 d finite dimensional behaviours this paper deals with the state space realization of autonomous autoregressive two dimensional 2 d systems in the context of the behavioural approach an arbitrary autoregressive 2 d system sig

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

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

on a class of quadratically convergent iteration formulae in this paper a new class of iterative formulae having quadratic convergence is presented furthermore these algorithms are comparable to the well known method of newton and the computed results support this theory

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

an acceleration of newton s method super halley method from a study of the convexity we give an acceleration for newton s method and obtain a new third order method then we use this method for solving non linear equations in banach spaces establishing conditions on convergence existence and uniqueness of solution as well as error estimates

two dimensional proper rational matrices and causal input output representations of two dimensional behavioral systems the concept of proper rational matrix is strictly connected with the representation of causal transfer matrices in the two dimensional 2d case there is much freedom in defining proper rational matrices this freedom is connected to the fact that past and future in the 2d case can be determined by a 2d cone in this way the concept of rational matrix which is proper with respect to a cone can be introduced moreover an algorithm that checks the properness of a rational matrix is proposed finally this algorithm is used for determining all possible causal input output i o representations of a behavior given by a kernel representation

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

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

an improvement of the euler chebyshev iterative method abstract we present a new method for the computation of the solutions of nonlinear equations when it is necessary to use high precision we improve the euler chebyshev iterative method which is a generalization of an improvement of newton s method a symbolic computation allows us to find the best coefficients respect to the local order of convergence the adaptation of the strategy presented here gives an additional iteration function with an additional evaluation of the function it provides a lower cost if we use adaptive multi precision arithmetics the numerical results computed using this system with a floating point representing a maximum of 210 decimal digits support this theory

the primal dual active set method for nonlinear optimal control problems with bilateral constraints the primal dual active set method has proved to be an efficient numerical tool in the context of diverse applications so far it has been investigated mainly for linear problems this paper is devoted to the study of global convergence of the primal dual active set method for nonlinear problems with bilateral constraints utilizing the close relationship between the primal dual active set method and semismooth newton methods local superlinear convergence of the method is investigated as well