almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

a numerical solution for non classical parabolic problem based on chebyshev spectral collocation method a numerical method based on chebyshev polynomials and local interpolating functions is proposed for solving the one dimensional parabolic partial differential equation subject to non classical conditions to assess its validity and accuracy the method is applied to solve several test problem

a survey of exact algorithms for the simple assembly line balancing problem in this survey paper we discuss the development of the simple assembly line balancing problem salbp modifications and generalizations over time present alternate 0 1 programming formulations and a general integer programming formulation of the problem discuss other well known problems related to salbp describe and comment on a number of exact i e optimum seeking methods and present a summary of the reported computational experiences all models discussed here are deterministic i e all input parameters are assumed to be known with certainty and all the algorithms discussed are exact the problem is termed simple in the sense that no mixed models subassembly lines zoning restrictions etc are considered due to the richness of the literature we exclude from discussion here a the inexact i e heuristic approximate algorithms for salpb and b the algorithms for the general assembly line balancing problem including the stochastic models

nonlocal reactive transport with physical and chemical heterogeneity localization errors the origin of nonlocality in macroscale models for subsurface chemical transport is illustrated it is argued that media that are either nonperiodic e g media with evolving heterogeneity or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean a metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal the intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations deng et al 1993 present a first order nonlocal eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions hu et al this issue extended these results to account for nonequilibrium linear sorption with random partition coefficient kd but deterministic constant reaction rate kr these theories are localized herein and comparisons are made between the fully nonlocal fnl nonlocal in time nlt and fully localized fl theories for conservative transport there is little difference between the first order fl and fnl models for spatial moments up to and including the third however for conservative transport the first order nlt model differs significantly from the fnl model in the third spatial moments for reactive transport all spatial moments differ between the fnl and fl models the second transverse horizontal and third longitudinal horizontal moments for the nlt model differ from the fnl model these results suggest that localized first order transport models for conservative tracers are reasonable if only lower order moments are desired however when the chemical reacts with its environment the localization approximation can lead to significant error in all moments and a fnl model will in general be required for accurate simulation

capacitated vehicle routing on trees t v e is a tree with nonnegative weights associated with each of its vertices a fleet of vehicles of capacity q is located at the depot represented by vertex v1 the capacitated vehicle routing problem on trees tcvrp consists of determining vehicle collection routes starting and ending at the depot such that the weight associated with any given vertex is collected by exactly one vehicle the sum of all weights collected by a vehicle does not exceed q a linear combination of the number of vehicles and of the total distance traveled by these vehicles is minimized the tcvrp is shown to be np hard this paper presents lower bounds for the tcvrp based on the solutions of associated bin packing problems we develop a linear time heuristic upper bound procedure and present a bound on its worst case performance these lower and upper bounds are then embedded in an enumerative algorithm numerical results follow

the solution of a parabolic differential equation with non local boundary conditions in the reproducing kernel space in this paper a general technique is proposed for solving the solution of a parabolic differential equation with integral boundary condition in the reproducing kernel space the representation of solution for the parabolic differential equation with integral boundary condition is given the solution is given by the form of series and its approximate solution is obtained by truncating the series numerical results show that the method employed in the paper is valid it is worthy to note that the method used in the paper can be generalized to solving linear ordinary or partial differential equation with initial condition and boundary conditions

numerical solution of a non classical parabolic problem an integro differential approach a numerical method based on an integro differential formulation and approximation by local interpolating functions is proposed for solving a one dimensional parabolic partial differential equation subject to non classical conditions some specific test problems are solved using the proposed method numerical results obtained indicate that it can give accurate solutions and that it is an interesting and viable alternative to existing numerical methods for solving the class of problems under consideration

branch and bound algorithms for the multi product assembly line balancing problem abstract this paper considers a flexible manufacturing system for several products each requiring a number of sequential nonpreemptive tasks some of which may be common to many products work stations are capable of handling any task the objective is to minimize the number of work stations necessary to manufacture all products so that tasks are performed in the proper order and so that all tasks assigned to any given station can be executed within a prescribed time frame a new branch and bound approach for this problem is presented it contains two main improvements over previously published tree search algorithms an improved lower bounding procedure and a more powerful partitioning scheme the algorithm can be used either as a heuristic in its truncated version or as an exact method in its full version tests performed on randomly generated problems confirm the effectiveness of the proposed improvements

fully implicit finite differences methods for two dimensional diffusion with a non local boundary condition three new fully implicit methods which are based on the 5 5 crank nicolson method the 5 5 n h noye hayman implicit method and the 9 9 n h implicit method are developed for solving the heat equation in two dimensional space with non local boundary conditions the latter is fourth order while the others are second order while the implicit methods developed here like the scheme based on the standard implicit backward time centered space btcs method use a large amount of central processor cpu time the high accuracy of the new fourth order fully implicit scheme is significant like the btcs method the new methods are also unconditionally stable

paper structural synthesis of multivariable controllers in this paper the problem of achieving a desired transfer function matrix h d s between external inputs and controlled outputs in a linear multivariable system by connecting proper stabilizing controllers between measured outputs and control inputs is solved in both transfer function and state space settings the class h d of achievable transfer functions is directly and constructively characterized via the theory of transfer function valuations for each h d s eh d the class of synthesizing controllers is determined similar valuation conditions are given for the asymptotic tracking and disturbance rejection problem in which h d s is only partially specified these results extend and complement earlier results the state space geometric solution of the problem of achieving a desired h d s is obtained by formulating it as an equivalent output feedback disturbance rejection problem a constructive solvability condition in terms of a pair of measurement and control invariant subspaces is given this requires a nontrivial generalization of the notion of c a b pairs an h d s is shown to be admissible if and only if it induces an appropriate pair of invariant subspaces the signal flow structure and certain factorizability conditions for a robust synthesis of the output feedback disturbance rejection problem are also given which extend earlier results on robust state feedback disturbance rejection it is shown that every compensator corresponds to a state feedback control law implemented by an unknown input observer the results of this paper are expected to be useful in the development of parameter optimization or other computer aided design algorithms where response specifications are to be traded against other design criteria such as sensitivity or stability margins since they explicitly characterize the algebraic variety h d in which the response transfer function may lie

transfer function conditions for c a b pairs state space conditions for the solution of the output feedback disturbance rejection problem have been given in a recent paper 1 in terms of c a b pairs this paper develops constructive and simple solvability conditions for this problem in terms of the rational matrix equation g s x s n s h s

exact and approximation algorithms for the min max k traveling salesmen problem on a tree abstract given k identical salesmen where k 2 is a constant independent of the input size the min max k traveling salesmen problem on a tree is to determine a set of k tours for the salesmen to serve all customers that are located on a tree shaped network so that each tour starts from and returns to the root of the tree with the maximum total edge weight of the tours minimized the problem is known to be np hard even when k 2 in this paper we have developed a pseudo polynomial time exact algorithm for this problem with any constant k 2 closing a question that has remained open for a decade along with this we have further developed a 1 approximation algorithm for any 0

the operational matrices of bernstein polynomials for solving the parabolic equation subject to specification of the mass abstract some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications in this paper a numerical method which employs the bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions the properties of bernstein polynomials and the operational matrices for integration differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the new technique

a spline collocation approach for a generalized parabolic problem subject to non classical conditions abstract a numerical method based on finite differences and spline collocation is proposed for solving a generalized parabolic problem subject to non classical conditions it is shown numerically that this scheme has second order convergence the stability conditions relating the time step size to mesh sizes were examined through numerical examples it turns out that the stability of the scheme in certain cases is subjected to a stability condition and depends on the nature of the problem a number of specific test problems are solved to assess the efficiency and performance of the method the numerical results obtained indicate that the approach is viable and yields accurate solutions when compared with the existing closed form solutions and generates results consistent with recent numerical approaches

solving rational matrix equations in the state space with applications to computer aided control system design a method of solving a class of linear matrix equations over various rings is proposed using results from linear geometric control theory an algorithm successfully implemented is presented along with non trivial numerical examples applications of the method to the algebraic control system design methodology are discussed

a class of non linear non classical parabolic equations abstract in this paper we deal with a class of non classical parabolic equations which arise from some practical problems such as the identification of unknown parameters free boundary problems etc one of the characteristics of this kind of equation is that the maximum principle is no longer valid we will first study the local solvability for a general equation of this kind via an auxiliary problem and the schauder fixed point principle then some a priori estimates are derived to establish the global existence of the solution moreover uniqueness and continuous dependence of the solution are proved finally some examples are given to illustrate the applications of the theory

solving parabolic integro differential equations by an explicit integration method vasudeva murthy a s and j g verwer solving parabolic integro differential equations by an explicit integration method journal of computational and applied mathematics 39 1992 121 132 the subject of this note is the numerical integration in time of nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain the integral terms may appear in the boundary conditions and or in the governing partial differential equation itself these terms generally complicate the application of standard methods our purpose here is to draw attention to an explicit method the runge kutta chebyshev method whose application remains straightforward in many cases of practical interest we discuss two numerical examples to illustrate this a comparison is made respectively with the implicit bdf code dassl and with the keller box scheme

a galerkin procedure for the diffusion equation subject to the specification of mass continuous and discrete galerkin procedures are derived and analyzed for the numerical solution of the diffusion equation u 1 u xx 0 x 1 0 t leqq t u x 0 f x u x 1 t g t and the specification of the mass int 0 b u x t dx m t 0 b 1 some numerical experiments are also presented

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

computing the asymptotic worst case of bin packing lower bounds abstract this paper addresses the issue of computing the asymptotic worst case of lower bounds for the bin packing problem we introduce a general result that allows to bound the asymptotic worst case performance of any lower bound for the problem and to derive for the first time the asymptotic worst case of the well known bound l 3 by martello and toth we also show that the general result allows to easily derive the asymptotic worst case of several lower bounds proposed in the literature

the three dimensional bin packing problem the problem addressed in this paper is that of orthogonally packing a given set of rectangular shaped items into the minimum number of three dimensional rectangular bins the problem is strongly np hard and extremely difficult to solve in practice lower bounds are discussed and it is proved that the asymptotic worst case performance ratio of the continuous lower bound is an exact algorithm for filling a single bin is developed leading to the definition of an exact branch and bound algorithm for the three dimensional bin packing problem which also incorporates original approximation algorithms extensive computational results involving instances with up to 90 items are presented it is shown that many instances can be solved to optimality within a reasonable time limit

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

diffusion subject to the specification of mass abstract existence uniqueness and continuous dependence upon the data are demonstrated for the solution u u x t of the diffusion equation ut uxx s x t 0 u x 0 x 0 ux 1 t g t 0 0 b 0 x dx m 0 a numerical procedure is discussed and results of numerical experiments are presented

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

the mixed and multi model line balancing problem a comparison abstract the balancing problem deals with the assignment of tasks to work stations we can distinguish two approaches in the literature on the mixed model line balancing problem that both transform this problem into a single model line balancing problem these approaches use combined precedence diagrams and adjusted task processing times respectively an experiment was carried out to compare several heuristics based on the combined precedence diagram a new optimisation method has been developed the results indicate that the position of common tasks in the precedence diagram of the different models has a significant effect on both the cpu time and the unequal distribution of the total work content of single models among work stations moreover good solutions with respect to the number of required stations go together with long cpu times for several instances we decreased the cpu times considerably without deteriorating the performance of the methods by using a reversed combined precedence diagram

conditioned invariant subspaces disturbance decoupling and solutions of rational matrix equations conditioned invariant subspaces are introduced both in terms of output injection and in terms of state estimation various properties of these subspaces are explored and the problem of disturbance decoupling by output injection oip is defined it is then shown that oip is equivalent to the problem of disturbance decoupled estimation as introduced in willems 1982 and willems and commault 1980 both solvability conditions and a description of solutions for a class of rational matrix equations of the form x s m s q s on several ways are given in state space form finally the problem of output stabilization with respect to a disturbance is briefly addressed

new lower bounds for the three dimensional orthogonal bin packing problem abstract in this paper we consider the three dimensional orthogonal bin packing problem which is a generalization of the well known bin packing problem we present new lower bounds for the problem from a combinatorial point of view and demonstrate that they theoretically dominate all previous results from the literature the comparison is also done concerning asymptotic worst case performance ratios the new lower bounds can be more efficiently computed in polynomial time in addition we study the non oriented model which allows items to be rotated

the boundary penalty method for the diffusion equation subject to the specification of mass in this paper we concern with a new boundary penalty finite element method for the one dimensional time dependent diffusion equation subject to the specification of mass instead of solving the diffusion equation directly we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi discrete and fully discrete approximations when the penalty parameter e is sufficiently small the numerical experiments show that our method is very effective and supports our analysis