on the boundedness and continuity of the spectral factorization mapping it is known that the spectral factorization mapping is bounded but not continuous on l infty the space of essentially bounded measurable functions on the unit circle in this article we study the spectral factorization mapping on decomposing banach algebras the most important example of a decomposing banach algebra is the wiener algebra the space of all absolutely convergent fourier series it is shown that the spectral factorization mapping is locally lipschitz continuous but not bounded on all decomposing banach algebras in consideration an application is given to the construction of approximate normalized coprime factorization

a connection between normalized coprime factorizations and linear quadratic regulator theory given a transfer matrix described by a minimal state space triple a method is given for computing state space realizations for the numerator and denominator of a normalized stable right coprime factorization for the transfer matrix the method involves the solution of an algebraic riccati equation it allows the use of existing computational state space algorithms in finding normalized stable right coprime factorizations and avoids explicit calculations of spectral factors

controllability and observability of linear retarded systems in banach spaces abstract controllability and observability are studied for linear retarded systems in a banach space x both the x exact and the x approximate controllability are considered and a number of necessary and sufficient conditions in terms of the coefficient operators and or the fundamental solution are presented various corresponding concepts for observability are introduced and the duality between controllability and observability is clarified an applicable condition for the x approximate controllability and the x observability is established under the assumptions that the control is a finite dimensional one and the system of generalized eigenspaces is projectively complete the condition is stated in terms of eigenvectors and controllers

a cubic b spline semi analytical algorithm for simulation of 3d steady state convection diffusion reaction problems abstract in this work a new cubic b spline based semi analytical algorithm is presented for solving 3d anisotropic convection diffusion reaction cdr problems in the inhomogeneous medium the mathematical model is expressed by the quasi linear second order elliptic partial differential equations epde with mixed derivatives and variable coefficients the final approximation is obtained as a sum of the rough primary solution and the modified spline interpolants with free parameters the primary solution mathematically satisfies boundary conditions thus the free parameters of interpolants are chosen to satisfy the governing equation in the solution domain the numerical examples demonstrate the high accuracy of the proposed method in solving 3d cdr problems in single and multi connected domains

boundary optimal lq control of coupled hyperbolic pdes and odes this contribution addresses the development of a linear quadratic lq regulator for a set of hyperbolic pdes coupled with a set of odes through the boundary the approach is based on an infinite dimensional hilbert state space description of the system and the well known operator riccati equation ore in order to solve the optimal control problem the ore is converted to a set of matrix riccati equations the feedback operator is found by solving the resulting matrix riccati equations the performance of the designed control policy is assessed by applying it to a system of interconnected continuous stirred tank reactor cstr and a plug flow reactor pfr through a numerical simulation

generating alternative mixed integer programming models using variable redefinition dropping the complicating constraints in a mixed integer linear program often yields a special structure subproblem that can be reformulated using a different set of decision variables once the new variables have been identified the entire problem can be reformulated in terms of the new variables we develop a theory of variable redefinition based on relating the two sets of decision variables by a linear transformation and describe methods for reformulating the special structure problem the reformulated models have a more accurate linear relaxation than the problems from which they were derived an important property within the context of linear programming based branch and bound modeling approaches

robust stabilization bibo stability distance notions and robustness optimization abstract this paper studies robust stabilization of both linear shift invariant causal systems in an lp setting and linear time invariant causal continuous time systems in an lp p 1 or setting two key technical results in the paper establish the existence of lp and lp stable normalized coprime factorizations for discrete time and continuous time systems respectively which have coprime factorizations as lp and lp stable operators several distance measures for systems are then introduced including the graph metric the function the gap between the graphs of the systems and the projection gap it is shown that these distance measures lead to the weakest convergence notions for systems for which closed loop stability is a robust property the function can be computed using the dahleh pearson theory for l1 l1 optimal control robustness optimization in a directed p function is shown to be closely related to robustness optimization for bibo stable normalized coprime factor perturbations this result connects the stability margin of dahleh for coprime factor perturbations to the function these considerations are further supported by a robustness result in terms of the projection gap

brief papers new results in state estimation and regulation an alternative technique to design linear state estimators and regulators is presented this technique is based on transfer matrix considerations the optimal regulator or estimator gain is obtained via spectral factorization and the solution of a simple equation in polynomial matrices this approach provides further insight displays the duality of estimation and control nicely and bridges the state space and frequency domain techniques the resulting design procedure is computationally attractive and particularly simple for system matrices in the observer or controller canonical form

comparison of preconditioning techniques for solving linear systems arising from the fourth order approximation of the three dimensional elliptic equation in this article we introduce a suitable preconditioner for the class of fourth order approximations employed for solving the elliptic partial differential equations auxx buyy cuzz f x y z u ux uy uz where a b and c are constants numerical results are given to show the efficiency of the presented approach

robustness analysis for feedback interconnections of distributed systems via integral quadratic constraints a framework is established for directly accommodating feedback interconnections of unstable distributed parameter transfer functions in robust stability analysis via integral quadratic constraints iqcs this involves transfer function homotopies that are continuous in a gap metric sense as such the development includes the extension of gap metric concepts to an irrational setting and the study of uncertainty set connectedness in these terms the main iqc based robust stability result is established for constantly proper transfer functions in the callier desoer algebra i e finitely many unstable poles and a constant limit at infinity problems of structured robust stability analysis and robust performance analysis are considered to illustrate use of the main result several numerical examples are also presented these include stability analysis of an autonomous system with uncertain time delay and a closed loop control system accounting for both the gain and phase characteristics of the distributed parameter uncertainty associated with the nominal rational plant model used for controller synthesis

spectral factorization and lq optimal regulation for multivariable distributed systems a necessary and sufficient condition is proved for the existence of a bistable spectral factor with entries in the distributed proper stable transfer function algebra x1d49c in the context of distributed multivariable convolution systems with no delays a by product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system with entries in the algebra of fractions over x1d49c we next study semigroup state space systems sgb with bounded sensing and control having a transfer function with entries in and consider its standard lq optimal regulation problem having an optimal state feedback operator k0 for a system sgb a formula is given relating any spectral factor of a transfer function coprime fraction power spectral density to k0 a by product is the description of any normalized coprime fraction of the transfer function in terms of k0 finally we describe an alternative way of finding the solution operator k0 of the lq problem using spectral factor

skew hermitian based iterations for nine point approximations of convection diffusion problems we study skew hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high order compact hoc approximations of two dimensional convection diffusion problems such discretisations lead to nine point system matrices with block tridiagonal structures the nonsymmetric discretisation matrix for a constant coefficient grid aligned flow problem is shown to be positive definite and thus hermitian and skew hermitian splitting hss methods can be considered for the solution of the corresponding linear system for the solution of this linear system using the hss iteration we derive an analytical expression for the optimal value for the upper bound of the contraction factor comparison of the hss iteration with triangular and skew hermitian splitting tss methods and their corresponding block variants btss are carried out for various test problems cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given our results show that the combination of high order compact discretisations and skew hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two dimensional convection diffusion equations

on discrete spectral factorizations a unified approach state space formulas for all key spectral factorizations appearing in the discrete time h sup 2 h sup infinity optimization are summarized the factorization problems are categorized into three groups solutions are constructed by finding special coprime factors of a given transfer matrix using the associated discrete algebraic riccati equation solution procedures for the three groups are in general the same which may lead to yield to a unified approach

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

multigrid solution of high order discretisation for three dimensional biharmonic equation with dirichlet boundary conditions of second kind abstract in this paper we propose two compact finite difference approximations for three dimensional biharmonic equation with dirichlet boundary conditions of second kind in these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques the unknown solution and its second derivatives are carried as unknowns at grid points we derive second order and fourth order approximations on a 27 point compact stencil classical iteration methods such as gauss seidel and sor for solving the linear system arising from the second order and fourth order discretisation suffer from slow convergence in order to overcome this problem we use multigrid method which exhibit grid independent convergence and solve the linear system of equations in small amount of computer time the fourth order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions

analytic controllability of time dependent quantum control systems the question of controllability is investigated for a quantum control system in which the hamiltonian operator components carry explicit time dependence which is not under the control of an external agent we consider the general situation in which the state moves in an infinite dimensional hilbert space a drift term is present and the operators driving the state evolution may be unbounded however considerations are restricted by the assumption that there exists an analytic domain dense in the state space on which solutions of the controlled schrodinger equation may be expressed globally in exponential form the issue of controllability then naturally focuses on the ability to steer the quantum state on a finite dimensional submanifold of the unit sphere in hilbert space and thus on analytic controllability a relatively straightforward strategy allows the extension of lie algebraic conditions for strong analytic controllability derived earlier for the simpler time independent system in which the d

lq optimal control of infinite dimensional systems by spectral factorization abstract we consider the standard lq optimal control of an exponentially stabilizable and detectable infinite dimensional semigroup state space system with bounded sensing and control it is reported that the reachable restriction of the lq optimal state feedback operator can be identified 1 by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper stable transfer function algebra and 2 by obtaining any constant solution of an operator diophantine equation over the properties of the restricted solution feedback operator are discussed it is shown in particular that the lq optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable these theoretical results are applied to a simple model of heat diffusion with collocated sensor and actuator leading to an approximation procedure converging exponentially fast to the lq optimal state feedback operator this procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus

robust stabilizability of normalized coprime factors the infinite dimensional case the problem of robustly stabilizing a linear system subject to h bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite dimensional systems this class has possible unbounded finite rank input and output operators which include many delay and distributed systems the optimal stability margin is expressed in terms of the solutions of the control and filter algebraic riccati equations

optimal linear quadratic control of coupled parabolic hyperbolic pdes abstractthis paper focuses on the optimal control design for a system of coupled parabolic hypebolic partial differential equations by using the infinite dimensional state space description and the corresponding operator riccati equation some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear quadratic lq optimal control problem a state lq feedback operator is computed by solving the operator riccati equation which is converted into a set of algebraic and differential riccati equations thanks to the eigenvalues and the eigenvectors of the parabolic operator the results are applied to a non isothermal packed bed catalytic reactor the lq optimal controller designed in the early portion of the paper is implemented for the original nonlinear model numerical simulations are performed to show the controller performances

optimal lq feedback regulation of a nonisothermal plug flow reactor model by spectral factorization the linear quadratic lq optimal temperature and reactant concentration regulation problem is studied for a partial differential equation model of a nonisothermal plug flow tubular reactor by using a nonlinear infinite dimensional hilbert state space description first the dynamical properties of the linearized model around a constant temperature equilibrium profile along the reactor are studied it is shown that it is exponentially stable and approximately reachable next the general concept of lq feedback is introduced it turns out that any lq feedback is optimal from the input output viewpoint and stabilizing for the plug flow reactor linearized model a state lq feedback operator is computed via the solution of a matrix riccati differential equation mrde in the space variable thanks to the reachability property the computed lq feedback is actually the optimal one then the latter is applied to the nonlinear model and the resulting closed loop system dynamical performances are analyzed a criterion is given which guarantees that the constant temperature equilibrium profile is an asymptotically stable equilibrium of the closed loop system moreover under the same criterion it is shown that the control law designed previously is optimal along the nonlinear closed loop model with respect to some cost criterion the results are illustrated by some numerical simulations

an implementation of the look ahead lanczos algorithm for non hermitian matrices the nonsymmetric lanczos method can be used to compute eigenvalues of large sparse non hermitian matrices or to solve large sparse non hermitian linear systems however the original lanczos algorithm is susceptible to possible breakdowns and potential instabilities an implementation of a look ahead version of the lanczos algorithm is presented that except for the very special situation of an incurable breakdown overcomes these problems by skipping over those steps in which a breakdown or near breakdown would occur in the standard process the proposed algorithm can handle look ahead steps of any length and requires the same number of matrix vector products and inner products as the standard lanczos process without look ahead

continuity of the spectral factorization on a vertical strip the continuity of the mapping which associates a spectral factor to a spectral density is investigated this mapping can be defined on several classes of spectral densities and spectral factors for the usual largest class of spectral densities i e essential bounded functions on the imaginary axis that are bounded away from zero it is known that this mapping is not continuous it is shown here that for slightly smaller but still generic class the mapping becomes continuous

a simple form for the fourth order difference method for 3 d elliptic equations abstract in 1992 jain et al m k jain r k jain r k mohanty fourth order finite difference method for three dimensional elliptic equations with nonlinear first derivative terms numer meth part differ equat 8 1992 575 591 proposed a fourth order finite difference scheme for the 3 d elliptic equation in this paper we present a simple and new form of 19 point fourth order difference method for the nonlinear second order 3 d elliptic difference equation au xx bu yy cu zz f x y z u u x u y u z where a b and c are constants on a cubic region w subject to the dirichlet boundary conditions on w

brief paper lq control design of a class of hyperbolic pde systems application to fixed bed reactor a general linear controller design method for a class of hyperbolic linear partial differential equation pdes systems is presented this is achieved by using an infinite dimensional hilbert state space description with infinite dimensional distributed input and output a state lq feedback operator is computed via the solution of a matrix riccati differential equation in the space variable the proposed method is applied to a fixed bed chemical reactor control problem where one elementary reaction takes place an optimal controller is designed for linearized fixed bed reactor model it is applied to the original nonlinear model and the resulting closed loop stability is analyzed numerical simulations are performed to show the performance of the designed controller

distributed system transfer functions of exponential order o is a real number we construct a transfer function algebra of fractions viz f o for modelling possibly unstable distributed systems such that i fcirc in f o is holomorphic in re s o i e is o stable iff fcirc is o exponentially stable and ii we allow delay in the direct input output transmission of the system this algebra is a a restriction of the algebra b o developed by callier and desoer 1978 1980 a b an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained the algebra f o can be used for modelling and feedback system design it is shown that standard semigroup systems are better modelled by a transfer function in f o rather than b o

preconditioned techniques for solving large sparse linear systems arising from the discretization of the elliptic partial differential equations abstract in this paper we use the bicg bicgstab methods as preconditioned techniques also we compare the preconditioned krylov subspace methods such as gmres gmres m qmr bicg cgs bicgstab for solving linear systems arising from a class of fourth order approximations for solving the elliptic partial differential equation au xx bu yy f x y u u x u y where a and b are constants numerical results are given to show the efficiency of the proposed preconditioned bicgstab method

spectral factorization by symmetric extraction for distributed parameter systems the spectral factorization problem of a scalar coercive spectral density is considered in the framework of the callier desoer algebra of distributed parameter system transfer functions criteria are given for the infinite product representation of a meromorphic coercive spectral density of finite order and for the convergence of infinite product representations of spectral factors i e for the convergence of the symmetric extraction method for solving the spectral factorization problem of such spectral density these convergence criteria are applied to the solution of the linear quadratic optimal control problem by spectral factorization for a specific class of semigroup hilbert state space systems with a riesz spectral generator the speed of convergence of the symmetric extraction method is also considered as an example a damped vibrating string model is handled

lq boundary control of a diffusion convection reaction system in this work the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations pdes with spatially varying coefficients is studied an infinite dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model the evolutionary model which incorporates the spatially varying coefficients of the underlying set of the pdes is used for subsequent linear quadratic regulator synthesis the formulated linear quadratic state feedback controller is applied to a nonlinear model of the reactor and its performance is studied

necessary and sufficient conditions for j spectral factorizations with a j lossless property for infinite dimensional systems in continuous and discrete time necessary and sufficient conditions for j spectral factorizations are given in terms of the existence of a self adjoint stabilizing solution of an appropriate riccati equation in infinite dimensions furthermore it is shown that a certain j lossless property holds if and only if the stabilizing solution to the riccati equation is nonnegative definite we prove these results first for the continuous time case and thereby derive analogous results for the discrete time case by means of a bilinear transformation of the complex plane

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

linear state space systems in infinite dimensional space the role and characterization of joint stabilizability detectability several fundamental results from the theory of linear state space systems in finite dimensional space are extended to encompass a class of linear state space systems in infinite dimensional space the results treated are those pertaining to the relationship between input output and internal stability the problem of dynamic output feedback stabilization and the concept of joint stabilizability detectability a complete structural characterization of jointly stabilizable detectable systems is obtained the generalized theory applies to a large class of linear state space systems assuming only that i the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators ii the state space is hilbert space iii the input and output spaces are finite dimensional and iv the sensing and control operators are bounded general conclusions regarding the fundamental structure of control theoretic problems in infinite dimensional space can be drawn from these results

nehari problems and equalizing vectors for infinite dimensional systems for a class of infinite dimensional systems we obtain a simple frequency domain solution for the suboptimal nehari extension problem the approach is via j spectral factorization and it uses the concept of an equalizing vector moreover the connection between the equalizing vectors and the nehari extension problem is given

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

analytic system problems and j lossless coprime factorization for infinite dimensional linear systems abstract this paper extends the coprime factorization approach to the synthesis of internally stabilizing controllers satisfying an h norm bound to a class of systems with irrational transfer matrices using the coprime factorization description the h control problem can be reduced to two stable analytic system problems such problems have solutions if and only if a certain j lossless factorization exists the full h synthesis problem is shown to be equivalent to the solution of two nested j lossless factorizations if the irrational transfer matrix has a state space realization then the known state space formulas for the h control problem may be recovered using the relationship between j lossless factorizations and solutions of riccati equations however the results derived here are valid for a larger class of infinite dimensional systems

reconciling nu gap metric and iqc based robust stability analysis this technical note elaborates on the flexibility of an approach that combines gap metric and integral quadratic constraint iqc based analysis in the study of uncertain feedback interconnections of distributed parameter transfer functions it is established that a standard gap ball robust stability result can be recovered within the blended iqc gap framework which only requires the existence of gap continuous paths within the uncertainty set of interest this is achieved in part by showing that sufficiently small gap balls are pathwise connected in the graph topology a linear fractional characterisation of the gap is a key ingredient this characterisation is underpinned by a certain j spectral factorisation also shown to exist herein

approximation and stabilization of distributed systems by lumped systems in this note we study two questions regarding linear time invariant distributed systems namely 1 when can the given system be approximated by a lumped system 2 when can the given system be stabilized by a lumped system in each case we obtain a complete answer to the question in terms of the behaviour of the purely atomic parts of the stable numerator and denominator of the plant

a rbf based technique for 3d convection diffusion reaction problems in an anisotropic inhomogeneous medium abstract we present a rbf based semi analytical technique for solving 3d convection diffusion reaction cdr equations to model transport in an anisotropic inhomogeneous medium the mathematical model is expressed as the boundary value problem for elliptic partial differential equation epde main feature of the presented technique is the separately satisfaction of the conditions on the boundary of the domain and the epde inside to be more precise we transform the original epde to the equation with homogeneous boundary condition bc and seek the approximate solution as a sum of the modified rbfs mrbfs the mrbfs satisfy the homogeneous bc of the problem so any linear combination also satisfies the homogeneous bc the rbfs of three types are used in the framework of the method the multiquadric mq rbf the gaussian rbf and the conical one the coefficients of the linear combination are determined so that it satisfies the governing equation of the epde ten numerical examples demonstrate the high effectiveness of the presented technique in solving 3d cdr problems in single and double connected domains