banach valued holomorphic functions on the maximal ideal space of h we study banach valued holomorphic functions defined on open subsets of the maximal ideal space of the banach algebra h of bounded holomorphic functions on the unit disk mathbb d subset mathbb c with pointwise multiplication and supremum norm in particular we establish vanishing cohomology for sheaves of germs of such functions and solving a banach valued corona problem for h prove that the maximal ideal space of the algebra h mathrm comp infty a of holomorphic functions on mathbb d with relatively compact images in a commutative unital complex banach algebra a is homeomorphic to the direct product of maximal ideal spaces of h and a

parametrization of all strictly causal stabilizing controllers within the framework of the factorization approach we present parametrizations of all strictly causal stabilizing controllers

on stabilization for discrete linear time varying systems abstract in this paper we consider the stabilization problem for discrete linear time varying systems in an operator theoretic framework by using the complete finiteness of a certain discrete nest algebra we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation this result extends the youla parametrization theorem by only requiring a strong left or a strong right representation but not both the strong stabilization problem is also discussed

optimal distributed controllers realizable over arbitrary networks in this paper we provide a general networked controller design methodology for networked plants and apply it to solve the optimal mathcal h 2 networked control problem both the plant and the controller are interconnected systems interacting over the same arbitrary directed network with noiseless and delay free communication links we introduce the notions of network implementability and network realizability and analyze the structure of network implementable and realizable systems based on the structural property of network implementable systems under certain network related constraints we characterize the set of all stabilizing controllers that are implementable over the given network using the state space version of youla parametrization moreover we provide a constructive procedure to implement the controllers as sub systems interacting over the given network without affecting the stability of the feedback networked system the distributed mathcal h 2 control problem is then cast as a convex optimization problem and its solution is shown to provide the optimal distributed controller over the given network in terms of its network interacting components the results of this paper allow one to apply many classical results and approaches of multi variable robust control theory to networked systems

spectral methods for tempered fractional differential equations in this paper we first introduce fractional integral spaces which possess some features i when 0 1 functions in these spaces are not required to be zero on the boundary ii the tempered fractional operators are equivalent to the riemann liouville operator in the sense of the norm spec tral galerkin and petrov galerkin methods for tempered fractional advection problems and tempered fractional diffusion problems can be developed as the classical spectral galerkin and petrov galerkin methods error analysis is provided and numerically confirmed for the tempered fractional advection and diffusion problems

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

feedback stabilization over commutative rings the matrix case this paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring for stabilizable plants certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other in the case of rings with irreducible spectrum this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs if the maxspectrum of the ring is noetherian and of zero krull dimension then this result shows that the stabilizable plants have doubly coprime fractions over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant certain invariants of these matrices known as elementary factors are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring this condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty a formula for the class of all stabilizing controllers is then developed that generalizes the previous well known formula in factorization theory for multidimensional transfer functions these results provide concrete conditions for stabilizabilty finally it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions

the determination of minimal projections and extensions in equations are derived which are shown to be necessary and sufficient for finite rank projections in l1 to be minimal more generally these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite dimensional subspace of l1 and thus may be viewed as an extension of the hahn banach theorem to higher dimensions in the l1 setting these equations are solved in terms of an l1 best approximation problem and the required orthogonality conditions moreover this solution has a simple geometric interpretation questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions including the minimal l1 projection onto the quadratics

on the stabilization of nonlinear systems we extend the applicability of the global q parametrization method of controller design to a large class of unstable nonlinear plants the main result is a two step compensation theorem analogous to that of zames for unstable linear plants if p l e2 rightarrow l e1 is a nonlinear possibly unstable plant and f 0 is any incrementally stable controller such that p 1 p i f 0 p 1 is incrementally stable then the class of controllers f which yields an f g stable closed loop system in the unity feedback configuration for p is globally parametrized by finite gain stable maps q l e1 rightarrow l e2 with f f 0 q i p 1 q 1

simultaneous controller design for time varying linear systems abstract in this paper we establish the parametrization of all simultaneously stabilizing controllers for two time varying linear systems a new approach to the simultaneous stabilization of three time varying linear systems is also provided

sufficient conditions for the projective freeness of banach algebras let r be a unital semi simple commutative complex banach algebra and let m r denote its maximal ideal space equipped with the gelfand topology sufficient topological conditions are given on m r for r to be a projective free ring that is a ring in which every finitely generated projective r module is free several examples are included notably the hardy algebra h x of bounded holomorphic functions on a riemann surface of finite type and also some algebras of stable transfer functions arising in control theory

on a general structure of the stabilizing controllers based on stable range in this paper we prove that some stabilizing controllers of a plant which admits a left right coprime factorization have a special form where their stable and unstable parts are separated the dimension of the unstable part depends on the algebraic concept of stable range of the ring a of siso stable plants moreover we prove that if the stable range of a is equal to 1 then every plant defined by a transfer matrix with entries in the quotient field of a and admitting a left right coprime factorization can be stabilized by a stable controller strong stabilization in particular using a result of treil proving that the stable range of h infty mathbb d is equal to 1 we show that every stabilizable plant defined by a transfer matrix with entries in the quotient field of h infty mathbb d or h infty mathbb c is strongly stabilizable and equivalently every couple of stabilizable plants can be simultaneously stabilized by a controller simultaneous stabilization finally using the fact that the topological stable range of h infty mathbb d is equal to 2 a result due to suarez we show that every unstabilizable siso plant defined by a transfer function with entries in the quotient field of h infty mathbb d is as close as we want to a stabilizable plant in the product topology

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

constrained traffic equilibrium in routing we study noncooperative routing in which each user is faced with a multicriterion optimization problem formulated as the minimization of one criterion subject to constraints on others we address the questions of existence and uniqueness of equilibrium we show that equilibria indeed exist but uniqueness may be destroyed due to the multicriteria nature of the problem we obtain uniqueness in some weaker sense under appropriate conditions we show that the link utilizations are uniquely determined at equilibrium we further study the normalized constrained equilibrium and apply it to pricing

robust regulation theory for transfer functions with a coprime factorization classical frequency domain results of robust regulation are extended by requiring only a right or a left coprime factorization of a plant but not both the famous internal model principle is generalized first which leads to a necessary and sufficient solvability condition of the robust regulation problem and to a parametrization of all robustly regulating controllers in addition a procedure for constructing robustly regulating controllers is proposed

controller parameterization of anantharam s example we consider the controller parameterization of anantharam s example 1985 it is shown that the parameterization is achieved by using only one parameter even though the plant does not have a coprime factorization

a stochastic differential equation model with jumps for fractional advection and dispersion the path of a tracer particle through a porous medium is typically modeled by a stochastic differential equation sde driven by brownian noise this model may not be adequate for highly heterogeneous media this paper extends the model to a general sde driven by a levy noise particle paths follow a markov process with long jumps their transition probability density solves a forward equation derived here via pseudo differential operator theory and fourier analysis in particular the sde with stable driving noise has a space fractional advection dispersion equation fade with variable coefficients as the forward equation this result provides a stochastic solution to anomalous diffusion models and a solid mathematical foundation for particle tracking codes already in use for fractional advection equations

mixedk functionals a measure of smoothness for blending type approximation we consider uniform approximation of bivariate functions f c 2 2 or f c i 2 i e continuous real valued functions which are 2 periodic with respect to both variables or defined on the compact square i 2 we are interested in quantitative results for blending type approximation more precisely we want to approximate f by continuous pseudo polynomials of degree m n we introduce mixed k functionals which will serve similar purposes as the k functionals in the univariate theory

kramers escape problem for fractional klein kramers equation with tempered α stable waiting times in this paper we extend the subdiffusive klein kramers model in which the waiting times are modeled by the stable laws to the case of waiting times belonging to the class of tempered stable distributions we introduce a generalized version of the klein kramers equation in which the fractional riemman liouville derivative is replaced with a more general integro differential operator this allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled taking advantage of the corresponding langevin equation we study some properties of the tempered dynamics in particular we approximate solutions of the tempered klein kramers equation via monte carlo methods also we study the distribution of the escape time from the potential well and compare it to the classical results in the kramers escape theory finally we derive the analytical formula for the first passage time distribution for the case of free particles we show that the well known sparre andersen scaling holds also for the tempered subdiffusion

robustness analysis of uncertain linear singular systems with output feedback control the robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered the problem is transformed into a robust nonsingularity problem based on the linear fractional transformation lft approach the robustness bounds to preserve regularity impulse immunity and stability are found in terms of the structured singular value spl mu with respect to parametric uncertainties the lft approach provides a unified framework for robustness analysis of both uncertain linear continuous discrete time singular systems

a characterization of convex problems in decentralized control ast we consider the problem of constructing optimal decentralized controllers we formulate this problem as one of minimizing the closed loop norm of a feedback system subject to constraints on the controller structure we define the notion of quadratic invariance of a constraint set with respect to a system and show that if the constraint set has this property then the constrained minimum norm problem may be solved via convex programming we also show that quadratic invariance is necessary and sufficient for the constraint set to be preserved under feedback these results are developed in a very general framework and are shown to hold in both continuous and discrete time for both stable and unstable systems and for any norm this notion unifies many previous results identifying specific tractable decentralized control problems and delineates the largest known class of convex problems in decentralized control as an example we show that optimal stabilizing controllers may be efficiently computed in the case where distributed controllers can communicate faster than their dynamics propagate we also show that symmetric synthesis is included in this classification and provide a test for sparsity constraints to be quadratically invariant and thus amenable to convex synthesis

distributed order fractional diffusions on bounded domains fractional derivatives can be used to model time delays in a diffusion process when the order of the fractional derivative is distributed over the unit interval it is useful for modeling a mixture of delay sources this paper provides explicit strong solutions and stochastic analogues for distributed order time fractional diffusion equations on bounded domains with dirichlet boundary conditions

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

a convex characterization of multidimensional linear systems subject to sqi constraints this technical note proposes a convex characterization of the set of all stable closed loop linear systems that are obtained from a given plant which may be multidimensional by interconnecting it in feedback with controllers that satisfy a certain pre selected constraint we take an approach that is particularly useful when a doubly coprime factorization of the plant is difficult to obtain and a stable stabilizing controller may not exist the plant is not strongly stabilizable or when one may be difficult to find most related work requires one of these we adopt the so called coordinate free approach which unlike youla s parametrization does not rely on a doubly coprime factorization of the plant we show that if constraints which satisfy a condition called strong quadratic invariance sqi are imposed on the controllers then the set of all stable closed loop multidimensional linear systems has a convex representation and norm optimal control problems can be cast in convex form although the sqi condition is in general slightly stronger than quadratic invariance qi which was developed in related work they are equivalent for common classes of problems arising in decentralized control

stability and stabilization of multidimensional input output systems in this paper we discuss stability and stabilization of continuous and discrete multidimensional input output io behaviors of dimension r which are described by linear systems of complex partial differential resp difference equations with constant coefficients where the signals are taken from various function spaces in particular from those of polynomial exponential functions stability is defined with respect to a disjoint decomposition of the r dimensional complex space into a stable and an unstable region with the standard stable region in the one dimensional continuous case being the set of complex numbers with negative real part a rational function is called stable if it has no poles in the unstable region an io behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region this is equivalent to the stability of its transfer matrix and an additional condition the system is called stabilizable if there is a compensator io system such that the output feedback system is well posed and stable we characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable io system parametrization the theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional io maps as developed for instance by bose guiver shankar sule xu lin ying zerz and quadrat and of course the seminal papers of vidyasagar youla and others in the one dimensional case in contrast to the existing literature the theorems and proofs of this paper do not need or employ the so called fractional representation approach i e various matrix fraction descriptions of the transfer matrix thus avoiding the often lengthy matrix computations and seeming to be of interest even for one dimensional systems at least to the author an important mathematical tool new in systems theory is gabriel s localization theory which only in the case of ideal convex shankar sule unstable regions coincides with the usual one algorithmic tests for stability stabilizability and ideal convexity and the algorithmic construction of stabilizing compensators are addressed but still encounter many difficulties see in particular the open problems listed by xu e

parameter estimation for fractional transport a particle tracking approach 1 space fractional advection dispersion models provide attractive alternatives to the classical advection dispersion equation for model applications that exhibit early arrivals and plume skewness this paper develops a flexible method for estimating the parameters of the fractional transport model on the basis of spatial plume snapshots or temporal breakthrough curve data a particle tracking approach provides error bars for the parameter estimates and a general method for model fitting and comparison via optimal weighted least squares a simple model of concentration variance based on the particle tracking approach identifies the optimal weights

youla like parametrizations subject to qi subspace constraints consider that a linear time invariant lti plant is given and that we wish to design a stabilizing controller for it admissible controllers are lti and must belong to a pre selected subspace that may impose structural restrictions such as sparsity constraints the subspace is assumed to be quadratically invariant qi with respect to the plant which from prior results guarantees that there is a convex parametrization of all admissible stabilizing controllers provided that an initial admissible stable stabilizing controller is provided this paper addresses the previously unsolved problem of extending youla s classical parametrization so that it admits qi subspace constraints on the controller in contrast with prior parametrizations the one proposed here does not require initialization and it does not require the existence of a stable stabilizing controller the main idea is to cast the stabilizability constraint as an exact model matching problem with stability restrictions which can be tackled using existing methods furthermore we show that when it exists the solution of the exact model matching problem can be used to compute an admissible stabilizing controller applications of the proposed parametrization on the design of norm optimal controllers via convex methods are also explored an illustrative example is provided and a special case is discussed for which the exact model matching problem has a unique and easily computable solution

feedback stabilization of mimo 3 d linear systems the authors solve the open problem of the existence of double coprime factorizations for a large class of multi input multi output mimo three dimensional 3 d linear systems it is proven that if all the unstable zeros of the contents associated with left and right matrix fraction descriptions of a given feedback stabilizable causal mimo 3 d plant are simple then the plant has a double coprime factorization the authors then give a parameterization of all stabilizing compensators for a mimo 3 d system in this class the key result developed in the paper is a novel and constructive technique of replacing an unstable polynomial with a stable polynomial step by step an illustrative example is also provided

a generalization of serre s conjecture and some related issues abstract several topics concerned with multivariate polynomial matrices like unimodular matrix completion matrix determinantal or primitive factorization matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original serre problem conjecture solved nonconstructively in 1976 and constructively more recently this generalized serre conjecture is proved to be equivalent to several other unsettled conjectures and therfore all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

on syzygy modules for polynomial matrices abstract in this paper we apply the theory of multivariate polynomial matrices to the study of syzygy modules for a system of homogeneous linear equations with multivariate polynomial coefficients several interesting structural properties of syzygy modules are presented and illustrated with examples

feedback stabilization over commutative rings further study of the coordinate free approach this paper is concerned with the coordinate free approach to control systems the coordinate free approach is a factorization approach but does not require the coprime factorizations of the plant we present two criteria for feedback stabilizability for multi input multi output mimo systems in which transfer functions belong to the total rings of fractions of commutative rings both of them are generalizations of sule s results in siam j control optim 32 1994 pp 1675 1695 the first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal it shows that if the plant is stabilizable the modules are projective the other criterion is expressed in terms of ideals called generalized elementary factors this gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors as an example a discrete finite time delay system is considered

fluid limit of the continuous time random walk with general levy jump distribution functions the continuous time random walk ctrw is a natural generalization of the brownian random walk that allows the incorporation of waiting time distributions psi t and general jump distribution functions eta x there are two well known fluid limits of this model in the uncoupled case for exponential decaying waiting times and gaussian jump distribution functions the fluid limit leads to the diffusion equation on the other hand for algebraic decaying waiting times psi approximately t 1 beta and algebraic decaying jump distributions eta approximately x 1 alpha corresponding to levy stable processes the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time however these are two special cases of a wider class of models here we consider the ctrw for the most general levy stochastic processes in the levy khintchine representation for the jump distribution function and obtain an integrodifferential equation describing the dynamics in the fluid limit the resulting equation contains as special cases the regular and the fractional diffusion equations as an application we consider the case of ctrws with exponentially truncated levy jump distribution functions in this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory long jumps and truncation effects in the intermediate asymptotic regime the dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as tauc approximately lambda alpha beta where 1 lambda is the truncation length scale the asymptotic behavior of the propagator green s function of the truncated fractional equation exhibits a transition from algebraic decay for t tauc

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

on a generalization of the youla kucera parametrization part i the fractional ideal approach to siso systems abstract in this paper we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for siso systems within this mathematical framework we give necessary and sufficient conditions so that a plant is internally strongly bistably stabilizable or admits a weak coprime factorization moreover we show how to generalize the youla kucera parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization this parametrization is generally affine in two free parameters

elementary proof of controller parametrization without coprime factorizability recently a controller parametrization method without coprime factorizability was presented by mori in this note we present this method using elementary matrix operations unlike in the earlier paper good knowledge of commutative algebra and module theory is not required

relationship between standard control problem and model matching problem without coprime factorizability we present the equivalence between a standard control problem and a model matching problem without coprime factorizability of plants this equivalence is well known if every stabilizable plant admit a doubly coprime factorization the result of this note does not require the coprime factorizability of plants

stabilizability representations and factorizations for time varying linear systems abstract we study the system representations factorizations and stabilizability of a discrete time time varying linear system in the framework of nest algebra we shown that every time varying linear system admits a normalized left factorization while it may possibly not have a right factorization the relationship between the system representation and factorization is studied and some new stabilizability criteria are established

the stabilization problem for discrete time varying linear systems in this paper we consider the strong stabilization problem for a time varying linear system which admits a coprime factorization we also study the simultaneous and robust stabilization problems for this class of systems

on the elementary factors of linear systems over unique factorization domains abstract the concept of elementary factors introduced by sule siam j control optim 32 1994 1675 1695 plays an important role in the output feedback stabilizability and stabilization of linear systems over unique factorization domains ufds this paper gives a simple proof that the elementary factors of t and w are identical where t and w are matrices associated with a right and a left matrix fraction description of a linear system over a ufd relationship between the reduced minors and the elementary factors is established an efficient method for computing the elementary factors based on the reduced minors is also proposed

minimal boolean sum and blending type projections and extensions in this paper useful characterization theorems are presented for minimal boolean sum and blending type projections and extensions 2000 elsevier science ltd all rights reserved

projective free algebras of continuous functions on compact abelian groups abstract it is proved that the wiener algebra of functions on a connected compact abelian group whose bohr fourier spectra are contained in a fixed subsemigroup of the additive dual group is projective free the semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite the projective free property is proved also for the algebra of continuous functions with the same condition on their bohr fourier spectra as an application the connected components of the set of factorable matrices are described the proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration

coprime factorization and dynamic stabilization of transfer functions it is known that a matrix valued transfer function p has a stabilizing dynamic controller q i e begin smallmatrix i q p i end smallmatrix 1 in rm h infty iff p has a right or left coprime factorization we show that the same result is true in the operator valued case thus the standard youla bongiorno parameterization applies to every dynamically stabilizable function we then derive further equivalent conditions one of them being that p has a stabilizing controller with internal loop this and some others are new even in the scalar valued case we also establish certain related results for example we extend the classical results on coprime factorization and partial feedback measurement feedback stabilization to nonrational transfer functions all our results apply in both discrete and continuous time settings except that in the latter it is not clear whether the controller q can always be chosen so that it is continuous time proper holomorphic and bounded on a right half plane unless e g p z rightarrow0 as operatorname re z rightarrow infty

random walk approximation of fractional order multiscaling anomalous diffusion random walks are developed to approximate the solutions of multiscaling fractional order anomalous diffusion equations the essential elements of the diffusion are described by the matrix order scaling indexes and the mixing measure which describes the diffusion coefficient in every direction two forms of the governing equation also called the multiscaling fractional diffusion equation based on fractional flux and fractional divergence are considered where the diffusion coefficient and the drift vary in space the particle tracking algorithm is also extended to approximate anomalous diffusion with a streamline dependent mixing measure using a streamline projection technique in this and other general cases the random walk method is the only known way to solve the nonhomogeneous equations five numerical examples demonstrate the flexibility simplicity and efficiency of the random walk method

minimal projections in tensor product spaces in this paper we present a condensed survey of recent results on minimal projections in tensor product spaces the idea of a minimal projection is an old one we have a banach space z and a complemented subspace w then the projection constant delta w z is defined by delta w z inf p p is a projection from z onto w

space fractional advection dispersion equations with variable parameters diverse formulas numerical solutions and application to the macrodispersion experiment site data 1 to model the observed local variation of transport speed an extension of the homogeneous space fractional advection dispersion equation fade to more general cases with space dependent coefficients drift velocity v and dispersion coefficient d has been suggested to provide a rigorous evaluation of this extension we explore the underlying physical meanings of two proposed and one other possible form of the fade by using the generalized mass balance law proposed by meerschaert et al 2006 when the classical fick s law is replaced by its generalized form in the first order mass conversation law the original fade with constant parameters extends to the advection dispersion equation ade with fractional flux denoted as ff ade when the net inflow of dispersive flux is from nonlocal concentration gradients following a fractional divergence we get the ade with fractional divergence fd ade when the total net inflow from both nonlocal advection and nonlocal concentration gradients follows a fractional form the fade contains fully fractional divergence ffd ade these three fades with constant parameters can also be obtained by proper choice of the two memory kernels in the nonlocal dispersive constitutive theory proposed by cushman et al 1994 while the space variable fades correspond to the conditional nonlocal theory proposed by neuman 1993 after specifying the general levy type functional form of the random distribution the corresponding langevin markov models can be found in many cases where the lagrangian stochastic processes can be conditioned directly on local aquifer properties at any practical measurable level and resolution the resulting lagrangian random walk particle tracking methods along with previous numerical solutions using implicit euler finite differences distinguish and elucidate the plume behavior described by these fades the fractional models are applied to fit the tritium plumes measured at the macrodispersion experiment test site when the local parameters gleaned from the hydraulic conductivity k distribution are varied even slightly i e a two zone model allowing the use of observed hydraulic conditions at the site the model fits are well within the variability of the data the extended fades describe the fast and space dependent leading edges of measured plumes in the regional scale alluvial system which was underestimated and could not be fully captured by the original fade with constant parameters applications also favor the fd ade model because of the ease of implementation and consistency with previous analysis of the k statistics

corrigendum feedback stabilization over commutative rings the matrix case in a previous paper 1 certain arguments in the proofs of proposition 1 proposition 4 and subsection 4 4 are erroneous these errors are highly regretted by this author and are corrected as follows these corrections however do not affect the main results of the paper 1 in definition 1 of a strictly causal matrix the second sentence should be corrected as a causal matrix m is called strictly causal if all entries of m are in j 2 in the sufficiency part of the proof of proposition 1 the two sentences starting in line 3 of the paragraph since it n j using units of r 1a should be replaced by the following from ny i x d and the assumption that all entries of p are in j and hence that of n since d is nonzerodivisor it follows that all entries of i x are in j hence detx det i i x is of the form 1 j where j is in j this implies that detx is a unit of r 1a as j is the jacobson s radical 3 the proof of the necessity part of proposition 4 is erroneous since this result is not needed further in the paper we replace it with the following version stating only the sufficient condition proposition 4 a strictly causal p nd 1 in f n is stabilizable if a b a

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

on using random walks to solve the space fractional advection dispersion equations the solution of space fractional advection dispersion equations fade by random walks depends on the analogy between the fade and the forward equation for the associated markov process the forward equation which provides a lagrangian description of particles moving under specific markov processes is derived here by the adjoint method the fade however provides an eulerian description of solute fluxes there are two forms of the fade based on fractional flux ff ade and fractional divergence fd ade the ff ade is derived by taking the integer order mass conservation of non local diffusive flux while the fd ade is derived by taking the fractional order mass conservation of local diffusive flux the analogy between the fade and the forward equation depends on which form of the fade is used and on the spatial variability of the dispersion coefficient d in the fade if d does not vary in space then the fades can be solved by tracking particles following a markov process with a simple drift and an stable levy noise with index that corresponds to the fractional order of the fade if d varies smoothly in space and the solute concentration at the upstream boundary remains zero the fd ade can be solved by simulating a markov process with a simple drift an stable levy noise and an additional term with the dispersion gradient and an additional levy noise of order 1 however a non markov process might be needed to solve the ff ade with a space dependent d except for specific d such as a linear function of space

monte carlo simulation of uncoupled continuous time random walks yielding a stochastic solution of the space time fractional diffusion equation we present a numerical method for the monte carlo simulation of uncoupled continuous time random walks with a levy stable distribution of jumps in space and a mittag leffler distribution of waiting times and apply it to the stochastic solution of the cauchy problem for a partial differential equation with fractional derivatives both in space and in time the one parameter mittag leffler function is the natural survival probability leading to time fractional diffusion equations transformation methods for mittag leffler random variables were found later than the well known transformation method by chambers mallows and stuck for levy stable random variables and so far have not received as much attention nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the mittag leffler distribution combining the two methods we obtain an accurate approximation of space and time fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes

on stabilization and the existence of coprime factorizations let k be an integral domain and f its quotient field we show there are plants over f which have no stable coprime factorizations but can be stabilized in a stable closed loop this answers a question posed by vidyasagar schneider and francis

a necessary and sufficient condition for feedback stabilization in a factorial ring the necessity for a coprime fractional representation of a plant in order to completely parametrize all stabilizing compensators by the youla theory is examined it is shown that the coprimeness is necessary for systems in a factorial ring a class inclusive of multidimensional scalar systems some observations on partial state transfer functions in the feedback configuration are found to be useful in showing the necessity

frequency domain robust regulation of signals generated by an infinite dimensional exosystem this paper deals with frequency domain robust regulation of signals generated by an infinite dimensional exosystem the problem is formulated and the stability types are chosen so that one can generalize the existing finite dimensional theory to more general classes of infinite dimensional systems and signals the main results of this article are extensions of the internal model principle of a necessary and sufficient solvability condition for the robust regulation problem and of davison s simple servo compensator for stable plants in the chosen algebraic framework

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

parameterization of stabilizing controllers with either right or left coprime factorization a parameterization method of stabilizing controllers that requires only one of right left coprime factorizations is presented as an illustrative model we will consider the multidimensional system with structural stability for which it is not yet known in general whether or not a doubly coprime factorization always exists a parameterization of stabilizing controllers for the model is presented