a boundary value problem for second order fuzzy differential equations abstract in this paper we interpret a two point boundary value problem for a second order fuzzy differential equation by using a generalized differentiability concept we present a new concept of solutions and utilizing the generalized differentiability we investigate the problem of finding new solutions some examples are provided for which the new solutions are found

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

an h minimax approach to the design of robust control systems abstract a realistic feedback design problem is posed based on the minimization of a weighted combination of the sensitivity and complementary sensitivity matrices a solution is obtained which makes use of the recently proposed methods for minimizing the sensitivity function alone

brauer type eigenvalue inclusion sets of stochastic irreducible tensors and positive definiteness of tensors abstract in this paper brauer type eigenvalue inclusion sets of stochastic tensors and irreducible tensors are given respectively some sufficient conditions of positive semi definiteness of stochastic tensors are obtained furthermore the stability of a high order nonlinear system is analysed by positive definiteness of tensors

some bounds for the spectral radius of nonnegative tensors in this paper we extend the well known column sum bound of the spectral radius for nonnegative matrices to the tensor case an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors we demonstrate that our new results improve existing results the other main results of this paper is to provide a sharper ky fan type theorem and a comparison theorem for nonnegative tensors finally we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors this result is similar to the weyl theorem for the matrix case

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

reduced order optimal control based on approximate inertial manifolds for nonlinear dynamical systems a reduced order method for optimal control problems in infinite dimensions based on approximate inertial manifolds is developed convergence of the cost optimal controls and optimal states of the finite dimensional reduced order optimal control problems to the original optimal control problem is analyzed special attention is given to the particular case when the dynamics are described by the navier stokes equations in dimension two

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

on the robust stability of linear time invariant plants with unstructured uncertainty we consider general families of linear time invariant plants described by a nominal plant model with unstructured uncertainty we show under a rather weak assumption that when a family of systems is not robustly stabilizable by any linear time invariant feedback controller then no nonlinear time varying controller can robustly stabilize the given family

tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose the oil palm frond a lignocellulosic material is a high yielding energy crop that can be utilized as a promising source of xylose it holds the potential as a feedstock for bioethanol production due to being free and inexpensive in terms of collection storage and cropping practices the aim of the paper is to calculate the concentration and yield of xylose from the acid hydrolysis of the oil palm frond through a fuzzy fractional kinetic model the approximate solution of the derived fuzzy fractional model is achieved by using a tau method based on the fuzzy operational matrix of the generalized laguerre polynomials the results validate the effectiveness and applicability of the proposed solution method for solving this type of fuzzy kinetic model a new fractional kinetic equation under uncertainty was addressed to depict the chemical reaction arising in palm oil frond an efficient numerical simulation based on a tau method was derived to solve the proposed kinetic equation different cases were solved to demonstrate the validity and efficiency of the proposed technique

balanced model reduction for linear time varying symmetric systems the goal of this paper is to develop balancing theory for the linear time varying ltv symmetric systems to this end first we extend the concept of symmetry in terms of the dual system then we define the cross gramian for the ltv systems for ltv symmetric systems we establish a connection among the controllability observability and cross gramians in particular if one of these three gramians is obtained the other two gramians can be constructed based on this fact we show that the symmetry structure is preserved under balanced truncation if the hankel singular values are pointwise distinct

m tensors and nonsingular m tensors abstract the m matrix is an important concept in matrix theory and has many applications recently this concept has been extended to higher order tensors 18 in this paper we establish some important properties of m tensors and nonsingular m tensors an m tensor is a z tensor we show that a z tensor is a nonsingular m tensor if and only if it is semi positive thus a nonsingular m tensor has all positive diagonal entries an m tensor regarding as the limit of a sequence of nonsingular m tensors has all nonnegative diagonal entries we introduce even order monotone tensors and present their spectral properties in matrix theory a z matrix is a nonsingular m matrix if and only if it is monotone this is no longer true in the case of higher order tensors we show that an even order monotone z tensor is an even order nonsingular m tensor but not vice versa an example of an even order nontrivial monotone z tensor is also given

a comparison of model reduction techniques from structural dynamics numerical mathematics and systems and control in this paper popular model reduction techniques from the fields of structural dynamics numerical mathematics and systems and control are reviewed and compared the motivation for such a comparison stems from the fact that the model reduction techniques in these fields have been developed fairly independently in addition the insight obtained by the comparison allows for making a motivated choice for a particular model reduction technique on the basis of the desired objectives and properties of the model reduction problem in particular a detailed review is given on mode displacement techniques moment matching methods and balanced truncation whereas important extensions are outlined briefly in addition a qualitative comparison of these methods is presented hereby focusing both on theoretical and computational aspects finally the differences are illustrated on a quantitative level by means of application of the model reduction techniques to a common example

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

gramian matrices and balanced model of generalized systems in this work the generalization of reachability and observability gramian matrices of control generalized systems have been studied the properties of these matrices to be solutions of lyapunov equations have been analyzed in addition an algorithm to obtain balanced generalized systems is given

fractional dynamics of populations abstract nature often presents complex dynamics which cannot be explained by means of ordinary models in this paper we establish an approach to certain fractional dynamic systems using only deterministic arguments the behavior of the trajectories of fractional non linear autonomous systems around the corresponding critical points in the phase space is studied in this work we arrive to several interesting conclusions for example we conclude that the order of fractional derivation is an excellent controller of the velocity how the mentioned trajectories approach to or away from the critical point such property could contribute to faithfully represent the anomalous reality of the competition among some species in cellular populations as cancer or hiv we use classical models which describe dynamics of certain populations in competition to give a justification of the possible interest of the corresponding fractional models in biological areas of research

time integral about solution of an uncertain fractional order differential equation and application to zero coupon bond model abstract uncertain fractional order differential equation is a significant tool for modeling the uncertain dynamic system first we consider solutions of an uncertain fractional order differential equation with the caputo type and investigate inverse uncertain distributions of their time integral on the basis of path two different time integral theorems for inverse uncertain distributions are given second in uncertain financial markets the interest rate is considered as an uncertain process as the application of the time integral we present a novel zero coupon bond model and derive a pricing formula of zero coupon bond under this model last by the predictor corrector method the numerical algorithm is designed analytic expressions and numerical calculations of the zero coupon bond price are illustrated for fractional order mean reverting model and standard deviation model respectively

a behavioral approach to balanced representations of dynamical systems abstract the behavioral approach to linear systems provides an alternative framework for studying the notion of balanced representations a new definition for balanced representations is proposed that is one to one related to a set of system invariants that is obtained by assuming a specific hilbert space structure on the system behavior this notion of balancing is more general than the prevailing notion of balancing in that it is well defined for nonstable systems and is independent of a particular input output representation of the system it is shown that lyapunov lqg and h balanced representations are obtained as a special case an application for the problem of model approximation is discussed

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

bounded real and positive real balanced truncation using σ normalised coprime factors in this article we will extend the method of balanced truncation using normalised right coprime factors of the system transfer matrix by meyer 1990 3 to balanced truncation with preservation of half line dissipativity special cases are preservation of positive realness and bounded realness we consider a half line dissipative input output system with quadratic supply rate given by a nonsingular symmetric matrix sigma with the property that its positive signature is equal to the number of input components of the system the transfer matrix of such system allows a sigma normalised right coprime factorisation we associate with such factorisation two lyapunov equations one of which is nonstandard involving the indefinite matrix sigma balancing will be based on making the unique solutions of these two lyapunov equations equal and diagonal the diagonal elements will be called the hankel sigma singular values because their squares are the nonzero eigenvalues of the composition of the graph hankel operator multiplication by sigma and the adjoint graph hankel operator this method of balanced truncation will be shown to preserve stability minimality and half line dissipativeness we will characterize the classical positive real and bounded real characteristic values in terms of the new hankel sigma singular values finally we will derive one step error bounds for the special case of balanced truncation of bounded real systems c 2009 elsevier b v all rights reserved

on the inverse of a tensor abstract in this paper we consider the left right inverse of a tensor we characterize the existence of any order k left right inverse of a tensor and show the expression of left right inverse of a tensor we also present a result for the similarity of tensors

a spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion wave equations in this paper an efficient and accurate spectral numerical method is presented for solving second fourth order fractional diffusion wave equations and fractional wave equations with damping the proposed method is based on jacobi tau spectral procedure together with the jacobi operational matrix for fractional integrals described in the riemann liouville sense the main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought for spectral approximations the validity and effectiveness of the method are demonstrated by solving five numerical examples numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier an efficient numerical scheme for time fractional diffusion wave equations is proposed a time space jacobi tau approximation is developed for such equations several tau spectral methods can be achieved as special cases the validity and applicability of the proposed method are demonstrated accurate numerical results are obtained by selecting limited collocation nodes

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

balanced realizations canonical form parametrization model reduction abstract balanced realizations are used to construct a canonical form for the set of minimal and asymptotically stable mimo systems of a given order a parametrization resutt is derived that allows us to show a truncation property of this canonical form

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

a general product of tensors with applications abstract we study a general product of two n dimensional tensors a and b with orders m 2 and k 1 this product satisfies the associative law and is a generalization of the usual matrix product using this product many concepts and known results of tensors can be simply expressed and or proved and a number of applications of it will be given using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables we define the similarity and congruence of tensors which are also the generalizations of the corresponding relations for matrices and prove that similar tensors have the same characteristic polynomials thus the same spectra we study two special kinds of similarity permutational similarity and diagonal similarity and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors we also define the direct product of tensors in matrix case it is also called the kronecker product and give its applications in the study of the spectra of two kinds of the products of hypergraphs we also give applications of this general product in the study of nonnegative tensors including a characterization of primitive tensors the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors

solution of a fuzzy system of linear equation abstract in this paper we study the general m n fuzzy linear system using a numerical method the weak fuzzy solutions to the fuzzy system are expressed by using the generalized inverses of the coefficient matrix conditions for the existence of a fuzzy solution are derived and a numerical procedure for calculating the solution is designed a numerical method for finding the least squares solution of an overdetermined fuzzy system of linear equation by applying an orthogonal matrix to coefficients matrix that needs simple calculations is given

exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method the purpose of this study is to introduce a modification of the homotopy perturbation method using laplace transform and pade approximation to obtain closed form solutions of nonlinear coupled systems of partial differential equations two test examples are given the coupled nonlinear system of burger equations and the coupled nonlinear system in one dimensional thermoelasticity the results obtained ensure that this modification is capable of solving a large number of nonlinear differential equations that have wide application in physics and engineering

solving differential equations of fractional order using an optimization technique based on training artificial neural network the current study aims to approximate the solution of fractional differential equations fdes by using the fundamental properties of artificial neural networks anns for function approximation in the first step we derive an approximate solution of fractional differential equation fde by using anns in the second step an optimization approach is exploited to adjust the weights of anns such that the approximated solution satisfies the fde different types of fdes including linear and nonlinear terms are solved to illustrate the ability of the method in addition the present scheme is compared with the analytical solution and a number of existing numerical techniques to show the efficiency of anns with high accuracy fast convergence and low use of memory for solving the fdes

algorithms for the fractional calculus a selection of numerical methods many recently developed models in areas like viscoelasticity electrochemistry diffusion processes etc are formulated in terms of derivatives and integrals of fractional non integer order in this paper we present a collection of numerical algorithms for the solution of the various problems arising in this context we believe that this will give the engineer the necessary tools required to work with fractional models in an efficient way

monotonicity of zeros of laguerre sobolev type orthogonal polynomials abstract denote by x n k m n k 1 n the zeros of the laguerre sobolev type polynomials l n m n x orthogonal with respect to the inner product p q 1 1 0 p x q x x e x d x m p 0 q 0 n p 0 q 0 where 1 m 0 and n 0 we prove that x n k m n interlace with the zeros of laguerre orthogonal polynomials l n x and establish monotonicity with respect to the parameters m and n of x n k m 0 and x n k 0 n moreover we find n 0 such that x n n m n 0 for all n n 0 where x n n m n is the smallest zero of l n m n x further we present monotonicity and asymptotic relations of certain functions involving x n k m 0 and x n k 0 n

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

system identification using balanced parametrizations some general issues in the black box identification of multivariable systems are first discussed it is then suggested that balanced parametrizations can be used to give identifiable forms a particular advantage is that balanced parametrizations are known for several useful classes of linear dynamic models including stable minimal models minimum phase models positive real models and normalized coprime factor models before optimizing the parameters of balanced parametrizations an initial model must be found we use realization based methods and so called subspace methods for this purpose these methods are very effective at finding accurate initial models without preliminary estimation of various structural indexes the paper ends with two simulation examples which compare the use of balanced parametrizations with more traditional ones and three real examples based on practical problems a distillation column an industrial dryer and the irrational spectrum or sea waves

dissipativity preserving balancing for nonlinear systems a hankel operator approach abstract in this paper we present a version of balancing for nonlinear systems which is dissipative with respect to a general quadratic supply rate that depends on the input and the output of the system we discuss an approach that allows us to apply the theory of balancing based upon hankel singular value analysis in order to do that we prove that the available storage and the required supply of the original system are the controllability and the observability functions of a modified asymptotically stable system then hankel singular value theory can be applied and the axis singular value functions of the modified system equal the nonlinear extensions of similarity invariants obtained from the required supply and available storage of the original system furthermore we also consider an extension of normalized comprime factorizations and relate the available storage and required supply with the controllability and observability functions of the factorizations the obtained relations are used to perform model order reduction based on balanced truncation yielding dissipative reduced order models for the original systems a second order electrical circuit example is included to illustrate the results

controller reduction by h sub infinity balanced truncation h sub infinity balanced truncation may be used to obtain reduced order plants or controllers the plant possibly unstable is compensated using a particular robustly stabilizing controller the two riccati equations involved are then used to define a set of closed loop input output invariants called the h sub infinity characteristic values that part of the plant or controller corresponding to small h sub infinity characteristic values is discarded to give a reduced order plant or controller by exploiting an intimate connection with coprime factorization a simple a priori test is derived for the ability of such a reduced order controller to stabilize the full order plant the performance of the resulting closed loop may also be bounded a priori i e in terms of the prespecified level of robustness and the discarded h sub infinity characteristic values

topology of the set of asymptotically stable minimal systems it is shown that the set of asymptotically stable and minimal single input single output systems of order n has n 1 connected components in the case of asymptotically stable and minimal multiple input multiple output systems however there is only one connected component

on a system of fuzzy fractional differential inclusions with projection operators abstract the aim of this paper is to introduce and study a class of fuzzy fractional differential inclusions with projection operators which captures the desired features of both fuzzy differential inclusions and fractional order projective dynamical systems within the same framework the existence of solutions for the open situation is proved by using continuous selection theorem moreover the existence of solutions for the closed situation is showed by employing lipschitzean selection theorem and fixed point theorem respectively furthermore an application and a numerical example are also given

a functional approach to lqg balancing the paper has as its theme a circle of problems related to lqg balancing with a special emphasis on the related problems of model reduction and robust stabilization the aim is to present a unified point of view to both previously described problems the unification is achieved by focusing on the study of three functions and the relationships between them and the various operators that are associated with these functions with an arbitrary transfer function g one can associate canonically two objects the conjugate inner function which is based on the nrcf and the r which is associated with the lqg controller of the function g the approach that is taken is functional emphasizing operators a balanced realization of a stable g arises as a matrix representation of the shift realization with respect to a basis made out of suitably normalized hankel singular vectors a similar result holds for lqg balanced realizations here the underlying hankel operator we study is h r where r u m v n and u v s

balanced canonical forms for minimal systems a normalized coprime factor approach abstract canonical forms are derived for the set of minimal systems of given order from a canonical form for a class of coinner transfer functions one of these canonical forms is in terms of so called riccati balanced coordinates the application of this work to model reduction is discussed

the inverse rank and product of tensors abstract in this paper we give some basic properties for the left right inverse rank and product of tensors the existence of order 2 left right inverses of tensors is characterized we obtain some equalities and inequalities on the tensor rank we also show that the rank of a uniform hypergraph is independent of the ordering of its vertices and the laplacian tensor and the signless laplacian tensor have the same rank for odd bipartite even uniform hypergraphs

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

the structure of characteristic values for classes of linear time invariant systems abstractin this paper a particular structure of the characteristic values found in some classes of systems is dealt with in particular for the linear time invariant systems in minimal form a b c d with transfer function matrix g s for which if i is a characteristic value then also 1 i is consequently when the system order is odd a singular value is equal to one the classes of systems for which this property holds are characterised by the existence of specific state space representations moreover the same structure of the characteristic values can be also retrieved for the class of even g s gt s and odd g s gt s systems several examples related to multi input multi output mimo systems are reported also referring to real physical systems

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

the fiedler vector of a laplacian tensor for hypergraph partitioning based on recent advances in spectral hypergraph theory l qi and z luo tensor anaysis spectral theory and special tensors siam philadelphia 2017 we explore the fiedler vector of an even uniform hypergraph which is the z eigenvector associated with the second smallest z eigenvalue of a normalized laplacian tensor arising from the hypergraph then we develop a novel tensor based spectral method for partitioning vertices of the hypergraph for this purpose we extend the normalized laplacian matrix of a simple graph to the normalized laplacian tensor of an even uniform hypergraph the corresponding fiedler vector is related to the cheeger constant of the hypergraph then we establish a feasible optimization algorithm to compute the fiedler vector according to the normalized laplacian tensor the convergence of the proposed algorithm and the probability of obtaining the fiedler vector of the hypergraph are analyzed theoretically finally preliminary numerical experiments illustrate that the new a

normalised coprime factorizations for nonstrictly proper systems some earlier results of d g meyer and g f franklin see ibid vol ac 32 pp 227 8 march 1987 are extended and a procedure is derived for obtaining directly a normalized right coprime factorization of a system that may not be strictly proper

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

a tau method for the one dimensional parabolic inverse problem subject to temperature overspecification in this work an approximational technique based on shifted legendre tau ideas is presented for the one dimensional parabolic inverse problem with a control parameter the method consists of expanding the required approximate solution as the elements of a shifted legendre polynomial using the operational matrices we reduce the problem to a set of algebraic equations a numerical example is included to demonstrate the validity and applicability of the technique and a comparison is made with existing results the method is easy to implement and produces very accurate results

a survey on fuzzy fractional differential and optimal control nonlocal evolution equations abstract we survey some representative results on fuzzy fractional differential equations controllability approximate controllability optimal control and optimal feedback control for several different kinds of fractional evolution equations optimality and relaxation of multiple control problems described by nonlinear fractional differential equations with nonlocal control conditions in banach spaces are considered

positive and negative solution of dual riccati equations by matrix sign function iteration abstract for a matrix algebraic riccati equation it is shown that the positive definite stabilizing solution the negative definite anti stabilizing solution and the positive and negative solutions of the corresponding dual equation may all be computed from a single matrix sign function iteration this is potentially far more efficient than four separate schur vector solutions moreover necessary and sufficient conditions are given for the existence of unique stabilizing or antistabilizing solutions of an algebraic riccati equation in terms of the matrix sign of an associated hamiltonian matrix

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

the solvability of fuzzy fractional partial differential equations under caputo gh differentiability the foundation of the concepts of fuzzy fractional integral and caputo gh partial for fuzzy valued multivariable functions is defined as a result fuzzy fractional partial differential equations are considered and the appropriateness of local boundary value problems for hyperbolic equations is proved we present two new results on the existence of two kinds of gh weak solutions of these problems the first result is based on the banach fixed point theorem with the lipschitz condition of functions on the right hand side of equations the second result is based on the nonlinear alternative of the schauder type for fuzzy valued continuous functions without the lipschitz condition moreover we indicate the boundedness and continuous dependence of solutions on the initial data of the problems our models are embedded in the sense of caputo gh differentiability some examples are presented to illustrate the results

a conjecture on the primitive degree of tensors abstract in this paper we prove the following let a be a nonnegative primitive tensor with order m and dimension n then its primitive degree a n 1 2 1 and the upper bound is sharp this confirms a conjecture of shao 5

on balanced realizations of bounded real and positive real functions based on normalized coprime factorizations with respect to indefinite metrics and the construction of suitable characteristic functions the ober balanced canonical forms for the classes of bounded real and positive real functions are derived this uses a matrix representation of the shift realization with respect to a basis related to sets of orthogonal polynomials

balanced realizations near stable invariant manifolds it is shown that the notion of balancing a state space realization about a stable isolated equilibrium point can be generalized to systems possessing stable invariant regular submanifolds of arbitrary dimension for example a stable limit cycle ycle

properties of optimally robust controllers abstract the robustness properties of controllers which are optimally robust with respect to normalized coprime factor uncertainty are analysed it is shown that such optimally robust controllers admit perturbations which are in size larger than the robustness measure it is shown how such perturbations can be constructed by solving hankel norm approximation problems

the bounds on spectral radius of nonnegative tensors via general product of tensors abstract in 1 j y shao proposed the general product of tensors based on this product the minc type bound on spectral radius of nonnegative tensors is given and the generalized minimax theorem of nonnegative tensors is presented moreover we give some bounds on the spectral radius of the general product of tensors

linear quadratic gaussian balancing for discrete time infinite dimensional linear systems in this paper we study the existence of linear quadratic gaussian lqg balanced realizations for discrete time infinite dimensional systems lqg balanced realizations are those for which the smallest nonnegative self adjoint solutions of the control and filter riccati equations are equal we show that the control filter riccati equation has a nonnegative self adjoint solution if and only if the system is output input stabilizable our main result is that the transfer function of a discrete time linear system has an approximately controllable and observable lqg balanced realization iff it has an input and output stabilizable realization the corresponding control and filter riccati equations have unique nonnegative self adjoint solutions moreover approximately controllable and observable lqg balanced realizations are unique up to a unitary state space transformation finally we show that the spectrum of the product of the smallest nonnegative self adjoint solutions of the control and filter riccati equations is independent of the particular realization

fractional spectral collocation methods for linear and nonlinear variable order fpdes while several high order methods have been developed for fractional pdes fpdes with fixed order there are no such methods for fpdes with field variable order these equations allow multiphysics simulations seamlessly e g from diffusion to sub diffusion or from wave dynamics transitioning to diffusion by simply varying the fractional order as a function of space or time we develop an exponentially accurate fractional spectral collocation method for solving linear nonlinear fpdes with field variable order following the spectral theory developed in 1 for fractional sturm liouville eigenproblems we introduce a new family of interpolants called left right sided and central fractional lagrange interpolants we employ the fractional derivatives of left right sided riemann liouville and riesz type and obtain the corresponding fractional differentiation matrices by collocating the field variable fractional orders we solve several fpdes including time and space fractional advection equation time and space fractional advection diffusion equation and finally the space fractional burgers equation to demonstrate the performance of the method in addition we develop a spectral penalty method for enforcing inhomogeneous initial conditions our numerical results confirm the exponential like convergence of the proposed fractional collocation methods

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

random fuzzy fractional integral equations theoretical foundations abstract this paper presents mathematical foundations for studies of random fuzzy fractional integral equations which involve a fuzzy integral of fractional order we consider two different kinds of such equations their solutions have different geometrical properties the equations of the first kind possess solutions with trajectories of nondecreasing diameter of their consecutive values on the other hand the solutions to equations of the second kind have trajectories with nonincreasing diameter of their consecutive values firstly the existence and uniqueness of solutions is investigated this is showed by using a method of successive approximations an estimation of error of n th approximation is given also a boundedness of the solution is indicated to show well posedness of the considered theory we prove that solutions depend continuously on the data of the equations some concrete examples of random fuzzy fractional integral equations are solved explicitly

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

reduced order control based on approximate inertial manifolds a reduced order method based on approximate inertial manifolds is applied to optimal control problems in infinite dimensional state spaces a detailed analysis of the method is given for the linear quadratic regulator problem the method can also be applied to higher order control systems with an appropriate decomposition of the state space in terms of slow and fast exponential decay

solving interval valued fractional initial value problems under caputo gh fractional differentiability abstract in this paper interval valued fractional differential equations ifdes under caputo generalized hukuhara differentiability are introduced we present existence and uniqueness results for ifdes with a krasnoselskii krein type condition the solution to interval valued fractional initial value problems under caputo type interval valued fractional derivatives by a modified fractional euler method mfem and a modified adams bashforth moulton method mabmm is also presented

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

symmetric nonnegative tensors and copositive tensors abstract we first prove two new spectral properties for symmetric nonnegative tensors we prove a maximum property for the largest h eigenvalue of a symmetric nonnegative tensor and establish some bounds for this eigenvalue via row sums of that tensor we show that if an eigenvalue of a symmetric nonnegative tensor has a positive h eigenvector then this eigenvalue is the largest h eigenvalue of that tensor we also give a necessary and sufficient condition for this we then introduce copositive tensors this concept extends the concept of copositive matrices symmetric nonnegative tensors and positive semi definite tensors are examples of copositive tensors the diagonal elements of a copositive tensor must be nonnegative we show that if each sum of a diagonal element and all the negative off diagonal elements in the same row of a real symmetric tensor is nonnegative then that tensor is a copositive tensor some further properties of copositive tensors are discussed

some variational principles for z eigenvalues of nonnegative tensors many important spectral properties of nonnegative matrices have recently been successfully extended to higher order nonnegative tensors for example see chang et al 2008 2011 friedland et al in press lim 2005 liu et al 2010 ng et al 2010 qi et al 2007 yang and yang 2010 2 3 9 17 23 24 27 28 however most of these results focus on the h eigenvalues introduced by qi 2005 2007 25 26 the key results of this paper reveal some similarities as well as some crucial differences between z eigenvalues and h eigenvalues of a nonnegative tensor in particular neither the positive z eigenvalue nor the associated positive z eigenvector of an irreducible nonnegative tensor has to be unique in general as demonstrated by example 2 7 furthermore the collatz type min max characterizations of the largest positive z eigenvalue of an irreducible nonnegative tensor is only partially true in general as seen in theorem 4 7 and example 4 8