necessary and sufficient conditions for robust stability of discrete systems with coefficients depending continuously on two interval parameters abstract let a real polynomial in a complex variable whose coefficients are any given continuous functions of two real interval parameters be given necessary and sufficient conditions are derived for the polynomial to have all its zeros outside or inside the unit circle of the complex variable plane in a multi polynomial dependence of the coefficients on parameters the conditions involve in addition to a finite number of algebraic steps only resolving single variable fixed coefficients real polynomials

frequency domain criterion for robust stability of interval time delay systems abstract in this paper we characterize the boundary f b of the image f b of a box b in r m under a nonlinear mapping f r m c we generalize results recently reported by polyak and kogan 1993 necessary and sufficient conditions for robust stability of multiaffine systems mathematics research report 93 06 university of maryland baltimore county for multiaffine mappings and provide computationally tractable necessary and sufficient robust stability conditions for quasipolynomials with interval coefficients and interval delays a numerical stability verification for a quasipolynomial family with two interval delays is presented

characterization of zero locations of polytopes of real polynomials given a stability region of a polytope of polynomials the edge theorem can be used to verify certain root clustering properties of the polytope the authors consider the inverse problem starting with a polytope of real polynomials they present a result that allows the construction of the smallest set of regions in the complex plane which characterizes its zero location

stability of polynomials under coefficient perturbation let the real polynomial a 0 x n a 1 x n 1 a n 1 x a n be stable and let the real numbers b k c k geq 0 0 leq k leq n be given we present a simple determinant criterion for finding the largest t 0 geq 0 such that the polynomial alpha 0 x n alpha 1 x n 1 alpha n 1 x alpha n is stable for all alpha k in a k b k t 0 a k c k t 0 cup a k 0 leq k leq n several further observations allow us to reduce the computational cost considerably

extended kronecker summation for cluster treatment of lti systems with multiple delays a new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant lti dynamics with multiple delays these hypersurfaces as they have very recently been introduced in a concept paper r sipahi and n olgac automatica 41 2005 pp 1413 1422 form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots ctcr in fact these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots to determine the kernel and offspring we use the extraordinary features of the extended kronecker summation operation in this paper the end result is that the infinite dimensional problem reduces to a finite dimensional one and preferably into an eigenvalue problem following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces we demonstrate these concepts via some example case studies one of the examples treats a 3 delay system for this case another interesting perspective called the building block is also utilized to display the kernel in three dimensional space in the domain of spectral delays

numerical approximation of a time fractional black scholes equation abstract in this paper a time fractional black scholes equation is examined we transform the initial value problem into an equivalent integral differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization a rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first order convergence with respect to the time variable is proved for overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the black scholes differential operator we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization it is proved that the scheme is stable and second order convergent with respect to the spatial variable numerical experiments support these theoretical results

wellposedness and regularity of the variable order time fractional diffusion equations abstract it was proved that the solution to a time fractional partial differential equation lacks regularity at the initial time t 0 which is often inconsistent with the physical problem being modeled even though the nature of the physical problem away from t 0 is well captured by a time fractional partial differential equation we argue that the fundamental reason why this phenomenon occurs lies in the incompatibility between the nonlocality of the time fractional partial differential equations and the locality of the classical initial condition at t 0 we show that the variable order time fractional partial differential equations in which the order varies in time as t goes to 0 to accommodate the impact of the local initial condition at time t 0 turn out to be a natural candidate to eliminate the nonphysical singularity of the solutions to the constant order time fractional partial differential equations and open up opportunities for modeling multiphysics phenomena from nonlocal to local dynamics and vice versa we prove the wellposedness of a variable order linear time fractional diffusion partial differential equation in multiple space dimensions we also prove that the regularity of its solution depends on the behavior of the variable order and its derivatives at time t 0 in addition to the usual smoothness assumptions more precisely we prove that its solution has full regularity like its integer order analogue if the variable order has an integer limit at t 0 or exhibits singular behaviors at t 0 like in the case of the constant order time fractional partial differential equations if the variable order has a non integer value at time t 0

minimality stabilizability and strong stabilizability of uncertain plants this paper considers a set of uncertain transfer functions whose numerator and denominators belong to independent polytopes it shows that i the members of this set are free from pole zero cancellations iff all the ratios of numerator edges and denominator edges are free from pole zero cancellations and the numerator and denominator corners evaluated at a finite number of points satisfy certain phase conditions ii the members of this set are free from pole zero cancellations in the closed right half plane iff all the ratios of numerator edges and denominator edges are free from pole zero cancellations in the closed right half plane and the numerator and denominator corners evaluated at a finite number of points satisfy certain phase conditions and iii in the strictly proper case all plants in the set are strongly stabilizable iff all plants avoid pole zero cancellations in the closed right half plane and all the corner ratios are strongly stabilizable a counter example is presented to show that this last result does not extend to biproper plants

numerical methods and analysis for a class of fractional advection dispersion models in this paper a class of fractional advection dispersion models fadms is considered these models include five fractional advection dispersion models i e the time fadm the mobile immobile time fadm with a time caputo fractional derivative 0 c 1 the space fadm with two sides riemann liouville derivatives the time space fadm and the time fractional advection diffusion wave model with damping with index 1 c 2 these equations can be used to simulate the regional scale anomalous dispersion with heavy tails we propose computationally effective implicit numerical methods for these fadms the stability and convergence of the implicit numerical methods are analysed and compared systematically finally some results are given to demonstrate the effectiveness of theoretical analysis

a necessary and sufficient condition for the stability of nonnegative interval discrete systems the stability of nonnegative discrete systems with interval uncertainty is discussed for this class of system a necessary and sufficient condition for the stability is provided when all elements of the system matrix are assumed to be specified by intervals an illustrative example is used as a guideline to demonstrate the effectiveness of the result simple computational tests are suggested and their usefulness in this framework is discussed the possible extension of this result to more general classes of systems is addressed

extremal robustness properties of multilinear interval systems this paper deals with robustness properties of a control system subject to parameter uncertainty when the parameters appear affine multilinearly in the coefficients of the characteristic polynomial the robust hurwitz stability of such a multilinear interval polynomial family reduces to that of the stability of a set of manifolds derived from the kharitonov polynomials several extremal properties of these stability manifolds are derived these properties are useful in determining the worst case parametric h and nonlinear sector bounded stability margins for control systems containing interval parametric uncertainty they generalize previous results obtained by the authors for the linear case

optimal regularity conditions for elliptic problems via l p_ delta spaces we consider positive solutions of the dirichlet problem for nonlinear elliptic equations where the nonlinearity is assumed to satisfy the polynomial upper growth condition f x u c 1 up it is known from a classical work of brezis and turner that the condition p pbt for a suitable nonlinearity f x u a x up with a x 0 and a l we prove the existence of an unbounded weak solution we next consider the case of systems and show that the polynomial growth conditions for a priori estimates recently obtained by quittner and the author are also optimal our results are strongly connected with the regularity theory of the laplace operator in the spaces l p delta omega the lebesgue spaces weighted by the distance to the boundary as a by product we in turn establish the optimality of the known linear l p delta regularity estimates our proofs are based on the construction of a solution of the laplace equation with a suitable boundary singularity with conical support and we use recent results on the boundary behavior of heat kernels

robust controller synthesis using kharitonov s theorem the authors present a method based upon kharitonov s theorem for synthesizing robustly stabilizing feedback controllers the approach is based upon a multiple plant model formulation with a quadratic cost functional sufficient conditions are obtained for characterizing robustly stabilizing static output feedback proportional controllers for mimo multiple input multiple output plants with denominator polynomial uncertainty

convergence and stability of compact finite difference method for nonlinear time fractional reaction diffusion equations with delay abstract this paper is concerned with numerical solutions of nonlinear time fractional reaction diffusion equations with time delay a linearized compact finite difference scheme is proposed to solve the equations in terms of a new developed fractional gronwall type inequality convergence and stability of the proposed scheme are obtained numerical experiments are given to illustrate the theoretical results

robust strict positive realness new results for interval plant controller families a frequency domain approach is used to derive several robust strict positive realness results for interval plants and interval plant plus controller families of transfer functions based on simple frequency domain properties of transfer functions the approach provides a framework for obtaining new results and constructing easy proofs of several important existing results on robust strict positive realness the main new result states that the minimum of the real part of the transfer functions belonging to an interval plant controller family is achieved on one of the 32 kharitonov segments of the interval plant the argument used in the proof is of wider interest and suggests easy ways of proving that robustness of other different frequency properties of interval plant plus controller families of transfer functions such as robust stability of h sub infinity norm computation can be deduced from a fixed number of segments of transfer functions of the family

finite element analysis for coupled time fractional nonlinear diffusion system abstract this paper is concerned with the l 1 galerkin finite element analysis of the coupled time fractional nonlinear diffusion system existence and uniqueness results are discussed at continuous as well as discrete levels two different methods for linearizing the nonlinear fully discrete problems are used we provide a priori bounds and convergence estimates in l 2 norm for fully discrete problems numerical results based on the presented finite element methods are provided to validate the theoretical estimates

on the nyquist envelope of an interval plant family in this note we study the envelope of the nyquist plots generated by an interval plant family and show that this boundary is not always contained in the nyquist plots of the kharitonov plants with this motivation in mind we give a sufficient condition for an envelope point to be contained in the nyquist plot of a kharitonov plant and use it to generate large and critical portions of the nyquist envelope e g we show that the outer nyquist envelope of a stable interval plant is generated by the nyquist plots of the kharitonov plants another by product of this sufficient condition is a framework for developing new extreme point results for interval feedback systems this framework is useful in computing the phase margin and the maximal peaking in the sensitivity and complementary sensitivity functions and in stating a robust version of the circle criterion we also use this framework to easily explain existing extreme point results for the gain margin the h sub spl infin norm and the positive realness of interval plants one conclusion which emerges is this seemingly all important properties of an interval feedback system are deducible from the kharitonov plants

robust stability and performance of interval plants abstract in this paper we consider the problem of robust performance for systems with unstructured and parametric perturbations the parametric uncertainty is modeled through a single input single output interval system description and the unstructured uncertainty through a norm bounded perturbation for this type of parametric robust performance we show that in order to synthesize a controller that achieves robust performance for a family of interval systems it is sufficient to synthesize that controller for a much smaller subset of the interval family regardless of the number of uncertain parameters in the original problem this reduced set is composed of thirty two one parameter families of systems a special class of controllers with stronger results is also presented

a fast finite volume method for conservative space time fractional diffusion equations discretized on space time locally refined meshes abstract a fast finite volume method was developed for conservative space time fractional diffusion partial differential equations in two space dimensions in the method a locally refined composite spatial mesh is used to resolve the boundary layers of the solutions and a graded temporal mesh is used to resolve the singularity of the solutions near the initial time a fast numerical method was developed by fully exploring the structure of the stiffness matrices numerical results show the utility of the method

on the robust popov criterion for interval lur e system the classical popov criterion is generalized to include problems with parametric uncertainty as well as nonlinear sector bounded perturbations the parametric dependence is modeled by an interval plant description a robust popov criterion is obtained for a plant controller configuration stronger external results are obtained for a special class of controllers an example illustrating the numerical implementation of these results is presented

the divergence equation in rough spaces abstract we aim at extending the existence theory for the equation div v f in a bounded or exterior domain with homogeneous dirichlet boundary conditions to a class of solutions which need not have a trace at the boundary typically the weak solutions that we shall consider will belong to some besov space b p q s with s 1 1 p 1 p after generalizing the notion of a solution for this equation we propose an explicit construction by means of the classical bogovski formula this construction enables us to keep track of a marginal information about the trace of solutions in particular it ensures that the trace is zero if f is smooth enough we expect our approach to be of interest for the study of rough solutions to systems of fluid mechanics

error analysis of a finite difference method on graded meshes for a time fractional diffusion equation a reaction diffusion problem with a caputo time derivative of order alpha in 0 1 is considered the solution of such a problem is shown in general to have a weak singularity near the initial time t 0 and sharp pointwise bounds on certain derivatives of this solution are derived a new analysis of a standard finite difference method for the problem is given taking into account this initial singularity this analysis encompasses both uniform meshes and meshes that are graded in time and includes new stability and consistency bounds the final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme so one can choose an optimal mesh grading numerical results are presented that confirm the sharpness of the error analysis

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

regularity theory for time fractional advection diffusion reaction equations abstract we investigate the behavior of the time derivatives of the solution to a linear time fractional advection diffusion reaction equation allowing space and time dependent coefficients as well as initial data that may have low regularity our focus is on proving estimates that are needed for the error analysis of numerical methods the nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic pde in our analysis we rely on novel energy methods in combination with a fractional gronwall inequality and certain properties of fractional integrals

a compact exponential difference method for multi term time fractional convection reaction diffusion problems with non smooth solutions abstract this paper is concerned with a numerical method for a class of one dimensional multi term time fractional convection reaction diffusion problems where the differential equation contains a sum of the caputo time fractional derivatives of different orders between 0 and 1 in general the solutions of such problems typically exhibit a weak singularity at the initial time a compact exponential finite difference method using the well known l1 formula for each time fractional derivative and a fourth order compact exponential difference approximation for the spatial discretization is proposed on a mesh that is generally nonuniform in time and uniform in space taking into account the initial weak singularity of the solution the stability and convergence of the method is proved and the optimal error estimate in the discrete l2 norm is obtained by developing a discrete energy analysis technique which enables us to overcome the difficulties caused by the nonsymmetric discretization matrices the error estimate shows that the method has the spatial fourth order convergence and reveals how to select an appropriate mesh parameter to obtain the temporal optimal convergence the extension of the method to two dimensional problems is also discussed numerical results confirm the theoretical convergence result and show the applicability of the method to convection dominated problems

kharitonov like theorems for robust performance of interval systems abstract the sensitivity function of a control system is an important concept in performance analysis classical filter design as well as modern robust h control for an interval system we prove that the maximal h norm of its sensitivity function is achieved at twelve out of sixteen kharitonov vertices similar kharitonov like results are established for the complementary sensitivity function and strict positive realness of interval systems these results are useful in robust performance analysis and h control design for dynamic systems under parametric perturbations

comments on extreme point results for robust stabilization of interval plants with first order compensators the authors point out a simple alternative proof of the main result theorem 1 of the paper by barmish hollot kraus and tempo 1992 this proof follows immediately on applying the vertex result of hollot and yang 1990 to the generalization of kharitonov s theorem given by chapellat and bhattacharyya 1988 1989 the authors also point out that more general vertex results are known

an overview of extremal properties for robust control of interval plants abstract this paper provides an overview of recent results on the problem of robust stability and performance of feedback control systems in the presence of plant perturbations the widely studied class of interval plants is considered and an effort is made to cover different types of plant perturbations by incorporating in addition to the interval plant description unstructured norm bounded perturbations extremality results for these classes of uncertain systems are provided in a unifying framework robust performance of control systems is addressed in the paper and a number of extremal results are given for the computation of the structured singular value function for feedback systems with interval plants also classical frequency response analysis for interval plant controller families of open loop transfer functions is surveyed with the aim of providing extremal results for the computation of specification parameters such as phase or gain margins and sensitivity and complementary sensitivity peaks

using hierarchical matrices in the solution of the time fractional heat equation by multigrid waveform relaxation abstract this work deals with the efficient numerical solution of the time fractional heat equation discretized on non uniform temporal meshes non uniform grids are essential to capture the singularities of typical solutions of time fractional problems we propose an efficient space time multigrid method based on the waveform relaxation technique which accounts for the nonlocal character of the fractional differential operator to maintain an optimal complexity which can be obtained for the case of uniform grids we approximate the coefficient matrix corresponding to the temporal discretization by its hierarchical matrix h matrix representation in particular the proposed method has a computational cost of o k n m log m where m is the number of time steps n is the number of spatial grid points and k is a parameter which controls the accuracy of the h matrix approximation the efficiency and the good convergence of the algorithm which can be theoretically justified by a semi algebraic mode analysis are demonstrated through numerical experiments in both one and two dimensional spaces

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

a critical functional framework for the inhomogeneous navier stokes equations in the half space abstract this paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous navier stokes equations in the half space in the case of small data with critical regularity in dimension n 3 we state that if the initial density 0 is close to a positive constant in l w n 1 r n and the initial velocity u 0 is small with respect to the viscosity in the homogeneous besov space b n 1 0 r n then the equations have a unique global solution the proof strongly relies on new maximal regularity estimates for the stokes system in the half space in l 1 0 t b p 1 0 r n interesting for their own sake

subdiffusion with time dependent coefficients improved regularity and second order time stepping this article concerns second order time discretization of subdiffusion equations with time dependent diffusion coefficients high order differentiability and regularity estimates are established for subdiffusion equations with time dependent coefficients using these regularity results and a perturbation argument of freezing the diffusion coefficient we prove that the convolution quadrature generated by the second order backward differentiation formula with proper correction at the first time step can achieve second order convergence for both nonsmooth initial data and incompatible source term numerical experiments are consistent with the theoretical results

closed loop hyperstability of interval plants this note examines the conditions for robust bounded input bounded output closed loop stability of interval plants when there is a nonlinear element in the feedback path specifically the authors present the robust generalized circle criterion and popov s criterion

on robust stability of interval control systems the authors develop results on the robust stability of a nonlinear control system containing both parametric as well as unstructured uncertainty the basic system considered is that of the classical lur e problem of nonlinear control theory a robust version of the lur e problem consisting of a family of linear time invariant systems subjected simultaneously to bounded parameter variations and feedback perturbations from a family of sector bounded nonlinear gains is presently treated by using the kharitonov theorem to develop some extremal results on positive realness of interval transfer functions i e a family of rational transfer functions with bounded independent coefficient perturbations the authors determine the size of a sector of nonlinear feedback gains for which absolute stability can be guaranteed these calculations amount to the determination of the stability margin of the system under joint parametric and nonlinear feedback perturbations

absolute stability of systems with parametric uncertainty and nonlinear feedback this paper gives an extension of the absolute stability analysis of systems whose forward path contains a linear time invariant subsystem with uncertain coefficients and whose feedback path contains a conic nonlinearity the results provide a simple frequency domain interpretation of stability that can be of importance in compensator design and robust stability analysis in nonlinear systems

a galerkin finite element method for time fractional stochastic heat equation abstract in this study a galerkin finite element method is presented for time fractional stochastic heat equation driven by multiplicative noise which arises from the consideration of heat transport in porous media with thermal memory with random effects the spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained numerical techniques are developed by the standard galerkin finite element method in spatial direction and gorenflo mainardi moretti paradisi scheme is applied in temporal direction the convergence error estimates for both semi discrete and fully discrete schemes are established finally numerical example is provided to verify the theoretical results

optimal spatial h1 norm analysis of a finite element method for a time fractional diffusion equation abstract a time fractional initial boundary value problem d t u u f where d t is a caputo fractional derivative of order 0 1 is considered on the space time domain 0 t where r d d 1 is a bounded lipschitz domain typical solutions u x t of such problems have components that behave like a multiple of t as t 0 so the integer order temporal derivatives of u blow up at t 0 the numerical method of the paper uses a standard finite element method in space on a quasiuniform mesh and considers both the l1 discretisation and alikhanov s l2 1 discretisation of the caputo derivative on suitably graded temporal meshes optimal error bounds in h 1 are proved no previous analysis of a discretisation of this problem using finite elements in space has established such a bound furthermore the optimal grading of the temporal mesh can be deduced from our analysis numerical experiments show that our error bounds are sharp

convexity properties of polynomials with assigned root location this paper presents some geometrical results on the domain of generalized stability for a family of nth order polynomials regions of root location such that the convex hull of the corresponding domain of stability in the coefficient space is a polyhedron are investigated and specific regions for which the convex hull is an n 1 vertex polyhedron are derived the discrete time stability domain falls in the latter class of regions implications of the results for the design of filters solving the robust strict positive realness for families of rational transfer functions with uncertainty in the numerator are also developed

expansion of det a b and robustness analysis of uncertain state space systems in this note an expansion for the determinant of the sum of two matrices det a b is given a robustness analysis approach for state space systems with uncertain parameters is investigated by using kharitonov s theorem some useful results are obtained several illustrative examples are given

robust strict positive real stabilization and asymptotic hyperstability robustness abstract this paper is devoted to the study of robust strict positive real stabilization of siso uncertain systems i e finding a single controller that not only robustly stabilizes a plant with uncertainties under unity feedback but also makes the closed loop system robustly strictly positive real it is proved that there must exist a stable controller achieving robust strict positive real stabilization of uncertain systems with both parameter variations and unstructured uncertainties under some reasonable assumptions the propemess of the controller is guaranteed when the largest relative degree of an uncertain system is retricted by applying these results to nonlinear systems some significant conclusions on asymptotic hyperstability robustness are given

superconvergence analysis of nonconforming finite element method for time fractional nonlinear parabolic equations on anisotropic meshes abstract in this paper we prove a novel result of the consistency error estimate with order o h 2 for e q 1 r o t element see lemma 2 on anisotropic meshes then a linearized fully discrete galerkin finite element method fem is studied for the time fractional nonlinear parabolic problems and the superclose and superconvergent estimates of order o h 2 in broken h 1 norm on anisotropic meshes are derived by using the proved character of e q 1 r o t element which improve the results in the existing literature numerical results are provided to confirm the theoretical analysis

a second order and nonuniform time stepping maximum principle preserving scheme for time fractional allen cahn equations abstract in this work we present a second order nonuniform time stepping scheme for the time fractional allen cahn equation we show that the proposed scheme preserves the discrete maximum principle and by using the convolution structure of consistency error we present sharp maximum norm error estimates which reflect the temporal regularity as our analysis is built on nonuniform time steps we may resolve the intrinsic initial singularity by using the graded meshes moreover we propose an adaptive time stepping strategy for large time simulations numerical experiments are presented to show the effectiveness of the proposed scheme this seems to be the first second order maximum principle preserving scheme for the time fractional allen cahn equation

a class of shifted high order numerical methods for the fractional mobile immobile transport equations abstract in this article we apply the generalized bdf2 to the fractional mobile immobile transport equations for its temporal discretization and the finite element method in the spatial direction to derive the stability estimates and obtain the optimal error convergence rate some properties of the convolution weights are proved based on which we show the scheme is unconditionally stable with an error of o 2 h r 1 where and h represent the temporal and spatial mesh size respectively we conduct exhaustive numerical tests to further confirm our theoretical analysis and to overcome the initial singularity of the time fractional derivative we adopt the generalized bdf2 with starting parts in accordance with the framework of the shifted convolution quadrature scq

a novel high order approximate scheme for two dimensional time fractional diffusion equations with variable coefficient abstract based on the spatial quasi wilson nonconforming finite element method and temporal l 2 1 formula a fully discrete approximate scheme is proposed for a two dimensional time fractional diffusion equations with variable coefficient on anisotropic meshes in order to demonstrate the stable analysis and error estimates several lemmas are provided which focus on high accuracy about projection and superclose estimate between the interpolation and projection unconditionally stable analysis are derived in l 2 norm and broken h 1 norm moreover convergence result of accuracy o h 2 2 and superclose property of accuracy o h 2 2 are deduced by combining interpolation with projection where h and are the step sizes in space and time respectively and then the global superconvergence is presented by employing interpolation postprocessing operator finally numerical results are provided to demonstrate the validity of the theoretical analysis

robust strict positive real stabilization criteria for siso uncertain systems abstract this paper is devoted to the study of robust strict positive real stabilization criteria for siso uncertain systems i e considering the problem of when a single controller not only robustly stabilizes a plant with uncertainty under unity feedback but also enables the closed loop system to be robustly strictly positive real the uncertain system models with both parameter variation and unstructured perturbation are considered and some necessary and sufficient conditions for the problem are derived

parametric absolute stability of lur e systems the concept of parametric stability is extended to include lur e type nonlinear control systems with uncertain parameters and constant reference inputs conditions for parametric absolute stability are derived which guarantee that the system remains stable despite uncertainty of equilibrium location caused by parametric uncertainties and values of the reference input the conditions can be tested by computing value sets using the polygon interval arithmetic algorithm

space time finite element method for the multi term time space fractional diffusion equation on a two dimensional domain abstract in this paper we propose a space time finite element method for the multi term time space fractional diffusion equation first some fractional derivative spaces are listed some properties of the fractional derivatives and fractional derivative spaces are given some definitions and properties of finite element spaces are introduced and a space time finite element fully discrete scheme for the considered problem is developed second the existence uniqueness and stability of the obtained numerical scheme are discussed third under the hypothesis about singular behavior of exact solution near t 0 the convergence is investigated in detail based on the suitable graded time mesh at last some numerical tests are given to verify the rationality and effectiveness of our method

a higher order numerical method for time fractional partial differential equations with nonsmooth data abstract gao et al 11 2014 introduced a numerical scheme to approximate the caputo fractional derivative with the convergence rate o k 3 0 1 by directly approximating the integer order derivative with some finite difference quotients in the definition of the caputo fractional derivative see also lv and xu 20 2016 where k is the time step size under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth lv and xu 20 2016 proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate o k 3 0 1 uniformly with respect to the time variable t however in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate o k 3 0 1 uniformly with respect to the time variable t in this paper we first obtain a similar approximation scheme to the riemann liouville fractional derivative with the convergence rate o k 3 0 1 as in gao et al 11 2014 by approximating the hadamard finite part integral with the piecewise quadratic interpolation polynomials based on this scheme we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using laplace transform methods that the time discretization scheme has the convergence rate o k 3 0 1 for any fixed t n 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases numerical examples are given to show that the theoretical results are consistent with the numerical results

robust parametric classical control design this paper extends well known graphical design tools such as the nyquist plot bode plot and nichols charts to the domain of control systems containing parameter uncertainty we have specifically considered linear interval control systems namely those where the uncertain parameters appear linearly in the characteristic polynomial and vary in intervals because in these cases exact construction of frequency envelopes is possible using these developments we calculate various extremal stability margins for such systems and these are used with standard classical control design techniques to deal with robustness with respect to parametric uncertainty in addition we give an effective technique to iteratively select values of the design parameters within given ranges so that the closed loop system produces improved gain or phase margin

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

robust stability under structured and unstructured perturbations the problem of robust stability for linear time invariant single output control systems subject to both structured parametric and unstructured h sub infinity perturbations is studied a generalization of the small gain theorem which yields necessary and sufficient conditions for robust stability of a linear time invariant dynamic system under perturbations of mixed type is presented the solution involves calculating the h sub infinity norm of a finite number of extremal plants the problem of calculating the exact structured and unstructured stability margins is then constructively solved a feedback control system containing a linear time invariant plant which is subject to both structured and unstructured perturbations is considered the case where the system to be controlled is interval is treated and a nonconservative easily verifiable necessary and sufficient condition for robust stability is given the solution is based on the extremal of a finite number of line segments in the plant parameter property of a finite number of line segments in the plant parameter space along which the points closest to instability are encountered

control of discrete time hybrid stochastic systems a realistic stochastic control problem for hybrid systems with markovian jump parameters can have switching parameters in both the state and the measurement equations furthermore both the base and jump states in general are not perfectly observed there are only two existing controllers for this problem both with complexity exponentially increasing with time the authors present another control algorithm for stochastic systems with markovian jump parameters this algorithm is derived through the use of stochastic dynamic programming and is designed to be used for realistic stochastic control problems i e with noisy state observations this scheme has fixed computational requirements at each stage and a natural parallel implementation simulation results are used to compare the algorithm with previous schemes

necessary and sufficient conditions on negative imaginariness for interval siso transfer functions and their interconnection this article is concerned with the study of the negative imaginariness of interval transfer functions necessary and sufficient conditions for strictly negative imaginary interval transfer functions subject to parameter uncertainty are first derived based on kharitonov s theorem then certain properties of the interconnection system with negative imaginary interval transfer functions are analyzed finally an example is provided to illustrate the main results of the article

a discrete comparison principle for the time fractional diffusion equation abstract we prove a discrete comparison principle equivalent to a discrete maximum principle for the l1 discretisation of the caputo time derivative and the standard 3 point discretisation of the spatial derivative in the time fractional initial boundary value problem d t u p u c x t u f where p 0 but no assumption is made on the sign of c previously any discrete comparison principle relied on the assumption that c 0 the result reveals why one needs a certain condition on the temporal mesh that has been assumed by several authors when analysing numerical methods for this problem then this comparison principle is used to give a new error analysis for the case c 0 by means of a barrier function all the analysis can be extended to certain other discretisations of the fractional derivative

a root distribution criterion for interval polynomials the problem of finding the conditions under which an interval polynomial has a given number of roots in the open left half plane and the other roots in the open right half plane irrespective of the values of its coefficients is considered a simple criterion is provided to test interval polynomials for the root distribution invariance viewed as an extension of kharitonov s theorem the goal is to provide an alternative theorem and then give an efficient means of checking the root distribution invariance

besov spaces on domains in rd we study besov spaces 5 lp i2 0 p q a oo on do mains i2 in rd we show that there is an extension operator w which is a bounded mapping from b lp u onto b lp rd this is then used to derive various properties of the besov spaces such as interpolation theorems for a pair of b lp il atomic decompositions for the elements of 5 lp f2 and a description of the besov spaces by means of spline approximation

a finite difference scheme on graded meshes for time fractional nonlinear korteweg de vries equation abstract in this paper a finite difference scheme for time fractional nonlinear korteweg de vries kdv problems with the caputo type fractional derivative is presented to deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer the well known l1 scheme on graded meshes has been used for time discretization meanwhile a nonlinear finite difference approximation on uniform meshes is proposed for spatial discretization the existence stability and convergence of the numerical solution are studied by the energy method it is proved that the scheme is of min 2 r order convergence in time and of first order convergence in space where is the order of fractional derivative and r is a parameter of graded meshes furthermore in order to increase computational efficiency the corresponding fast algorithm of the presented scheme has been considered in numerical simulation a valid approach based on linear interpolation is designed to test convergence rate for the scheme on temporal graded meshes numerical results demonstrate sharpness of the theoretical convergence estimate and effectiveness of the fast algorithm

on bogovskiĭ and regularized poincare integral operators for de rham complexes on lipschitz domains we study integral operators related to a regularized version of the classical poincare path integral and the adjoint class generalizing bogovski s integral operator acting on differential forms in mathbb r n we prove that these operators are pseudodifferential operators of order 1 the poincare type operators map polynomials to polynomials and can have applications in finite element analysis for a domain starlike with respect to a ball the special support properties of the operators imply regularity for the de rham complex without boundary conditions using poincare type operators and with full dirichlet boundary conditions using bogovski type operators for bounded lipschitz domains the same regularity results hold and in addition we show that the cohomology spaces can always be represented by fancyscript c infty functions

error analysis of a finite element method with gmmp temporal discretisation for a time fractional diffusion equation abstract a time fractional initial boundary value problem is considered where the spatial domain has dimension d 1 2 3 the spatial differential operator is a standard elliptic operator and the time derivative is a caputo derivative of order 0 1 to discretise in space we use a standard piecewise polynomial finite element method while for the temporal discretisation the gmmp scheme a variant of the grunwald letnikov scheme is used on a uniform mesh the analysis of the gmmp scheme for solutions that exhibit a typical weak singularity at the initial time t 0 has not previously been considered in the literature a global convergence result is derived in l l 2 then a more delicate analysis of the error in this norm shows that away from t 0 the method attains optimal rate convergence numerical results confirm the sharpness of the theoretical error bounds

some results on quadratic stability of switched systems with interval uncertainty in this paper we present some results on the quadratic stability of switched systems with uncertain parameters we show that the quadratic stability of a class of switched uncertain systems may be readily verified using simple algebraic conditions

existence of weak solutions for compressible navier stokes equations with entropy transport we consider the compressible navier stokes system with variable entropy the pressure is a nonlinear function of the density and the entropy potential temperature which unlike in the navier stokes fourier system satisfies only the transport equation we provide existence results within three alternative weak formulations of the corresponding classical problem our constructions hold for the optimal range of the adiabatic coefficients from the point of view of the nowadays existence theory

a second order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel abstract a second order accurate numerical method with graded meshes is proposed and analyzed for an evolution equation with a weakly singular kernel the graded meshes are employed to compensate for the singular behavior of the exact solution at t 0 for the time discretization the product integration rule is used to approximate the riemann liouville fractional integral a generalized crank nicolson time stepping is considered and shown that the error is of order k 2 where k denotes the maximum time step a fully discrete difference scheme is constructed with space discretization by compact difference method numerical experiment is carried out to support the theoretical results the comparison between the method on uniform grids and graded grids shows the efficiency of our method

robust stability manifolds for multilinear interval systems the stability of a class of multilinearly perturbed families of systems is considered it is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved the extremal property of these subsets is established the results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces

vertices and segments of interval plants are not sufficient for step response analyses abstract interval plants are of interest in control theory as models of uncertain systems they are useful because many worst case analyses of these models are simple to carry out for example robust stability of an interval plant can be determined by investigating only the four kharitonov vertices of the denominator polynomial also the maximum peak of the bode magnitude plot can be found using just 16 special plant vertices these 16 vertices are connected by 32 special line segments most stability and frequency domain analyses that cannot be done using only the special vertices can be carried out using just the segments from these results it is tempting to conjecture that the 16 vertices or at least the 32 segments are adequate for step response analyses this paper presents examples showing that these conjectures are not true

root clustering for convex combination of complex polynomials in some important applications such as the edge theorem it is required that a polynomial is alpha stable along a parameter segment using the critical constraint the necessary and sufficient conditions for root clustering alpha stability of convex combinations of complex polynomials are presented the approach is general and requires that a certain real polynomial has no zeros in the open interval

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

incompressible flows with piecewise constant density we investigate the incompressible navier stokes equations with variable density the aim is to prove existence and uniqueness results in the case of discontinuous initial density in dimension n 2 3 assuming only that the initial density is bounded and bounded away from zero and that the initial velocity is smooth enough we get the local in time existence of unique solutions uniqueness holds in any dimension and for a wider class of velocity fields in particular all those results are true for piecewise constant densities with arbitrarily large jumps global results are established in dimension two if the density is close enough to a positive constant and in n dimensions if in addition the initial velocity is small the lagrangian formulation for describing the flow plays a key role in the analysis that is proposed in the present paper

technical communique a note on absolute stability of uncertain systems the absolute stability of nonlinear systems is an important robustness issue which has been studied intensively since its first formulation by lur e recently extensions of the solutions to different versions of this problem have been developed for cases with either structured or unstructured uncertainties in the linear part in this note the circle criterion and the popov criterion are recast in an even more general setting one which accommodates both types of perturbations in the linear plant numerical examples are provided to illustrate the application of these frequency domain tools

modified output error identification elimination of the spr condition in this note we present a modified adaptive output error algorithm for identification we refer to both the continuous time and the discrete time cases in the standard algorithm stability requires that a certain transfer function should be strictly positive real spr however in the present algorithm the spr condition which is difficult to satisfy is eliminated this is achieved by adding a fixed feedback gain when there exists an a priori partial knowledge about the parameters of the plant we provide a simple procedure to design that gain an illustrative example comparing the modified algorithm to the standard one is provided

finite difference spectral approximation for a time space fractional equation on two and three space dimensions abstract we consider the numerical method for solving a time space fractional equation whose solution has a weak initial singularity l 1 scheme on graded mesh is used to capture the initial weak singularity it is shown that the method can recover order 2 convergence by choosing some appropriate grading parameters where 0 1 is the order of temporal caputo fractional derivative a fully discrete scheme is proposed by combining spectral approximation for the discretization of space in 2d and 3d cases stability and convergence of the fully discrete scheme are rigorously established numerical results are presented to show the sharpness of the error estimate