global existence for the einstein vacuum equations in wave coordinates we prove global stability of minkowski space for the einstein vacuum equations in harmonic wave coordinate gauge for the set of restricted data coinciding with the schwarzschild solution in the neighborhood of space like infinity the result contradicts previous beliefs that wave coordinates are unstable in the large and provides an alternative approach to the stability problem originally solved for unrestricted data in a different gauge and with a precise description of the asymptotic behavior at null infinity by d christodoulou and s klainerman

global existence and asymptotic behavior for some multidimensional quasilinear hyperbolic systems abstract for the cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self interaction we show the global existence of classical solutions with small initial data we also prove that the global solution will converge to a solution of the linearized system as time tends to infinity and the convergence rate is also determined

the model matching problem for periodic discrete time systems in this paper the model matching problem via state feedback is studied for discrete time periodic systems it is shown that under some zero matching conditions it is possible to assign a target input output periodic system by means of a periodic state feedback control law furthermore this entails that a periodic system can be converted into a closed loop time invariant system via state feedback

remarks on energy approach for global existence of some one dimension quasilinear hyperbolic systems abstract we show that luli yang and yu s weighted energy estimate with positive weights 8 together with some variants of alinhac s ghost weight energy approach 1 can be used to prove the global existence of classical solutions to some one dimension quasilinear hyperbolic systems of diagonal form with small initial data

scattering for the quasilinear wave equations with null conditions in two dimensions abstract we give a detailed description of the scattering theory of solutions to the quasilinear wave equations with null conditions and small initial data in two dimensions the scattering profile is shown to be the same as the one conjectured by alinhac 1 1995 our proof is based on a framework of fourier transform by deng and pusateri 6 2018 and an expansion of oscillatory integral together with the global well posedness of the solutions by alinhac 2 2001

on the assignment of invariant factors by time varying feedback strategies abstract in this paper the realization of arbitrary invariant factors or cyclic components of the closed loop system matrix is discussed it is shown that the use of time varying feedback strategies enables one to realize arbitrary invariant factors in contrast with time invariant feedback control

almost global existence for 2 d incompressible isotropic elastodynamics author s lei z sideris tc zhou y abstract 2015 american mathematical society we consider the cauchy problem for 2 d incompressible isotropic elastodynamics standard energy methods yield local solutions on a time interval 0 t for initial data of the form u0 where t depends only on some sobolev norm of u0 we show that for such data there exists a unique solution on a time interval 0 exp t provided that is sufficiently small this is achieved by careful consideration of the structure of the nonlinearity the incompressible elasticity equation is inherently linearly degenerate in the isotropic case in other words the equation satisfies a null condition this is essential for time decay estimates the pressure which arises as a lagrange multiplier to enforce the incompressibility constraint is estimated in a novel way as a nonlocal nonlinear term with null structure the proof employs the generalized energy method of klainerman enhanced by weighted l2 estimates and the ghost weight introduced by alinhac

temps de vie des solutions regulieres des equations d euler compressibles axisymetriques en dimension deux nous considerons le systeme des equations d euler isentropiques en dimension deux pour des donnees initiales invariantes par rotation et perturbations de taille e d un etat de repos on etablit un equivalent du temps de viet e de la solution classique lim 2 t 2 de plus on donne pour t leqq frac a 2 varepsilon 2 a tau une estimation de la vraie solution en calculant la taille de son ecart a une solution approchee construite dans un precedent travail

computation of minimal periodic realizations of transfer function matrices we present a numerical approach to compute a minimal periodic state space realization of a transfer function matrix corresponding to a lifted state space representation the proposed method determines a realization with time varying state dimensions by using exclusively orthogonal transformations the new method is numerically reliable computationally efficient and thus well suited for robust software implementations

noncyclic geometric phase and its non abelian generalization we use the theory of dynamical invariants to yield a simple derivation of noncyclic analogues of the abelian and non abelian geometric phases this derivation relies only on the principle of gauge invariance and elucidates the existing definitions of the abelian noncyclic geometric phase we also discuss the adiabatic limit of the noncyclic geometric phase and compute the adiabatic non abelian noncyclic geometric phase for a spin 1 magnetic or electric quadrupole interacting with a precessing magnetic electric field

a note on robust pole assignment for periodic systems in this note a robust pole assignment algorithm is proposed for linear periodic discrete time systems with time varying dimensions of the state and or input spaces the algorithm deduces a periodic state feedback law by the minimization of the condition numbers of the eigenvector matrices of the closed loop system numerical examples are provided to show the performances of the algorithm

global existence of null form wave equations on small asymptotically euclidean manifolds abstract we prove the global existence of the small solutions to the cauchy problem for quasilinear wave equations satisfying the null condition on r 3 g where the metric g is a small perturbation of the flat metric and approaches the euclidean metric like 1 x 2 2 with 1 global and almost global existence for systems without the null condition are also discussed for certain small time dependent perturbations of the flat metric in appendix a

noncyclic and nonadiabatic geometric phase for counting statistics we propose a general framework of the geometric phase interpretation for counting statistics counting statistics is a scheme to count the number of specific transitions in a stochastic process the cumulant generating function for the counting statistics can be interpreted as a phase and it is generally divided into two parts the dynamical phase and a remaining one it has already been shown that for cyclic evolution the remaining phase corresponds to a geometric phase such as the berry phase or aharonov anandan phase we here show that the remaining phase has also an interpretation as a geometric phase even in noncyclic and nonadiabatic evolution

uniform bound of sobolev norms of solutions to 3d nonlinear wave equations with null condition abstract this article concerns the time growth of sobolev norms of classical solutions to the 3d quasi linear wave equations with the null condition given initial data in h s h s 1 with compact supports the global well posedness theory has been established independently by klainerman 13 and christodoulou 3 respectively for a relatively large integer s however the highest order sobolev energy namely the h s energy of solutions may have a logarithmic growth in time in this paper we show that the h s energy of solutions is also uniformly bounded for s 5 the proof employs the generalized energy method of klainerman enhanced by weighted l 2 estimates and the ghost weight introduced by alinhac

the global smooth symmetric solution to 2 d full compressible euler system of chaplygin gases abstract for one dimensional or multidimensional compressible euler system of polytropic gases it is well known that the smooth solution will generally develop singularities in finite time however for three dimensional chaplygin gases due to the crucial role of null condition in the potential equation which is derived by the irrotational and isentropic flow p godin in 9 has proved the global existence of a smooth 3 d spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a constant state it is noted that there are some clear differences for the global solution or blowup problems between 2 d and 3 d hyperbolic equations or systems in this paper we will focus on the global symmetric solution problem of 2 d full compressible euler system of chaplygin gases through carrying out involved analysis and finding an appropriate weight we can derive some uniform weighted energy estimates on the small symmetric solution to 2 d compressible euler system of chaplygin gases and further establish the global existence of the smooth solution by the continuous induction method

disturbance localization by measurement feedback for linear periodic discrete time systems abstract the recent notions of inner controllable outer controllable and outer reconstructible subspaces together with the extension of the notion of c a b pair of subspaces to the class of linear periodic discrete time systems are the main tools used in this paper in order to study the disturbance localization problem by measurement feedback for the same class of systems by means of these concepts the necessary and sufficient solvability conditions of such a problem and synthesis procedures of the solutions are found for three different cases i e with and without an additional requirement of output or state dead beat control

computing the zeros of periodic descriptor systems in this paper we give a numerically reliable algorithm to compute the zeros of a periodic descriptor system the algorithm is a variant of the staircase algorithm applied to the system pencil of an equivalent lifted time invariant state space system and extracts a low order pencil which contains the zeros both finite and infinite as well as the kronecker structure of the periodic descriptor system the proposed algorithm is efficient in terms of complexity by exploiting the structure of the pencil and is exclusively based on orthogonal transformations which ensures some form of numerical stability

on the real and complex geometric phases some of the existing definitions of the real and complex geometric phases are unified and generalized the relationship of the real to the complex geometric phase is discussed it is shown why these two phases are in general completely independent in particular a simple explicit example is given which shows that the real geometric phase is not in general the real part of the complex geometric phase

brief paper assigning closed loop invariant polynomials over polytopes for linear periodic systems this paper provides a parameterization of all output feedback periodic controllers providing closed loop invariant polynomials varying over some assigned polytopes for linear discrete time periodic plants using the notion of time invariant system associated to a periodic system the parameterization is provided in terms of the general solution of a suitably defined matrix diophantine equation

brief minimal representations of mimo time varying systems and realization of cyclostationary models this paper describes a family of minimally parameterized state space models for completely n step observable multi input multi output time varying systems and their algebraical links with equivalent input output models these results are then used to develop an algorithm for the realization of generic input output sequences generated by a periodic system a complete numerical example is also developed to illustrate all steps of the algorithm described in the paper

computation of transfer function matrices of periodic systems we present a numerical approach to evaluate the transfer function matrices of a periodic system corresponding to lifted state space representations as constant systems the proposed pole zero method determines each entry of the transfer function matrix in a minimal zeros poles gain representation a basic computation is the minimal realization of special single input single output periodic systems for which both balancing related as well as orthogonal periodic kalman forms based algorithms can be employed the main computational ingredient to compute poles is the extended periodic real schur form of a periodic matrix this form also underlies the solution of periodic lyapunov equations when computing minimal realizations via balancing related techniques to compute zeros and gains numerically stable fast algorithms are proposed which are specially tailored to particular single input single output periodic systems the new method relies exclusively on reliable numerical computations and is well suited f

fault detection and isolation of linear discrete time periodic systems using the geometric approach this technical note solves the problem to detect and isolate faults in linear periodic discrete time systems a residual generator is designed through a suitable unknown input observer with the requirement that each residual should be sensitive only to one fault and simultaneously insensitive to the other faults that can affect the system and with the additional requirement of detectability that is the transition matrix of its relative error system has all eigenvalues smaller than in modulus for a given positive p 1 the solution is based on geometric tools and it makes use of the outer observable subspace notion

positive n periodic descriptor control systems this paper establishes the relationship between the positive n periodic descriptor system in discrete time and its associated invariant systems reachability and controllability properties of these kinds of systems are analyzed transmission of structural properties from the n periodic system to their associated invariant systems is studied finally some comments on the stability property are made

global well posedness of the incompressible magnetohydrodynamics this paper studies the cauchy problem of the incompressible magnetohydro dynamic systems with or without viscosity under the assumption that the initial velocity field and the displacement of the initialmagnetic field froma non zero constant are sufficiently small in certain weighted sobolev spaces the cauchy problem is shown to be globally well posed for all 0 and all spaces with dimension n 2 such a result holds true uniformly in nonnegative viscosity parameters the proof is based on the inherent strong null structure of the systems introduced by lei commun pure appl math 69 11 2072 2106 2016 and the ghost weight technique introduced by alinhac invent math 145 3 597 618 2001

deformations for linear periodic discrete time systems the adjoint normal form following the technique proposed by arnold for matrices time invariant autonomous systems and extended by tannenbaum to time invariant linear control systems with inputs and outputs a normal form for linear periodic discrete time systems depending on physical parameters is proposed the considered actions are deformations i e linear periodic transformations depending on the parameters being identity at the nominal values of them

codimension one stability of the catenoid under the vanishing mean curvature flow in minkowski space we study timelike hypersurfaces with vanishing mean curvature in the 3 1 dimensional minkowski space which are the hyperbolic counterparts to minimal embeddings of riemannian manifolds the catenoid is a stationary solution of the associated cauchy problem this solution is linearly unstable and we show that this instability is the only obstruction to the global nonlinear stability of the catenoid more precisely we prove in a certain symmetry class the existence in the neighborhood of the catenoid initial data of a codimension one lipschitz manifold transverse to the unstable mode consisting of initial data whose solutions exist globally in time and converge asymptotically to the catenoid

global smooth axisymmetric solutions to 2d compressible euler equations of chaplygin gases with non zero vorticity abstract for 2d compressible isentropic euler equations of polytropic gases s alinhac establishes that the axisymmetric smooth solution blows up in the present paper for 2d compressible isentropic euler equations of chaplygin gases we show that the small perturbed smooth solution exists globally when the rotationally invariant initial data are a perturbation of size e 0 of a rest state near the light cone 2d euler equations of chaplygin gases can be transformed into a second order quasilinear wave equation which satisfies both the null conditions this will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone however away from the light cone the hydrodynamical waves have no decay in time and strongly affect the related acoustical waves by introducing a nonlinear ode we can distinguish the fast decay part and non decay part so that the global energy estimates are derived

on the zeros of blocked time invariant systems abstract this paper studies the zero properties of blocked linear systems resulting from blocking of linear time invariant systems the main idea is to establish a relation between the zero properties of blocked systems and the zero properties of their corresponding unblocked systems in particular it is shown that the blocked system has a zero if and only if the associated unblocked system has a zero furthermore the zero properties of blocked systems under a generic setting i e a setting which parameter matrices a b c d assume generic values are examined it is demonstrated that nonsquare blocked systems i e blocked systems with number of outputs unequal to the number of inputs generically have no zeros however square blocked systems i e blocked systems with equal number of inputs and outputs generically have only finite zeros and these finite zeros have geometric multiplicity one

survey invariant representations of discrete time periodic systems the correspondence between periodic and time invariant systems can take a variety of forms the purpose of this survey paper is to present a complete and clear picture on time invariant representations for discrete time periodic systems the core of the treatment evolves around the concept of periodic transfer function

balancing related methods for minimal realization of periodic systems we propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time varying dimensions the first method belongs to the family of square root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations an alternative balancing free square root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems the key numerical computation in both methods is the solution of nonnegative periodic lyapunov equations directly for the cholesky factors of the solutions for this purpose a numerically reliable computational algorithm is proposed to solve nonnegative periodic lyapunov equations with time varying dimensions

optimal and robust controllers for periodic and multirate systems the problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete time periodic systems the solution consists of solving an equivalent time invariant standard l sup 1 optimization problem subject to an additional constraint this constraint assures the causality of the resulting periodic controller by the duality theory the problem is shown to be equivalent to a linear programming problem which is no harder than the standard l sup 1 problem also it is shown that the method of solution presented applies exactly to the problem of disturbance rejection in the case of multirate sampled data systems finally the results are applied to the problem of robust stabilization of periodic and multirate systems

output stabilization via pole placement of discrete time linear periodic systems the output stabilization problem for discrete time linear periodic systems is solved both the state feedback control law and the state predictor are based on a suitable time invariant state sampled reformulation associated with a periodic system preliminary concepts of periodic system theory are briefly recalled in particular the structural properties of a linear discrete time periodic system are properly related to those of a time invariant system associated with it by resorting to such a time invariant reformulation the output stabilization problem via pole placement is solved the stabilizing controller is constituted by a control law and an asymptotic state predictor both of which are shown to be periodic

on one dimension semi linear wave equations with null conditions abstract it is well known that in space dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data it is also known that in space dimension two such result still holds for a certain class of quasi linear wave equations with null conditions the proofs are based on the decay mechanism of linear waves however in one space dimension waves do not decay nevertheless we will prove that small data still lead to global solutions if the null condition is satisfied

the realization problem for linear periodic systems abstract the realization problem for discrete time linear periodic systems is considered deep insight on the various complicated aspects of minimal quasi minimal and uniform realization is provided based on the knowledge of the structural properties of discrete time periodic systems the results hinge on the lifting isomorphism between such systems and time invariant ones

an algebraic riccati equation for the discrete time periodic prediction problem abstract in this paper the periodic problem is reformulated as a prediction problem relative to a suitable time invariant system this enables one to establish a one to one correspondence between the solutions of the discrete time periodic riccati equation dpre and the solutions of a suitable discrete time and time invariant riccati equation in particular the constant solutions of this last equation are the periodic generators of the original periodic riccati equation this enables one to easily derive existence uniqueness and convergence results for the drpe from known results relative to the time invariant case

computational issues for linear periodic systems paradigms algorithms open problems linear periodic systems originate in various control fields involving periodic phenomena in the beginnings of algorithmic developments for periodic systems computational detours have been often employed to reduce the computational problems for periodic systems to those for standard systems well known techniques are the employment of lifted representations in discrete time or the use of periodic generators in continuous time new computational paradigms evolved later whose main ingredients are numerically reliable and efficient algorithms for manipulating matrix products without forming explicitly them reduction of large scale structured matrix pencils without building the underlying pencils computing with discrete time periodic system models with time varying state dimensions or solving periodic matrix differential equations using multiple shooting techniques the paper will give a succinct overview of all these new developments and will point out some of still existing open computational problems

brief a periodically time varying minimal partial realization algorithm based on twisting the 1975 paper of meyer and burrus gave necessary conditions on the coefficients of a periodically time varying difference equation to generate a prespecified set of impulse responses the main contribution of the present paper consists of an algorithm that constructs the coefficients of such a periodically time varying realization of minimal lag thus the algorithm solves the minimal partial realization problem for the periodically time varying scalar case we use a general technique twisting for associating a time invariant system with a periodically time varying system in such a way that the time step is preserved the technique plays an essential role in the ideas underlying the algorithm

integral fluctuation theorem for the housekeeping heat the housekeeping heat qhk is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states by analysing the evolution of its probability distribution we prove an integral fluctuation theorem exp qhk 1 valid for arbitrary driven transitions between steady states we discuss gaussian limiting cases and the difference between the new theorem and both the hatano sasa and the jarzynski relation

robust tracking and regulation of linear periodic discrete time systems in this paper the robust asymptotic tracking and disturbance rejection problem is studied for linear periodic discrete time systems and for general classes of disturbance functions and reference signals consisting in free responses of periodic processes the necessary and sufficient conditions for the existence of a solution which involve the periodic notion of invariant zero are found a constructive design procedure of the periodic compensator is given it is based on the internal model principle

remarks on global solutions for nonlinear wave equations under the standard null conditions abstract a combination of some weighted energy estimates is applied for the cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions alternative proofs for global solutions are shown including the exterior domain problems

zeros of discrete time linear periodic systems in this note the concept of zero for linear discrete time siso periodic systems is discussed the definition is based on a time invariant mimo equivalent representation an input output characterization of the zeros is provided which extends a well known property for time invariant systems

the small data solutions of general 3 d quasilinear wave equations ii abstract this paper is a continuation of the work in 8 where the authors established the global existence of smooth small data solutions to the general 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0 when the weak null condition holds in the present paper we show that the smooth small data solutions of equation i j 0 3 g i j u u i j 2 u 0 will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied moreover a precise blowup time is completely determined therefore collecting the main results in this paper and 8 we have given a basically complete study on the blowup or global existence of small data solutions to the 3 d quasilinear wave equation i j 0 3 g i j u u i j 2 u 0

the geometric approach for linear periodic discrete time systems abstract recently the disturbance localization problem and the problems of designing disturbance decoupled observers and of giving a geometric characterization of invariant zeros were solved for linear periodic discrete time systems through an extension of the geometric approach based on the notions of controlled invariant and conditionally invariant subspaces to periodic ones here the existing periodic geometric theory is supplemented with the notions of outer reachable subspace and controllability subspace and with some further results on the previously introduced notions of inner reachable controllable subspace and outer controllable subspace in addition it is shown that any such periodic geometric theory can be restated in an equivalent time invariant form which is just the same theory written for a suitable time invariant system having an extended state space

parametrized solution of the periodic dead beat control problem the dead beat control problem for linear periodic discrete time systems in presence of unmeasurable disturbances and external reference signals is considered using a polynomial method a parametrization of all causal output feedback controllers solving the problem is provided

global solutions for incompressible viscoelastic fluids we prove the existence of both local and global smooth solutions to the cauchy problem in the whole space and the periodic problem in the n dimensional torus for the incompressible viscoelastic system of oldroyd b type in the case of near equilibrium initial data the results hold in both two and three dimensional spaces the results and methods presented in this paper are also valid for a wide range of elastic complex fluids such as magnetohydrodynamics liquid crystals and mixture problems

global existence for systems of wave equations with nonresonant nonlinearities and null forms abstract we consider the cauchy problem for systems of nonlinear wave equations with different propagation speeds in three space dimensions we prove global existence of small amplitude solutions for systems with some nonresonant nonlinearities which may depend on both of the unknowns and their derivatives our method here can be also adopted to treat the null forms

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

minimal periodic realizations of transfer matrices periodic controllers designed based on the so called lifting technique are usually represented by transfer matrices real operations require that the controllers be implemented as periodic systems the problem of realizing an nn sub o nn sub i proper rational transfer matrix as an n sub i input n sub o output n periodic discrete time system is studied an algorithm for obtaining periodic realizations which have a minimal number of states is proposed the result can also be used to remove any redundant states that exist in a periodic system

a new class of shift varying operators their shift invariant equivalents and multirate digital systems a class of linear shift varying operators that generalize the notion of n periodicity is defined it is shown that shift invariant equivalents for these operators exist and that the equivalence is in a strong sense preserving both algebraic and analytic system properties it is shown that multirate sampled data systems although not generally periodic fall into this class kranc vector switch decomposition and block filter implementations for single input single output multirate systems are connected under the unifying framework of shift invariant equivalents and this framework provides a way to extend them both to multi input multi output systems

brief paper mpc of constrained discrete time linear periodic systems a framework for asynchronous control strong feasibility stability and optimality via periodic invariance state feedback model predictive control mpc of discrete time linear periodic systems with time dependent state and input dimensions is considered the states and inputs are subject to periodically time dependent hard convex polyhedral constraints first periodic controlled and positively invariant sets are characterized and a method to determine the maximum periodic controlled and positively invariant sets is derived the proposed periodic controlled invariant sets are then employed in the design of least restrictive strongly feasible reference tracking mpc problems the proposed periodic positively invariant sets are employed in combination with well known results on optimal unconstrained periodic linear quadratic regulation lqr to yield constrained periodic lqr control laws that are stabilizing and optimal one motivation for systems with time dependent dimensions is efficient control law synthesis for discrete time systems with asynchronous inputs for which a novel modeling framework resulting in low dimensional models is proposed the presented methods are applied to a multirate nano positioning system

the null condition for quasilinear wave equations in two space dimensions ii we consider quasilinear wave equations with small data of size e in two space dimensions assuming the null condition is satisfied we proved in a previous paper the conjectured large time existence result in this paper using the geometric blowup techniques introduced in our previous work we prove that under generic conditions on the data blowup does occur at the specified time more precisely the blowup can be described as a geometric blowup of cusp type uniformly in e in appropriate variables taking advantage of the techniques introduced in the paper we also prove the blowup result for the technically similar case of quasilinear wave equations without null condition in three space dimensions

from singular to nonsingular filtering of periodic systems filling the gap with the spectral interactor matrix with reference to periodic discrete time systems the concept of the periodic spectral interactor matrix operator psimo is introduced such an interactor enables one to transform the singular system into a nonsingular one with identical spectral properties this concept is useful to characterize the family of periodic solutions of the singular periodic riccati equation arising from the singular filtering problem the psimo is also a synthetic tool to represent the invariant zeros structure at infinity delays of periodic systems in an operational fashion

global solutions of nonlinear hyperbolic equations for small initial data on considere des systemes quasilineaires d equations hyperboliques d ordre 2 qui sont des deformations non lineaires de l equation d onde on etudie l existence globale et le comportement asymptotique des solutions du probleme de cauchy quand la donnee initiale est suffisamment petite

properties of blocked linear systems this paper presents a systematic study on the properties of blocked linear systems that have resulted from blocking discrete time linear time invariant systems the main idea is to explore the relationship between the blocked and the unblocked systems existing results are reviewed and a number of important new results are derived focus is given particularly on the zero properties of the blocked system as no such study has been found in the literature

a period specific realization of linear continuous time systems it is a well known fact that a weighting pattern matrix has a periodic realization if and only if the matrix is separable and periodic this fact however does not cope with a reasonable question of period specific realization problem for a given weighting pattern matrix and a given positive real number find a periodic realization corresponding to the given matrix and with a period equal to the given number this paper answers the question by showing that the period specific realization problem has a solution if and only if the given weighting pattern matrix is separable and has a period equal to the given positive real number to this end two types of period specific realizations are constructed one is with a constant a matrix whose dimension is not necessarily minimal among all possible period specific realizations the other is with a non constant and periodically time varying a matrix whose dimension turned out minimal among all possible period specific realizations