spectral theory of multiple intervals we present a model for spectral theory of families of selfadjoint operators and their corresponding unitary one parameter groups acting in hilbert space the models allow for a scale of complexity indexed by the natural numbers n for each n 2 n we get families of selfadjoint operators indexed by i the unitary matrix group u n and by ii a prescribed set of n non overlapping intervals take to be the complement in r of n fixed closed finite and disjoint intervals and let l 2 be the corresponding hilbert space moreover given b 2 u n then both the lengths of the respective intervals and the gaps between them show up as spectral parameters in our corresponding spectral resolutions within l 2 our models have two advantages one they encompass realistic features from quantum theory from acoustic wave equations and their obstacle scattering as well as from harmonic analysis secondly each choice of the parameters in our models n 2 n b 2 u n and interval configura tion allows for explicit computations and even for closed form formulas computation of spectral resolutions of generalized eigenfunctions in l 2 for the continuous part of spectrum and for scattering coecients our models further allow us to identify embedded point spectrum in the continuum corresponding for example to bound states in scattering to trapped states and to barriers in quantum scattering the possibilities for the discrete atomic part of spectrum includes both periodic and non periodic distributions

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

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

pod deim model order reduction for strain softening viscoplasticity we demonstrate a model order reduction technique for a system of nonlinear equations arising from the finite element method fem discretization of the three dimensional quasistatic equilibrium equation equipped with a perzyna viscoplasticity constitutive model the procedure employs the proper orthogonal decomposition galerkin pod g in conjunction with the discrete empirical interpolation method deim for this purpose we collect samples from a standard full order fem analysis in the offline phase and cluster them using a novel kk means clustering algorithm the pod and the deim algorithms are then employed to construct a corresponding reduced order model in the online phase a sample from the current state of the system is passed at each time step to a nearest neighbor classifier in which the cluster that best describes it is identified the force vector and its derivative with respect to the displacement vector are approximated using deim and the system of nonlinear equations is projected onto a lower dimensional subspace using the pod g the constructed reduced order model is applied to two typical solid mechanics problems showing strain localization a tensile bar and a wall under compression and a three dimensional square footing problem

the spectrum of differential operators and square integrable solutions abstract we give a comprehensive account of the relationship between the square integrable solutions for real values of the spectral parameter and the spectrum of self adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by weidmann in his well known 1987 monograph according to a well known result if one endpoint is regular and for some real value of the spectral parameter the number of linearly independent square integrable solutions is less than d then is in the essential spectrum of every self adjoint realization of the equation weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied here we prove this result without the additional condition

multivariate mimo fir inverses the problem of computing exact finite impulse response fir inverses for multivariate multiple input multiple output mimo fir systems is considered necessary and sufficient conditions for invertibility are given along with computation techniques random systems and structured systems are defined necessary and sufficient conditions for the almost sure invertibility of structured random systems are derived bounds on the orders of the inverse filters are computed

on the control of finite support behaviors abstract in the paper the main properties of finite support behaviors are introduced and characterized in terms of the polynomial matrices involved in their representation necessary and sufficient conditions for the existence of an input output possibly non anticipating description of a finite support behavior are given the control problem is also addressed and properly restated in the behavior theory context as an interconnection problem conditions for the existence of controllers that make the resulting connected system observable as well as a complete description of the class of characteristic polynomials that the resulting controlled system possibly exhibits are given finally the problem of controlling a dynamical system in input output from by means of a controller with the same structure is investigated and solved

a construction of mds 2d convolutional codes of rate 1 n based on superregular matrices abstract in this paper two dimensional convolutional codes with finite support are considered i e convolutional codes whose codewords have compact support indexed in n 2 and take values in f n where f is a finite field the main goal of this work is to analyze the free distance properties of this type of codes of rate 1 n and degree we first establish an upper bound on the maximum possible distance for these codes we then present particular constructions of two dimensional convolutional codes with finite support of rate 1 n and degree that attain such a bound and therefore have the maximum distance among all two dimensional convolutional codes with finite support with the same rate and degree we call such codes maximum distance separable two dimensional convolutional codes

the dynamics of group codes dual abelian group codes and systems fundamental results concerning the dynamics of abelian group codes behaviors and their duals are developed duals of sequence spaces over locally compact abelian lca groups may be defined via pontryagin duality dual group codes are orthogonal subgroups of dual sequence spaces the dual of a complete code or system is finite and the dual of a laurent code or system is anti laurent if c and c sup spl perp are dual codes then the state spaces of c act as the character groups of the state spaces of c sup spl perp the controllability properties of c are the observability properties of c sup spl perp in particular c is strongly controllable if and only if c sup spl perp is strongly observable and the controller memory of c is the observer memory of c sup spl perp the controller granules of c act as the character groups of the observer granules of c sup spl perp examples of minimal observer form encoder and syndrome former constructions are given finally every observer granule of c is an end around controller granule of c

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

spectral theory of hamiltonian systems with almost constant coefficients abstract we derive the spectral theory for general linear hamiltonian systems the coefficients are assumed to be asymptotically constant and satisfy certain smoothness and decay conditions these latter constraints preclude the appearance of singular continuous spectra the results are thus far reaching extensions of earlier theorems of the authors two three and four dimensional systems are studied in greater detail the results also apply to the case of the dirichlet index and dirichlet spectrum

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

multidimensional filter bank signal reconstruction from multichannel acquisition we study the theory and algorithms of an optimal use of multidimensional signal reconstruction from multichannel acquisition by using a filter bank setup suppose that we have an channel convolution system referred to as analysis filters in dimensions instead of taking all the data and applying multichannel deconvolution we first reduce the collected data set by an integer uniform sampling matrix and then search for a synthesis polyphase matrix which could perfectly reconstruct any input discrete signal first we determine the existence of perfect reconstruction pr systems for a given set of finite impulse response fir analysis filters second we present an efficient algorithm to find a sampling matrix with maximum sampling rate and to find a fir pr synthesis polyphase matrix for a given set of fir analysis filters finally once a particular fir pr synthesis polyphase matrix is found we can characterize all fir pr synthesis matrices and then find an optimal one according to design criteria including robust reconstruction in the presence of noise

exact image deconvolution from multiple fir blurs we address the problem of restoring an image from its noisy convolutions with two or more blur functions channels deconvolution from multiple blurs is in general better conditioned than from a single blur and can be performed without regularization for moderate noise levels we characterize the problem of missing data at the image boundaries and show that perfect reconstruction is impossible even in the no noise case almost surely unless there are at least three channels conversely when there are at least three channels we show that perfect reconstruction is not only possible almost surely in the absence of noise but also that it can be accomplished by finite impulse response fir filtering such fir reconstruction is vastly more efficient computationally than the least squares solution and is suitable for low noise levels even in the high noise case the estimates obtained by fir filtering provide useful starting points for iterative least squares algorithms we present results on the minimum possible sizes of such deconvolver filters we derive expressions for the mean square errors in the fir reconstructions and show that performance comparable to that of the least squares reconstruction may be obtained with relatively small deconvolver filters finally we demonstrate the fir reconstruction on synthetic and real data

multidimensional multichannel fir deconvolution using grobner bases we present a new method for general multidimensional multichannel deconvolution with finite impulse response fir convolution and deconvolution filters using grobner bases previous work formulates the problem of multichannel fir deconvolution as the construction of a left inverse of the convolution matrix which is solved by numerical linear algebra however this approach requires the prior information of the support of deconvolution filters using algebraic geometry and grobner bases we find necessary and sufficient conditions for the existence of exact deconvolution fir filters and propose simple algorithms to find these deconvolution filters the main contribution of our work is to extend the previous grobner basis results on multidimensional multichannel deconvolution for polynomial or causal filters to general fir filters the proposed algorithms obtain a set of fir deconvolution filters with a small number of nonzero coefficients a desirable feature in the impulsive noise environment and do not require the prior information of the support moreover we provide a complete characterization of all exact deconvolution fir filters from which good fir deconvolution filters under the additive white noise environment are found simulation results show that our approaches achieve good results under different noise settings

generic invertibility of multidimensional fir filter banks and mimo systems in this paper we study the invertibility of m variate laurent polynomial n times p matrices such matrices represent multidimensional systems in various settings such as filter banks multiple input multiple output systems and multirate systems given an n times p laurent polynomial matrix h z1 zm of degree at most k we want to find a p times n laurent polynomial left inverse matrix g z of h z such that g z h z j we provide computable conditions to test the invertibility and propose algorithms to find a particular inverse the main result of this paper is to prove that h z is generically invertible when n p ges m whereas when n p m then h z is generically noninvertible as a result we propose an algorithm to find a particular inverse of a laurent polynomial matrix that is faster than current algorithms known to us

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics summary a priori model reduction methods based on separated representations are introduced for the prediction of the low frequency response of uncertain structures within a parametric stochastic framework the proper generalized decomposition method is used to construct a quasi optimal separated representation of the random solution at some frequency samples at each frequency an accurate representation of the solution is obtained on reduced bases of spatial functions and stochastic functions an extraction of the deterministic bases allows for the generation of a global reduced basis yielding a reduced order model of the uncertain structure which appears to be accurate on the whole frequency band under study and for all values of input random parameters this strategy can be seen as an alternative to traditional constructions of reduced order models in structural dynamics in the presence of parametric uncertainties this reduced order model can then be used for further analyses such as the computation of the response at unresolved frequencies or the computation of more accurate stochastic approximations at some frequencies of interest because the dynamic response is highly nonlinear with respect to the input random parameters a second level of separation of variables is introduced for the representation of functions of multiple random parameters thus allowing the introduction of very fine approximations in each parametric dimension even when dealing with high parametric dimension copyright 2011 john wiley sons ltd

galerkin v least squares petrov galerkin projection in nonlinear model reduction least squares petrov galerkin lspg model reduction techniques such as the gauss newton with approximated tensors gnat method have shown promise as they have generated stable accurate solutions for large scale turbulent compressible flow problems where standard galerkin techniques have failed however there has been limited comparative analysis of the two approaches this is due in part to difficulties arising from the fact that galerkin techniques perform optimal projection associated with residual minimization at the time continuous level while lspg techniques do so at the time discrete level this work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators linear multistep schemes and runge kutta schemes we present a number of new findings including conditions under which the lspg rom has a time continuous representation conditions under which the two techniques are equivalent and time discrete error bounds for the two approaches perhaps most surprisingly we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the lspg rom instead the time step should be matched to the spectral content of the reduced basis in numerical experiments carried out on a turbulent compressible flow problem with over one million unknowns we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the lspg reduced order model by an order of magnitude

translation representations and scattering by two intervals studying unitary one parameter groups in hilbert space u t h we show that a model for obstacle scattering can be built up to unitary equivalence with the use of translation representations for l2 functions in the complement of two finite and disjoint intervals the model encompasses a family of systems u t h for each we obtain a detailed spectral representation and we compute the scattering operator and scattering matrix we illustrate our results in the lax phillips model where u t h represents an acoustic wave equation in an exterior domain and in quantum tunneling for dynamics of quantum states

perfect blind restoration of images blurred by multiple filters theory and efficient algorithms we address the problem of restoring an image from its noisy convolutions with two or more unknown finite impulse response fir filters we develop theoretical results about the existence and uniqueness of solutions and show that under some generically true assumptions both the filters and the image can be determined exactly in the absence of noise and stably estimated in its presence we present efficient algorithms to estimate the blur functions and their sizes these algorithms are of two types subspace based and likelihood based and are extensions of techniques proposed for the solution of the multichannel blind deconvolution problem in one dimension we present memory and computation efficient techniques to handle the very large matrices arising in the two dimensional 2 d case once the blur functions are determined they are used in a multichannel deconvolution step to reconstruct the unknown image the theoretical and practical implications of edge effects and weakly exciting images are examined finally the algorithms are demonstrated on synthetic and real data

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

filter bank frame expansions with erasures we study frames for robust transmission over the internet in our previous work we used quantized finite dimensional frames to achieve resilience to packet losses here we allow the input to be a sequence in l sub 2 z and focus on a filter bank implementation of the system we present results in parallel r sup n or c sup n versus l sub 2 z and show that uniform tight frames as well as newly introduced strongly uniform tight frames provide the best performance

proper orthogonal decomposition and its applications part i theory abstract in view of the increasing popularity of the application of proper orthogonal decomposition pod methods in engineering fields and the loose description of connections among the pod methods the purpose of this paper is to give a summary of the pod methods and to show the connections among these methods firstly the derivation and the performance of the three pod methods karhunen loeve decomposition kld principal component analysis pca and singular value decomposition svd are summarized then the equivalence problem is discussed via a theoretical comparison among the three methods the equivalence of the matrices for processing the objective functions the optimal basis vectors the mean square errors and the asymptotic connections of the three methods are demonstrated and proved when the methods are used to handle the pod of discrete random vectors

localized discrete empirical interpolation method this paper presents a new approach to construct more efficient reduced order models for nonlinear partial differential equations with proper orthogonal decomposition and the discrete empirical interpolation method deim whereas deim projects the nonlinear term onto one global subspace our localized discrete empirical interpolation method ldeim computes several local subspaces each tailored to a particular region of characteristic system behavior then depending on the current state of the system ldeim selects an appropriate local subspace for the approximation of the nonlinear term in this way the dimensions of the local deim subspaces and thus the computational costs remain low even though the system might exhibit a wide range of behaviors as it passes through different regimes ldeim uses machine learning methods in the offline computational phase to discover these regions via clustering local deim approximations are then computed for each cluster in the online computational phase machine

on essential spectra of singular linear hamiltonian systems abstract the paper is concerned with essential spectra of singular linear hamiltonian systems of arbitrary order with arbitrary equal defect indices several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding hamiltonian system and a sufficient and necessary condition for the essential spectral points is obtained for hamiltonian systems of even order with one singular endpoint an advantage of these results is that one can determine the essential spectral points of hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations in addition two illustrative examples are provided to show how to get some information about the essential spectrum by our results

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

real parameter square integrable solutions and the spectrum of differential operators abstract we continue to investigate the connection between the spectrum of self adjoint ordinary differential operators with arbitrary deficiency index d and the number of linearly independent square integrable solutions for real values of the spectral parameter we show that if for all in an open interval i there are d linearly independent square integrable solutions then there is no continuous spectrum in i this for any self adjoint realization with boundary conditions which may be separated coupled or mixed the proof is based on a new characterization of self adjoint domains and on limit point lp and limit circle lc solutions established in an earlier paper

hyper reduction of mechanical models involving internal variables we propose to improve the efficiency of the computation of the reduced state variables related to a given reduced basis this basis is supposed to be built by using the snapshot proper orthogonal decomposition pod model reduction method in the framework of non linear mechanical problems involving internal variables the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model this drawback is due to the fact that using a galerkin formulation the size of the reduced basis has no effect on the complexity of the constitutive equations in this paper we show how a reduced basis approximation and a petrov galerkin formulation enable to reduce the computational effort related to the internal variables the key concept is a reduced integration domain where the integration of the constitutive equations is performed the local computations being not made over the entire domain we extrapolate the computed internal variable over the full domain using pod vectors related to the internal variables this paper shows the improvement of the computational saving obtained by the hyper reduction of the elasto plastic model of a simple structure

multidimensional systems with finite support behaviors signal structure generation and detection the main features of finite multidimensional behaviors are introduced as properties of the trajectories supports and connected with the polynomial matrices adopted for their description observability and local detectability are shown to be equivalent to the kernel representation of a behavior via some parity check matrix ht the main properties of locally undetectable behaviors as well as their connections with the notion of constrained variables are investigated and a general representation result for finite support behaviors is derived the input output representation via generator matrices is finally discussed and some connections between matrix primeness and the constraints every trajectory imposes on the support of the corresponding input are analyzed