conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

optimizability and estimatability for infinite dimensional linear systems an infinite dimensional linear system described by dot x t ax t bu t t geq 0 is said to be optimizable if for every initial state x 0 an input u in l 2 can be found such that x in l 2 here a is the generator of a strongly continuous semigroup on a hilbert space and b is an admissible control operator for this semigroup in this paper we investigate optimizability also known as the finite cost condition and its dual estimatability we explore the connections with stabilizability and detectability we give a very general theorem about the equivalence of input output stability and exponential stability of well posed linear systems the two are equivalent if the system is optimizable and estimatable we conclude that a well posed system is exponentially stable if and only if it is dynamically stabilizable and input output stable we illustrate the theory by two examples based on pdes in two or more space dimensions the wave equation and a structural acoustics model

constrained candecomp parafac via the lasso the candecomp parafac cp model is a well known tool for summarizing a three way array by extracting a limited number of components unfortunately in some cases the model suffers from the so called degeneracy that is a solution with diverging and uninterpretable components to avoid degeneracy orthogonality constraints are usually applied to one of the component matrices this solves the problem only from a technical point of view because the existence of orthogonal components underlying the data is not guaranteed for this purpose we consider some variants of the cp model where the orthogonality constraints are relaxed either by constraining only a pair or a subset of components or by stimulating the cp solution to be possibly orthogonal we theoretically clarify that only the latter approach based on the least absolute shrinkage and selection operator and named the cp lasso is helpful in solving the degeneracy problem the results of the application of cp lasso on simulated and real life data show its effectiveness

on the non existence of optimal solutions and the occurrence of degeneracy in the candecomp parafac model the candecomp parafac cp model decomposes a three way array into a prespecified number of r factors and a residual array by minimizing the sum of squares of the latter it is well known that an optimal solution for cp need not exist we show that if an optimal cp solution does not exist then any sequence of cp factors monotonically decreasing the cp criterion value to its infimum will exhibit the features of a so called degeneracy that is the parameter matrices become nearly rank deficient and the euclidean norm of some factors tends to infinity we also show that the cp criterion function does attain its infimum if one of the parameter matrices is constrained to be column wise orthonormal

degeneracy in candecomp parafac explained for p p 2 arrays of rank p 1 or higher the candecomp parafac cp model decomposes a three way array into a prespecified number r of rank 1 arrays and a residual array in which the sum of squares of the residual array is minimized the practical use of cp is sometimes complicated by the occurrence of so called degenerate solutions in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large we consider the real valued cp model in which p p 2 arrays of rank p 1 or higher are decomposed into p rank 1 arrays and a residual array it is shown that the cp objective function does not have a minimum in these cases but an infimum moreover any sequence of cp approximations of which the objective value approaches the infimum will become degenerate this result extends ten berge kiers de leeuw 1988 who consider a particular 2 2 2 array of rank 3

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

on the minimal ranks of matrix pencils and the existence of a best approximate block term tensor decomposition abstract under the action of the general linear group with tensor structure the ranks of matrices a and b forming an m n pencil a b can change but in a restricted manner specifically with every pencil one can associate a pair of minimal ranks which is unique up to a permutation this notion can be defined for matrix pencils and more generally also for matrix polynomials of arbitrary degree in this paper we provide a formal definition of the minimal ranks discuss its properties and the natural hierarchy it induces in a pencil space then we show how the minimal ranks of a pencil can be determined from its kronecker canonical form for illustration we classify the orbits according to their minimal ranks under the action of the general linear group in the case of real pencils with m n 4 subsequently we show that real regular 2 k 2 k pencils having only complex valued eigenvalues which form an open positive volume set do not admit a best approximation in the norm topology on the set of real pencils whose minimal ranks are bounded by 2 k 1 our results can be interpreted from a tensor viewpoint where the minimal ranks of a degree d 1 matrix polynomial characterize the minimal ranks of matrices constituting a block term decomposition of an m n d tensor into a sum of matrix vector tensor products

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

pointwise pressure observations of a canonical structural acoustics model in this paper we study regularity properties of a physically motivated observation associated with a controlled partial differential equation pde model on a rectangular domain this particular pde describes the active control of acoustic pressure in a chamber by means of piezoelectric activation and involves a wave coupled to a structurally damped beam equation with the coupling being accomplished by boundary trace terms the observation here consists of pointwise in space measurements of the velocity of the wave component with this velocity representing the acoustic pressure of the system as defined then the observation of the system like the control imput is modeled by an unbounded operator the main result of this paper is the discovery of an energy space such that for initial data taken from this space the desired observation of the acoustic pressure is well defined

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

explicit candecomp parafac solutions for a contrived 2 2 2 array of rank three kruskal harshman and lundy have contrived a special 2 2 2 array to examine formal properties of degenerate candecomp parafac solutions it is shown that for this array the candecomp parafac loss has an infimum of 1 in addition the array will be used to challenge the tradition of fitting indscal and related models by means of the candecomp parafac process

three mode factor analysis by means of candecomp parafac a three mode covariance matrix contains covariances of n observations e g subject scores on j variables for k different occasions or conditions we model such an jk jk covariance matrix as the sum of a common covariance matrix having candecomp parafac form and a diagonal matrix of unique variances the candecomp parafac form is a generalization of the two mode case under the assumption of parallel factors we estimate the unique variances by minimum rank factor analysis the factors can be chosen oblique or orthogonal our approach yields a model that is easy to estimate and easy to interpret moreover the unique variances the factor covariance matrix and the communalities are guaranteed to be proper a percentage of explained common variance can be obtained for each variable condition combination and the estimated model is rotationally unique under mild conditions we apply our model to several datasets in the literature and demonstrate our estimation procedure in a simulation study

exponential stability of coupled beams with dissipative joints a frequency domain approach two examples of coupled euler bernoulli beams with a dissipative joint are considered the joint placements that lead to exponential stability for these systems are characterized the technique used shows input output stability of a related controlled observed system and then shows that in these examples input output stability implies exponential stability in the first example the energy dissipation arises from a discontinuity in the shear at the joint in the second example the energy dissipation arises from a discontinuity in the bending moment at the joint the analysis of this system involves a complete spectrum analysis of the zero dynamics of the associated controlled observed system

degeneracy in candecomp parafac and indscal explained for several three sliced arrays with a two valued typical rank the candecomp parafac cp method decomposes a three way array into a prespecified number r of rank 1 arrays by minimizing the sum of squares of the residual array the practical use of cp is sometimes complicated by the occurrence of so called degenerate sequences of solutions in which several rank 1 arrays become highly correlated in all three modes and some elements of the rank 1 arrays become arbitrarily large we consider the real valued cp decomposition of all known three sliced arrays i e of size p q 3 with a two valued typical rank these are the 5 3 3 and 8 4 3 arrays and the 3 3 4 and 3 3 5 arrays with symmetric 3 3 slices in the latter two cases cp is equivalent to the indscal model for a typical rank of m m 1 we consider the cp decomposition with r m of an array of rank m 1 we show that in most cases the cp objective function does not have a minimum but an infimum moreover any sequence of feasible cp solutions in which the objective value approaches the infimum will become degenerate we use the tools developed in stegeman 2006 who considers p p 2 arrays and present a framework of analysis which is of use to the future study of cp degeneracy related to a two valued typical rank moreover our examples show that cp uniqueness is not necessary for degenerate solutions to occur

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

well posedness of a model for structural acoustic coupling in a cavity enclosed by a thin cylindrical shell abstract a fully coupled mathematical model describing the interactions between a vibrating thin cylindrical shell and an enclosed acoustic field is presented because the model will ultimately be used in control applications involving piezoceramic actuators the loads and material contributions resulting from piezoceramic patches bonded to the shell are included in the discussion theoretical and computational issues lead to the consideration of a weak form of the modeling set of partial differential equations pdes and through the use of a semigroup formulation well posedness results for the system model are obtained

lack of time delay robustness for stabilization of a structural acoustics model in this paper we consider a natural robustness question for a model for structural acoustics this model which has been of great interest in recent years is represented by a wave equation in r 2 coupled to a kelvin voigt beam the coupling is natural physically and is represented mathematically by highly unbounded operators we assume that the observation consists of point evaluation of the beam position the beam velocity and the wave velocity we are interested in the effect of arbitrarily small delays in the feedback loop on a controller that uses these observations we show that it is not possible to construct a dynamic stabilizer of a very general form including static feedback such that the stabilization is robust with respect to delays in the feedback loop in order to do this we need to carefully analyze the input to output map finally we relate these results to already existing numerical results obtained for a galerkin approximation of the system

simplicity of core arrays in three way principal component analysis and the typical rank of p q 2 arrays abstract interpreting the solution of a principal component analysis of a three way array is greatly simplified when the core array has a large number of zero elements the possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand and by mathematical analysis on the other in the present paper it is shown that a p q 2 array with p q 2 can almost surely be transformed to have all but 2 q elements zero it is also shown that arrays of that form have three way rank p at most this has direct implications for the typical rank of p q 2 arrays also when p q when p 2q the typical rank is 2 q when q it is p and when p q the rank is typically almost surely p or p 1 these typical rank results pertain to the decomposition of real valued three way arrays in terms of real valued rank one arrays and do not apply in the complex setting where the typical rank of p q 2 arrays is also min p 2q when p q but it is p when p q

robustness and continuity of the spectrum for uncertain distribution parameter systems we analyze the spectrum of a class of abstract partial differential equations with boundary feedback control if we interpret the feedback as a perturbation then we obtain a robustness radius for spectrum placement we obtain continuity of the spectrum using this radius and rouche s theorem we exploit extensively recent developments in the representation theory for abstract linear systems especially regular systems we also analyze ill posed systems by interchanging the roles of input and output we apply these results to a wave equation and an ill posed beam equation

the special sign indeterminacy of the direct fitting parafac2 model some implications cautions and recommendations for simultaneous component analysis parafac2 is the most flexible simultaneous component analysis sca model that produces an essentially unique solution in this paper we discuss how parafac2 s special sign indeterminacy affects applications of sca and we reveal how an external criterion variable can be used to ensure that estimated parafac2 weights are meaningfully signed across the levels of the nesting mode we present an example with real data from clinical psychology that illustrates the importance of parafac2 s special sign indeterminacy as well as the effectiveness of our proposed solution we also discuss the implications of our results for general applications of sca

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

second order systems with acceleration measurements accelerometers are often used to measure the output of second order systems such as structural vibrations conditions under which these systems are well posed are obtained we also establish conditions under which these systems have minimum phase transfer functions

robustness with respect to small delays for exponential stability of non autonomous systems robustness of stability with respect to small delays e g motivated by feedback systems in control theory is of great theoretical and practical important but this property does not hold for many systems in this paper we introduce the conception of robustness with respect to small time varying delays for exponential stability of the non autonomous linear systems sufficient conditions are given for the non autonomous systems to be robust and examples are provided to illustrate that the conditions are satisfied for a large class of the non autonomous parabolic systems

well posedness of a structural acoustics control model with point observation of the pressure abstract we consider a controlled and observed partial differential equation pde which describes a structural acoustics interaction physically this pde describes an acoustic chamber with a flexible chamber wall the control is applied to this flexible wall and the class of controls under consideration includes those generated by piezoceramic patches the observation we consider is point measurements of acoustic pressure inside the cavity mathematically the model consists of a wave equation coupled through boundary trace terms to a structurally damped plate or beam equation and the point controls and observations for this system are modeled by highly unbounded operators we analyze the map from the control to the observation since the properties of this map are central to any control design which is based upon this observation we also show there exists an appropriate state space x so that if the initial state is in x and the control is in l2 then the state evolves continuously in x and the observation is in l2 the analysis of this system entails a microlocal analysis of the wave component of the system and the use of pseudodifferential machinery

kruskal s polynomial for 2 2 2 arrays and a generalization to 2 n n arrays a remarkable difference between the concept of rank for matrices and that for three way arrays has to do with the occurrence of non maximal rank the set ofn n matrices that have a rank less thann has zero volume kruskal pointed out that a 2 2 2 array has rank three or less and that the subsets of those 2 2 2 arrays for which the rank is two or three both have positive volume these subsets can be distinguished by the roots of a certain polynomial the present paper generalizes kruskal s results to 2 n n arrays incidentally it is shown that twon n matrices can be diagonalized simultaneously with positive probability

subtracting a best rank 1 approximation may increase tensor rank it has been shown that a best rank r approximation of an order k tensor may not exist when r 2 and k 3 this poses a serious problem to data analysts using tensor decompositions it has been observed numerically that generally this issue cannot be solved by consecutively computing and subtracting best rank 1 approximations the reason for this is that subtracting a best rank 1 approximation generally does not decrease tensor rank in this paper we provide a mathematical treatment of this property for real valued 2 2 2 tensors with symmetric tensors as a special case regardless of the symmetry we show that for generic 2 2 2 tensors which have rank 2 or 3 subtracting a best rank 1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank 2 and rank 3 sets hence for a typical tensor of rank 2 subtracting a best rank 1 approximation increases the tensor rank

on the generic and typical ranks of 3 tensors abstract we study the generic and typical ranks of 3 tensors of dimension l m n using results from matrices and algebraic geometry we state a conjecture about the exact values of the generic rank of 3 tensors over the complex numbers which is verified numerically for l m n 14 we also discuss the typical ranks over the real numbers and give an example of an infinite family of 3 tensors of the form l m n m 1 2 1 m 3 4 which have at least two typical ranks

low gain control of uncertain regular linear systems it is well known that closing the loop around an exponentially stable finite dimensional linear time invariant plant with square transfer function matrix bg s compensated by a controller of the form k s gamma 0 where k in bbb r and gamma 0 in bbb r mm will result in an exponentially stable closed loop system which achieves tracking of arbitrary constant reference signals provided that i all the eigenvalues of bg 0 gamma 0 have positive real parts and ii the gain parameter k is positive and sufficiently small in this paper we consider a rather general class of infinite dimensional linear systems called regular systems for which convenient representations are known to exist both in time and in frequency domain the purpose of the paper is twofold i we extend the above result to the class of exponentially stable regular systems and ii we show how the parameters k and gamma 0 can be tuned adaptively the resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms nor is the injection of probing signals required