applications of vector bundles to factorization of rational matrices abstract we show that wiener hopf s and forney s factorization theorems are immediate consequences of grothendieck s theorem on vector bundles

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

parallel concatenated convolutional codes from linear systems theory viewpoint abstract the aim of this work is to characterize two models of concatenated convolutional codes based on the theory of linear systems the problem we consider can be viewed as the study of composite linear system from the classical control theory or as the interconnection from the behavioral system viewpoint in this paper we provide an input state output representation of both models and introduce some conditions for such representations to be both controllable and observable we also introduce a lower bound on their free distances and the column distances

generalised autoregressive models and their trajectories the set of classical ar models is not compact so that a sequence of ar models may fail to converge to anything reasonable a compactification leads to ar models with arbitrary singularity at infinity these limit ar models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition it is shown that they produce trajectories in a very natural way then the classical equivalence theorem is extended to them stating that two ar models having the same behaviour are equivalent

descriptor representations without direct feedthrough term abstract descriptor representations are considered that are given by e a b c d with d 0 minimality under external equivalence is characterized in terms of the matrices e a b and c also transformations are given by which minimal e a b c representations are related under external equivalence the transformations turn out to be more simple than in the d 0 case algorithms for rewriting an e a b c d representation in e a b c form are also given finally a realization procedure is presented for obtaining a minimal e a b c representation for a system that is given in polynomial matrix fractional form

linear systems and taylor complexes a taylor complex is a graded collection of polynomials satisfying a few simple axioms we show that there is a natural one to one correspondence between the solution sets of linear constant coefficient differential equations and taylor complexes using this correspondence we then give a characterization of the solution sets of linear constant coefficient differential equations which follows closely willems characterization of the solution sets of linear constant coefficient difference equations we also characterize linear differential operators the presentation is given for the case of several variables

relative completeness and specifiedness properties of continuous linear dynamical systems the properties of relative completeness and specifiedness are introduced for continuous linear dynamical systems it is shown then that having these two properties and the differentiation invariance property is necessary and sufficient for a continuous linear dynamical system to be represented by means of a linear constant coefficient differential equation

input output structure of linear differential algebraic systems the problem of describing the input output structure of general linear differential algebraic systems is addressed explicit formulas are given in terms of the original parameters for systems with an arbitrary amount of redundancy these formulas allow one to establish whether the system determines an input output relation at all and if so they describe the rank of the transfer matrix and its pole zero structure at infinity the formulas may be seen as generalizations of a number of classical results on the input output structure of standard state space systems and descriptor systems satisfying certain constraints for the derivation the pencil representation rather than the descriptor representation is used

general properties of solutions to inhomogeneous black scholes equations with discontinuous maturity payoffs abstract we provide representations of solutions to terminal value problems of inhomogeneous black scholes equations and study such general properties as min max estimates gradient estimates monotonicity and convexity of the solutions with respect to the stock price variable which are important for financial security pricing in particular we focus on finding representation of the gradient with respect to the stock price variable of solutions to the terminal value problems with discontinuous terminal payoffs or inhomogeneous terms such terminal value problems are often encountered in pricing problems of compound like options such as bermudan options or defaultable bonds with discrete default barrier default intensity and endogenous default recovery our results can be used in pricing real defaultable bonds under consideration of existence of discrete coupons or taxes on coupons

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

common minimal multiples and divisors for rational matrix functions abstract we extend to rational matrix functions the theory of common multiples and common divisors of regular matrix polynomials developed in 12 attention is focused mainly on problems of existence uniqueness and construction from local data

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

first order representations of fliess models abstract fuhrmann s state space construction in its generalized form is used to obtain a general theory of first order representations of fliess models defined over an arbitrary noetherian commutative ring the case of arbitrary linear delay differential equations is involved

a new notion of equivalence for discrete time ar representations we present a new equivalence transformation termed divisor equivalence that has the property of preserving both the finite and the infinite elementary divisor structures of a square non singular polynomial matrix this equivalence relation extends the known notion of strict equivalence which dealt only with matrix pencils to the general polynomial matrix case it is proved that divisor equivalence characterizes in a closed form relation the equivalence classes of polynomial matrices that give rise to fundamentally equivalent discrete time auto regressive representations

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

equational properties of kleene algebras of relations with conversion abstract we describe a set of equational axioms for the variety generated by all algebras of binary relations with operations of union composition conversion and reflexive transitive closure and neutral elements 0 the empty relation and 1 the identity relation

on equations for union free regular languages in this paper we consider the variety uf generated by all algebras of binary relations equipped with the operations of composition reflexive transitive closure and the empty set and the identity relation as constants this variety coincides with the variety generated by the union free reducts of kleene algebras of languages and its free objects are formed by union free regular languages that is regular languages represented by regular expressions having no occurrence of we show that the variety uf is not finitely based the situation does not change if we consider the variety uf generated by the above algebras of binary relations equipped with the conversion operation

on dynamic algebras abstract dynamic algebras are the lindenbaum tarski algebras of dynamic logics these algebras can be considered as boolean algebras with some operators indexed by the elements of some kleene algebra in this paper we prove that there are infinitely many finitely generated varieties of dynamic algebras having undecidable equational theories all these varieties are generated by representable dynamic algebras

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

continuity of latent variable models the authors study the continuity of the behavior of dynamical systems as a function of the parameters in their behavioral equations a system is defined in terms of its behavior and continuity requires that this behavior be continuous in the limit the problem is demonstrated by means of an example involving an rlc circuit whose port behavior exhibits a surprising discontinuity as a function of the numerical values of the elements in the circuit the main result states that a system described by means of difference equations involving manifest external and latent internal variables will have a continuous behavior in the limit if the limit system is observable

the variety of kleene algebras with conversion is not finitely based given an arbitrary set a one obtains the full kleene algebra of binary relations over a by considering the operations of union composition reflexive transitive closure conversion and the empty set and the identity relation as constants such algebras generate the variety of kleene algebras with conversion as a result of a general analysis of identities satisfied by varieties having an involution operation we prove that the variety of kleene algebras with conversion has no finite equational axiomatization in our argument we make use of the fact that the variety of kleene algebras without conversion is not finitely based and that relatively to this variety the variety of kleene algebras with conversion is finitely axiomatized

on kleene algebras abstract in this paper we prove that the class of inversion free kleene algebras is not finitely based the main idea is to use a result of redko and salomaa for regular languages we also prove unsolvability of the word problem for kleene algebras and some other varieties of algebras

linear systems and arma and fliess models linear dynamical systems are central objects of study in linear system theory and arma and fliess models are two very important classes of models that are used to represent them this article is concerned with the question of what is a relation between them in case of higher dimensions it is shown that the category of linear systems the weak category of arma models and the category of fliess models are equivalent to each other

duality for differential difference systems over lie groups in modern mathematical systems theory there exist two consistent ways of defining and describing a linear system i in the behavioral approach a linear system is the kernel mathfrak b of a matrix valued operator r in a power of a signal space w ii in the module theoretic setting a linear system is the cokernel m of the above matrix r these two formulations have connections the minimal conditions under which they are equivalent are investigated in this paper the general theory is applied to differential difference systems over lie groups

the equational theory of kleene lattices languages and families of binary relations are standard interpretations of kleene algebras it is known that the equational theories of these interpretations coincide and that the free kleene algebra is representable both as a relation and as a language algebra we investigate the identities valid in these interpretations when we expand the signature of kleene algebras with the meet operation in both cases meet is interpreted as intersection we prove that in this case there are more identities valid in language algebras than in relation algebras exactly three more in some sense and representability of the free algebra holds for the relational interpretation but fails for the language interpretation however if we exclude the identity constant from the algebras when we add meet then the equational theories of the relational and language interpretations remain the same and the free algebra is representable as a language algebra too the moral is that only the identity constant behaves differently in the language and the relational interpretations and only meet makes this visible

on behavior homomorphisms and system equivalence in this paper we introduce the concept of a behavior homomorphisms and isomorphisms and use it to present a uni ed approach to the study of equivalence of di erent behavior representations concentrating on a special representation we call normalized arma narma representation c 2001 elsevier science b v all rights reserved

almost exact modelling assumption in adaptive linear control modem adaptive controllers are known to give bounded solutions when the system normalized unmodelled effects are bounded by a small constant this paper studies this unusual characterization of uncertainties we show it encompasses more classical approaches we discuss how feedback and by passing may allow this assumption to be satisfied we conclude by proposing the notion of almost exactly linearly modelled systems

canonical forms for polynomial and quadratic differential operators we define the canonical representative for the equivalence class consistingof all polynomial and quadratic differential operators that take the same values on a given subspaceof c

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

minimal realizations of linear systems the shortest basis approach given a discrete time linear system c a shortest basis for c is a set of linearly independent generators for c with the least possible lengths a basis b is a shortest basis if and only if it has the predictable span property i e has the predictable delay and degree properties and is non catastrophic or alternatively if and only if it has the subsystem basis property for any interval j the generators in b whose span is in j is a basis for the subsystem cj the dimensions of the minimal state spaces and minimal transition spaces of c are simply the numbers of generators in a shortest basis b that are active at any given state or symbol time respectively a minimal linear realization for c in controller canonical form follows directly from a shortest basis for c and a minimal linear realization for c in observer canonical form follows directly from a shortest basis for the orthogonal system c this approach seems conceptually simpler than that of classical minimal realization theory

analytical pricing of defaultable discrete coupon bonds in unified two factor model of structural and reduced form models abstract pricing formulae for defaultable corporate bonds with discrete coupons under consideration of the government taxes in the united two factor model of structural and reduced form models are provided the aim of this paper is to generalize the two factor structural model for defaultable corporate discrete coupon bonds considered in 1 into the unified model of structural and reduced form models in our model the bond holders receive the stochastic coupon which is the discounted value of a predetermined value at the maturity at predetermined coupon dates and the face value debt and the coupon at the maturity as well as the effect of government taxes which are paid on the proceeds of an investment in bonds is considered the expected default event occurs when the equity value is not sufficient to pay coupon or debt at the coupon dates or maturity and the unexpected default event can occur at the first jump time of a poisson process with the given default intensity provided by a step function of time variable we provide the model and pricing formula for equity value and using it calculate expected default barrier then we provide pricing model and formula for defaultable corporate bonds with discrete coupons and consider its duration

system theory for system identification abstract prediction and control problems require for their solution mathematical models in the form of dynamic systems in system theory the approach to this problem is to investigate realization problems for system identification novel in this paper is the emphasis on the realization problem for observation based realizations and on the realization problem for control a major aspect of these realization problems is computation and complexity algebraic approaches to realization problems may lead to the distinction of subclasses of systems with required finiteness properties to decompositions and to abstractions for hierarchical systems

invariants and canonical forms for systems structural and parametric identification the basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled with reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set then the set of polynomial input output models of the type p z y t q z u t and the equivalence relation induced by the premultiplication of p and q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced since the two sets of canonical forms share common sets of complete independent invariants very simple algebraical links between state space and input output canonical forms can be deduced the previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input output sequences generated by a linear discrete time invariant multivariable system the results obtained in the identification of a real process are then reported

the input output and controllable behavioural structures of a state space representation given the matrices of a linear state space representation we find an expression for a universal left annihilator of the matrix ctwovec zi a c and hence derive kernel representations for the input output behaviour and by duality the controllable part more generally in the discrete case we derive representations for the l completion for different values of l and the sub behaviours of trajectories reachable in a given time interval the representations are in certain trim canonical forms which are intimately connected with structure indices as a by product of the state elimination procedure we obtain a minimal state space representation for a given behaviour in terms of an arbitrary one

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

when is a storage function a state function in discrete time the purpose of this paper is to investigate when a storage function is a state function in discrete time as shown by trentelman and willems systems control lett 32 1997 pp 249 259 siam j control optim 36 1998 pp 1703 1749 every storage function is a state function in continuous time at first glance the same claim seems to hold also in discrete time contrary to this expectation this is not true in general in fact the discrete time counterpart involves not only some different but also some more difficult issues compared with the continuous time case this paper addresses these issues exactly and shows that every nonnegative storage function is a state function of a supply rate with a linear time invariant dynamical system in discrete time

a cell structure for the set of autoregressive systems the set of autoregressive systems generalizes the set of transfer functionsin a natural way we describe a topology for the set of all autoregressive systems of fixed size and bounded mcmillan degree we show that this topological space has the structure of a finite cw complex

on behaviors and convolutional codes it is well known that a convolutional code is essentially a linear system defined over a finite field in this paper we elaborate on this connection we define a convolutional code as the dual of a complete linear behavior in the sense of willems 1979 using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes as an application of these ideas we present a new algebraic construction for convolutional codes

controllability and observability indices for linear time invariant systems a new approach abstract this paper is devoted to an investigation of the controllability and observability indices for linear time invariant discrete time systems these indices are here introduced at an intrinsic level in the context of the behavioral approach to systems theory for the class of the complete controllable systems it is proven that the controllability indices can be interpreted as a minimal and complete set of invariants with respect to a very natural systems equivalence autonomous phenomena are also discussed

rational differential operators and their kernels abstract a linear differential integral operator is associated to a rational matrix in a natural way and the behavior is defined to be the kernel of this operator conditions under which two rational matrices have the same behavior are derived

characterization of linear differential systems in several variables abstract we show that a set of smooth trajectories is the solution set of a linear constant coefficient partial differential equation if and only if it is linear shift invariant and complete by completeness we mean exactly what willems called jet completeness in his automatica paper in 1986

diophantine equations in control a survey abstract this survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems these are linear equations in a ring and result from a fractional representation of the systems involved the cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant one can then choose in principle the best controllers for various applications these ideas evolved from early attempts to use polynomial equations in the design of discrete time linear systems by now they have been extended to continuous time infinite dimensional time varying and non linear systems

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

a behavioral approach to time varying linear systems part 1 general theory we develop a behavioral approach to linear time varying differential algebraic systems the analysis is almost everywhere in the sense that the statements hold on mathbb r setminus mathbb t where mathbb t is a discrete set controllability observability and autonomy are introduced and related to the behavior of the system classical results on the behavior of time invariant systems are studied in the context of time varying systems

alternative solution of discrete time kalman filter abstract this paper concerns the closed form solutions of the minimum variance and the optimal filter gain of the discrete time kalman filter we directly construct the closed form solution of riccati equations and the optimal gain of the filter from the system parameters the approach is based on the time series treatment of the dual system of the stochastic systems the closed form solution of the minimum variance yields the riccati equation

characterising solution sets of lti differential equations we prove that a set of smooth trajectories is ltid i e the solution set of a constant coefficient linear differential equation if and only if it is the direct sum of a linear time invariant closed controllable part and a linear time invariant finite dimensional part this characterisation does not directly involve derivatives in its formulation it solves a problem opened by willems 1991 we also characterise morphisms between ltid sets as linear time invariant maps which do not increase support

subspace based methods for the identification of linear time invariant systems subspace based methods for system identification have attracted much attention during the past few years this interest is due to the ability of providing accurate state space models for multivariable linear systems directly from input output data the methods have their origin in classical state space realization theory as developed in the 1960s the main computational tools are the qr and the singular value decompositions here an overview of existing subspace based techniques for system identification is given the methods are grouped into the classes of realization based and direct techniques similarities between different algorithms are pointed out and their applicability is commented upon we also discuss some recent ideas for improving and extending the methods a simulation example is included for comparing different algorithms the subspace based approach is found to perform competitive with respect to prediction error methods provided the system is properly excited

survey paper structured low rank approximation and its applications fitting data by a bounded complexity linear model is equivalent to low rank approximation of a matrix constructed from the data the data matrix being hankel structured is equivalent to the existence of a linear time invariant system that fits the data and the rank constraint is related to a bound on the model complexity in the special case of fitting by a static model the data matrix and its low rank approximation are unstructured we outline applications in system theory approximate realization model reduction output error and errors in variables identification signal processing harmonic retrieval sum of damped exponentials and finite impulse response modeling and computer algebra approximate common divisor algorithms based on heuristics and local optimization methods are presented generalizations of the low rank approximation problem result from different approximation criteria e g weighted norm and constraints on the data matrix e g nonnegativity related problems are rank minimization and structured pseudospectra

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

discrete time average positivity and spectral factorization in a behavioral framework abstract quite recently willems and trentelman siam j control opt 36 1998 1703 1749 have proposed quadratic differential forms for system analysis and synthesis from the behavioral point of view in this paper we develop their discrete time version called quadratic difference forms and some related fundamental concepts with respect to dissipativeness for a dynamical system which interacts with its environment along square summable trajectories moreover we provide an algorithm for spectral factorization of polynomial matrices as one example of quadratic difference forms a numerical example will be given to show the validity of our result

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

taylor approximations of multidimensional linear differential systems abstractthe taylor approximations of a multidimensional linear differential system are of importance as they contain a complete information about it it is shown that in order to construct them it is sufficient to truncate the exponential trajectories only a computation of the taylor approximations is provided using purely algebraic means without requiring explicit knowledge of the trajectories

controllability of 2 d systems the concept of controllability of 2 d systems is introduced in a behavioral framework this property is defined as a systems concept and is characterized in terms of system representations state space realizations for controllable 2 d systems are presented the structure of state space systems is analyzed and the notions of trimness reachability and observability are introduced and characterized these realizations do not depend on any assumptions concerning causality

linear quadratic optimization for systems in the behavioral approach in this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed given a linear time invariant and complete system find the set of trajectories of the system that optimize a quadratic type cost function and that satisfy some linear static constraints this formulation provides a unifying framework where several generalized versions of the classical lq optimal control can be stated and solved the existence of solutions is first discussed it is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by product of a reduction procedure translating the problem into an equivalent one of minimum complexity such a procedure is based on the theory of ell 2 systems in the behavioral setting once the complexity is reduced a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two step optimization procedure

autonomous subbehaviours and output nulling subspaces the aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours specifically given a behaviour we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems we use this to analyse the different module structures on some vectorial polynomial spaces subsequently we apply this analysis to the study and the polynomial characterization of output nulling subspaces controlled invariant subspaces and the natural feedback induced module structures in these spaces toeplitz operators and wiener hopf factorizations play an important role

column reduction of polynomial matrices a few years ago beelen developed an algorithm to determine a minimal basis for the kernel of a polynomial matrix in this paper we use a modified version of this algorithm to find a column reduced polynomial matrix unimodularly equivalent to a given polynomial matrix

trajectory control and interconnection of 1d and nd systems in this paper we examine the relationship between control viewed as concatenation of trajectories and control viewed as interconnection of systems we show that for one dimensional linear time invariant systems the ability to obtain a given subsystem by regular interconnection a prerequisite for any feedback type structure is equivalent to the ability to drive any trajectory into that subsystem however in the case of multidimensional systems the former is a stronger property than the latter trajectory controllability can however be expressed as a regular interconnection of behaviors in an extended variable space by introducing latent or auxiliary variables this leads as a by product to the notion of controlling a system by means of latent variables