admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

asymptotic stability of infinite dimensional discrete time balanced realizations the question of power and asymptotic stability of infinite dimensional discrete time state space systems is investigated it is shown that every balanced realization is asymptotically stable conditions are given for balanced input normal or output normal realizations to be asymptotically and or power stable

infinite dimensional continuous time linear systems stabilityand structure analysis the question of exponential and asymptotic stability of infinite dimensional continuous time state space systems is investigated it is shown that every par balanced realization is asymptotically stable conditions are given for par balanced input normal or output normal ealizations to be asymptotically and or exponentially stable the boundedness of the system operators is also studied examples of delay systems are given to illustrate the theory

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

on possibilistic correlation in 2004 fuller and majlender introduced the notion of covariance between fuzzy numbers by their joint possibility distribution to measure the degree to which they interact based on this approach in this paper we will present the concept of possibilistic correlation representing an average degree of interaction between marginal distributions of a joint possibility distribution as compared to their respective dispersions moreover we will formulate the classical cauchy schwarz inequality in this possibilistic environment and show that the measure of possibilistic correlation satisfies the same property as its probabilistic counterpart in particular applying the idea of transforming level sets of possibility distributions into uniform probability distributions we will point out a fundamental relationship between our proposed possibilistic approach and the classical probabilistic approach to measuring correlation

expected value of fuzzy variable and fuzzy expected value models this paper will present a novel concept of expected values of fuzzy variables which is essentially a type of choquet integral and coincides with that of random variables in order to calculate the expected value of general fuzzy variable a fuzzy simulation technique is also designed finally we construct a spectrum of fuzzy expected value models and integrate fuzzy simulation neural network and genetic algorithms to produce a hybrid intelligent algorithm for solving general fuzzy expected value models

bilinear transformation of infinite dimensional state space systems and balanced realizations of nonrational transfer functions the bilinear transform maps the open right half plane to the open unit disk and is therefore a suitable tool for carrying over results for continuous time systems to discrete time systems and vice versa corresponding state space formulae are widely used and well understood for the case of finite dimensional systems in this paper infinite dimensional generalizations of these formulae are studied for a general class of infinite dimensional state space systems in particular it is shown that reachability and observability are carried over and that the reachability and observability gramians are preserved under this transformation young showed that a wide class of nonrational discrete time transfer functions admit a balanced state space representation it is shown that this result carries over to the continuous time situation via the bilinear transformation

the interest rate for saving as a possibilistic risk abstract the paper is concerned with a saving model in which the interest rate risk is a fuzzy number our model can be seen as a possibilistic variant of a saving model due to rothschild and stiglitz in which the interest rate risk is a random variable a notion of possibilistic precautionary saving is defined as a measure of the optimal saving variation when moving from a certain model to the possibilistic interest rate risk model a first result of the paper characterizes the possibilistic precautionary saving i e generating extra saving by an inequality in which the relative prudence index associated with the consumer s utility function appears a second result compares the level of optimal saving from the rothschild and stiglitz with the level of optimal saving from the possibilistic model from the paper finally two propositions on the approximate calculation of possibilistic optimal saving are proved the two approximate calculation procedures are illustrated for particular cases of utility functions and fuzzy numbers

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

equivalence of a behavioral distance and the gap metric in this note it is shown that the topology of the gap metric is precisely the topology induced by the behavioral distance introduced in 10 when restricted to the class of stable state space systems with a fixed macmillan degree

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

admissible factorizations of hankel operators induce well posed linear systems abstract one of the basic axioms of a well posed linear system says that the hankel operator of the input output map of the system factors into the product of the input map and the output map here we prove the converse every factorization of the hankel operator of a bounded causal time invariant map from l 2 to l 2 which satisfies a certain admissibility condition induces a stable well posed linear system in particular there is a one to one correspondence between the set of all minimal stable well posed realizations of a given stable causal time invariant input output map or equivalently of a given h transfer function and all minimal stable admissible factorizations of the hankel operator of this input output map

coprime factorizations and well posed linear systems we study the basic notions related to the stabilization of an infinite dimensional well posed liner system in the sense of salamon and weiss we first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in h infty the converse is also true every function with a doubly coprime factorization in h infty is the transfer function of a jointly stabilizable and detectable well posed linear system we show further that a stabilizable and detectable system is stable if and only if its input output map is stable finally we construct a dynamic possibly non well posed stabilizing compensator the notion of stability that we use is the natural one for the quadratic cost minimization problem and it does not imply exponential stability

on possibilistic mean value and variance of fuzzy numbers dubois and prade introduced the mean value of a fuzzy number as a closed interval bounded by the expectations calculated from its upper and lower distribution functions in this paper introducing the notations of lower possibilistic and upper possibilistic mean values we definine the interval valued possibilistic mean and investigate its relationship to the interval valued probabilistic mean we also introduce the notation of crisp possibilistic mean value and crisp possibilistic variance of continuous possibility distributions which are consistent with the extension principle we also show that the variance of linear combination of fuzzy numbers can be computed in a similar manner as in probability theory

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

hankel norm approximation for well posed linear systems the sub optimal hankel norm approximation problem is solved for a well posed linear system with generating operators a b c and transfer function g satisfying some mild assumptions in the special case of the sub optimal nehari problem an explicit parameterization of all solutions is obtained in terms of the system parameters a b c and g 0

when are two commutative c algebras stably homotopy equivalent let x and y be finite connected cw complexes with base points and let k denote the c algebra of compact operators on a separable infinite dimensional complex hilbert space the purpose of this paper is to study the question of when c0 x k is homotopy equivalent to c0 y k here c0 x is as usual the c algebra of continuous complex valued functions which vanish at the base point recall that two c algebras a and b are said to be homotopy equivalent written a b if there are homomorphisms a b and b a for which and may be deformed by a path of endomorphisms to the identity maps ida a a and idb b b respectively two c algebras a and b are called stably homotopy equivalent if a k b k this should not be confused with the notion of stable homotopy of spaces used in topology so another way of stating the problem we consider is when are c0 x and c0 y stably homotopy equivalent to begin with we should remark that the analogous question for homotopy equivalence has a simple answer c0 x and c0 y are homotopy equivalent if and only if x and y are based homotopy equivalent the situation for stable homotopy equivalence is more complicated and is closely related to k theory for example if two c algebras are stably homotopy equivalent then they have the same k theoretic invariants the main purpose of this paper is to show that the converse is not true we give an example of two spaces x and y which cannot be distinguished by

possibilistic risk aversion risk theory is usually developed within probability theory the aim of this paper is to propose an approach of the risk aversion by possibility theory initiated by zadeh in 1978 the main notion studied in this paper is the possibilistic risk premium associated with a fuzzy number a and a utility function u under the hypothesis that the utility function u verifies certain hypotheses one proves a formula to evaluate the possibilistic risk premium in terms of u and of some possibilistic indicators

quadratic optimal control of well posed linear systems we study the infinite horizon quadratic cost minimization problem for a well posed linear system in the sense of salamon and weiss the quadratic cost function that we seek to minimize need not be positive but it is convex and bounded from below we assume the system to be jointly stabilizable and detectable and give a feedback solution to the cost minimization problem moreover we connect this solution to the computation of either a j s inner or an s normalized coprime factorization of the transfer function depending on how the problem is formulated we apply the general theory to get factorization versions of the bounded and positive real lemmas in the case where the system is regular it is possible to show that the feedback operator can be expressed in terms of the riccati operator and that the riccati operator is a stabilizing self adjoint solution of an algebraic riccati equation this riccati equation is nonstandard in the sense that the weighting operator in the quadratic term differs from the expected one and the computation of the correct weighting operator is a nontrivial task

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

on the discrete and continuous time infinite dimensional algebraic riccati equations the standard state space solution of the finite dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite dimensional problems with bounded or moderately unbounded control and observation operators however if these operators are allowed to be sufficiently unbounded then a strange change takes place in one of the coefficients of the algebraic riccati equation and the continuous time riccati equation begins to resemble the discrete time riccati equation to explain why this phenomenon must occur we discuss a particular hyperbolic pde in one space dimension with boundary control and observation a transmission line that can be formulated both as a discrete time system and as a continuous time system and show that in this example the continuous time riccati equation can be recovered from the discrete time riccati equation a particular feature of this example is that the riccati operator does not map the domain of the generator into the domain of the adjoint generator as it does in the standard case