a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

geometric design techniques for observers in singular systems abstract we present geometric design techniques for observer design in generalized state space or singular systems the unknown input conditioned invariant subspaces are defined for singular systems and are related to the solution of a generalized lyapunov equation the approach emphasizes solution procedures based on the lyapunov equation and a singular system structure algorithm this paper extends the singular system observer theory both by providing a geometric basis for it and showing the best that may be accomplished if the system is not observable

sobolev functions on infinite dimensional domains abstract we study extensions of sobolev and bv functions on infinite dimensional domains along with some positive results we present a negative solution of the long standing problem of existence of sobolev extensions of functions in gaussian sobolev spaces from a convex domain to the whole space

impulsive modes and causality in singular systems the concepts of impulsive modes and causality in singular systems are considered their properties relations and applications are studied matrix rank criteria are presented thus all controllabilities and observabilities and related concepts have matrix rank criteria based on the original system coefficient matrices

design of observers for time varying discrete time descriptor systems in this note linear discrete time time varying descriptor systems i e control systems of the form ek 1 xk 1 akxk bkuk yk ckxk are considered where ek 1 and ak 1 are constant not necessarily square matrices the problem is to design a reduced order observer for the system using matrix generalized inverses and singular value decomposition a procedure is developed which either leads to a routine observer design problem involving an ordinary discrete time state equation or proves that no observer exists for the given descriptor system the method of this note is partly based on an earlier paper of the authors on observer design for continuous time time invariant systems of the form e dx dt ax bu y cx where e and a are square matrices el tohami et al 1983 and makes use of some results of luenberger on discrete time time varying descriptor systems luenberger 1977

observers for descriptor systems abstract the design of observers is investigated for the estimation of the descriptor vector x t of descriptor systems of the form ex t ax t bu t y t cx t such systems are divided into two categories non causal impulsive systems and causal impulse free systems for non causal systems an observer of order rank e is proposed and termed a full order observer since the dynamical order of a descriptor system is regarded as rank e for causal systems it is shown that the number of independent outputs m does not exceed rank e and an observer of order rank e m is introduced and termed a minimal order observer the approach is based on the transformation of the system into particular design coordinates via a singular value decomposition of the matrix e illustrative examples are included

on the reduced order causal observer for generalized control systems abstract this paper considers the problem of a causal state observer for a square singular system necessary and sufficient conditions for solvability of the problem have been established the approach of this study leads to a new insight concerning synthesis of a causal observer and establishes new results with regard to the structure and the dimension of a reduced order observer for singular systems

on the dirichlet semigroup for ornstein uhlenbeck operators in subsets of hilbert spaces we consider a family of self adjoint ornstein uhlenbeck operators l in an infinite dimensional hilbert space h having the same gaussian invariant measure for all 0 2 0 1 we study the dirichlet problem for the equation l f in a closed set k with f 2 l 2 k we first prove that the variational solution trivially provided by the lax milgram theorem can be rep resented as expected by means of the transition semigroup stopped to k then we address two problems 1 the regularity of the solution which is by definition in a sobolev space w 1 2 k of the dirichlet problem 2 the meaning of the dirichlet boundary condition concerning regularity we are able to prove interior w 2 2 regularity results concerning the boundary condition we consider both irreg ular and regular boundaries in the first case we content to have a solution whose null extension outside k belongs to w 1 2 h in the second case we exploit the malliavin s theory of surface integrals which is recalled in the appendix of the pa per then we are able to give a meaning to the trace of at k and to show that it vanishes as it is natural

remarks on observability of implicit linear discrete time systems abstract some results concerning the observability of implicit linear discrete time systems of the form exk 1 fxk yk hxk are presented it is not assumed that the corresponding pencil se f is regular the concept of reachability with pulse introduced in this paper is shown to be dual to the notion of observability

observers for discrete singular systems the author considers the design of observers for the discrete singular system ex k 1 ax k bu k y k cx k placing special emphasis on the problems of state reconstruction and minimal time state reconstruction it is shown that for a singular system finite poles can be moved to infinity by state feedback and the state can be reconstructed by causal observers

traces of sobolev functions on regular surfaces in infinite dimensions abstract in a banach space x endowed with a nondegenerate gaussian measure we consider sobolev spaces of real functions defined in a sublevel set o x x g x 0 of a sobolev nondegenerate function g x r we define the traces at g 1 0 of the elements of w 1 p o for p 1 as elements of l 1 g 1 0 where is the surface measure of feyel and de la pradelle the range of the trace operator is contained in l q g 1 0 for 1 q p and even in l p g 1 0 under further assumptions if o is a suitable halfspace the range is characterized as a sort of fractional sobolev space at the boundary an important consequence of the general theory is an integration by parts formula for sobolev functions which involves their traces at g 1 0

design of a minimal order observer for singular systems this paper considers the design of an observer for generalized state space systems using the concept of generalized inverses in contrast to the luenberger observer theory a new method is proposed which does not presuppose the observer structure it is shown that under certain conditions it is possible to construct a minimal order observer for this class of systems illustrative examples are included

on the design of observers for generalized state space systems using singular value decomposition in this paper a method is developed for the design of luenberger type observers for linear time invariant control systems whose state equation is of the form ex ax bu where e is a singular square matrix the method is based on the singular value decomposition of the matrix e and on the reduction of the equation ex ax bu to a system consisting of a differential equation of the form w1 f1w1 f2 w2 g1u and an algebraic equation of the form h1w1 h2w2 g1u 0 if w2 can be eliminated from the differential equation by the aid of the algebraic equation and the original output equation of the system the method yields a reduced order observer for the generalized state space system

observer design for descriptor systems two detectability notions are introduced for the most general class of linear descriptor systems necessary and sufficient conditions are derived for the detectabilities and for the existence of observers depending on existence conditions satisfied the observers can be either in the familiar luenberger form or in a generalized form i e in descriptor form as the original system itself is a unified procedure is provided for the design of observers in both forms

observers for descriptor systems for a singular non causal descriptor system of order n a p k lh order observer is proposed where p n q q is the number of infinite non dynamic modes of the system and k is the rank of a matrix obtained from a transformed system derived by using the singular value decomposition some useful results are given which establish the existence of this observer and which resolve problems encountered with pth order observers proposed in the literature robustness is achieved by choosing optimum gain matrices which reduce the sensitivity of observer performance to perturbations

kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a hilbert space we consider the stochastic reflection problem associated with a selfadjoint operator a and a cylindrical wiener process on a convex set k with nonempty interior and regular boundary e in a hilbert space h we prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite dimensional kolmogorov equation with neumann boundary condition on e

hodge classes on abelian varieties of low dimension in this paper we study hodge classes on complex abelian varieties x if dimx then it is wellknown that every hodge class on x is a linear combination of products of divisor classes in the authors showed that if x is simple of dimension then every hodge class is a linear combination of products of divisor classes and weil classesif there are any the notion of a weil class shall be briey reviewed in the aim of this note is to extend this to arbitrary abelian varieties of dimension in order to state our main results let us describe some special cases a the abelian variety x is isogenous to a product x x where x is an elliptic curve with complex multiplication by an imaginary quadratic eld k and where x is a simple abelian threefold such that there exists an embedding k endx b the abelian variety x is simple of dimension such that endx is a eld containing an imaginary quadratic eld k which acts on the tangent space tx with multiplicities see x for further explanation c the abelian variety x is simple of dimension with d endx a denite quaternion algebra over q type iii in the albert classication note that for every d n q the subalgebra q d is an imaginary quadratic eld d the abelian variety x is simple of dimension with endx q