nonsmooth differential geometry and algebras of generalized functions algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in nonsmooth differential geometry this article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics

commuting toeplitz operators on the segal bargmann space abstract consider two toeplitz operators t g t f on the segal bargmann space over the complex plane let us assume that g is a radial function and both operators commute under certain growth condition at infinity of f and g we show that f must be radial as well we give a counterexample of this fact in case of bounded toeplitz operators but a fast growing radial symbol g in this case the vanishing commutator t g t f 0 does not imply the radial dependence of f finally we consider toeplitz operators on the segal bargmann space over c n and n 1 where the commuting property of toeplitz operators can be realized more easily

commutative c algebras of toeplitz operators and quantization on the unit disk abstract a family of recently discovered commutative c algebras of toeplitz operators on the unit disk can be classified as follows each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles the orthogonal trajectories to lines forming a pencil the c algebra generated by toeplitz operators with such symbols turns out to be commutative we show that these cases are the only possible ones which generate the commutative c algebras of toeplitz operators on each weighted bergman space

commuting toeplitz operators with quasi homogeneous symbols on the segal bargmann space abstract let t t z l z k with l k n 0 be a toeplitz operator with monomial symbol acting on the segal bargmann space over the complex plane we determine the symbols of polynomial growth at infinity such that t and t z l z k commute on the space of all holomorphic polynomials by using polar coordinates we represent as an infinite sum r e i j j r e i j then we are able to reduce the above problem to the case of quasi homogeneous symbols j e i j we obtain the radial part j r in terms of the inverse mellin transform of an expression which is a product of gamma functions and a trigonometric polynomial if we allow operator symbols of higher growth at infinity we point out that in some of the cases more than one toeplitz operator t j e i j exists commuting with t

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

commutants of toeplitz operators with radial symbols on the fock sobolev space abstract in the setting of the fock space over the complex plane bauer and lee have recently characterized commutants of toeplitz operators with radial symbols under the assumption that symbols have at most polynomial growth at infinity their characterization states if one of the symbols of two commuting toeplitz operators is nonconstant and radial then the other must be also radial we extend this result to the fock sobolev spaces

on algebras generated by toeplitz operators and their representations abstract we study banach and c algebras generated by toeplitz operators acting on weighted bergman spaces a 2 b 2 over the complex unit ball b 2 c 2 our key point is an orthogonal decomposition of a 2 b 2 into a countable sum of infinite dimensional spaces each one of which can be identified with a differently weighted bergman space a 2 d over the complex unit disk d moreover all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a toeplitz operator on a 2 d the symbols of the generating toeplitz operators are chosen to be suitable extensions to b 2 of families s of bounded functions on d symbol classes s that generate important classical commutative and non commutative toeplitz algebras in l a 2 d are of particular interest in this paper we discuss various examples in the case of s c d and s c d l 0 1 we characterize all irreducible representations of the resulting toeplitz operator c algebras their calkin algebras are described and index formulas are provided

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

scalar products of elementary distributions the schrodinger equation with point interaction in one dimension is revisited in a simple framework where the singular potential is defined as a symmetric operator in a natural way the main tool is a scalar product of the elementary distributions constructed after a commutative and well ordered field extension of r followed by complexification a contact with the hyper real numbers that arise in nonstandard analysis is possible but not essential our extensions of r and c being obtained by a quite elementary method

generalized pseudo riemannian geometry generalized tensor analysis in the sense of colombeau s construction is employed to introduce a nonlinear distributional pseudo riemannian geometry in particular after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo riemannian metric generalized connection and generalized curvature tensor we prove a fundamental lemma of pseudo riemannian geometry in this setting and define the notion of geodesics of a generalized metric finally we present applications of the resulting theory to general relativity

intrinsic characterization of manifold valued generalized functions the concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory we establish several characterization results which allow a global intrinsic formulation both of the theory of manifold valued generalized functions and of generalized vector bundle homomorphisms as a consequence a characterization of equivalence that does not resort to derivatives analogous to scalar valued cases of colombeau s construction is achieved these results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly finally a new concept of association adapted to the present setting is introduced

the boundedness of all products of a pair of matrices is undecidable we show that the boundedness of the set of all products of a given pair sigma of rational matrices is undecidable furthermore we show that the joint or generalized spectral radius rho sigma is not computable because testing whether rho sigma less than or equal to1 is an undecidable problem as a consequence the robust stability of linear systems under time varying perturbations is undecidable and the same is true for the stability of a simple class of hybrid systems we also discuss some connections with the so called finiteness conjecture our results are based on a simple reduction from the emptiness problem for probabilistic finite automata which is known to be undecidable c 2000 elsevier science b v all rights reserved

commuting toeplitz operators on bounded symmetric domains and multiplicity free restrictions of holomorphic discrete series abstract for any given bounded symmetric domain we prove the existence of commutative c algebras generated by toeplitz operators acting on any weighted bergman space the symbols of the toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain these subgroups include the maximal compact groups of biholomorphisms we prove the commutativity of the toeplitz operators by considering the bergman spaces as the underlying space of the holomorphic discrete series and then applying known multiplicity free results for restrictions to certain subgroups of the holomorphic discrete series in the compact case we completely characterize the subgroups that define invariant symbols that yield commuting toeplitz operators in terms of the multiplicity free property

stabilization of switched systems via composite quadratic functions this paper investigates stabilization of switched systems by using a lyapunov function composed of a family of continuously differentiable functions switching laws are constructed by using the directional derivatives along the trajectories of the subsystems conditions for stabilization are established with careful consideration of sliding modes and directional derivatives along sliding modes three types of composite quadratic lyapunov functions the max of quadratics the min of quadratics and the convex hull of quadratics are used for deriving matrix conditions of stabilization and for constructing switching laws dual stabilization results are established with respect to the pair of conjugate lyapunov functions the max of quadratics and the convex hull of quadratics conditions of stabilization are derived as bilinear matrix inequalities and solved with lmi based numerical tools relationship between the newly derived conditions and some existing conditions are investigated it is observed that the min of quadratics which is nondifferentiable and nonconvex may be a more convenient tool than the other two types of functions which are convex and or differentiable numerical examples are used to demonstrate the synthesis results and the advantage of the composite quadratic functions over existing multiple lyapunov functions in particular better results have been obtained when the number of quadratic functions is greater than the number of subsystems

optimal control of continuous time switched affine systems this paper deals with optimal control of switched piecewise affine autonomous systems where the objective is to minimize a performance index over an infinite time horizon we assume that the switching sequence has a finite length and that the decision variables are the switching instants and the sequence of operating modes we present two different approaches for solving such an optimal control problem the first approach iterates between a procedure that finds an optimal switching sequence of modes and a procedure that finds the optimal switching instants the second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur therefore providing a state feedback control law

a rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves the geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions 2 2 a solution concept for these equations based on embedding the distributional metric into the colombeau algebra of generalized functions is presented using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly the obtained limits are regularization independent and display the physically expected behavior

commuting toeplitz operators with pluriharmonic symbols on the fock space abstract in the setting of the bergman space over the disk or the ball it has been known that two toeplitz operators with bounded pluriharmonic symbols can semi commute only in the trivial cases in this paper we study the analogues on the fock space over the multi dimensional complex space as is the case in various other settings we are naturally led to the problem of characterizing a certain type of fixed points of the berezin transform for such fixed points we obtain a complete characterization by means of eigenfunctions of the laplacian we also obtain other characterizations in particular it turns out that there are many nontrivial cases on the fock space for semi commuting toeplitz operators with pluriharmonic symbols all in all our results reveal that the situation on the fock space appears to be much more complicated than that on the classical bergman space setting which partly is caused by the unboundedness of the operator symbols some of our results are restricted to the one variable case and the corresponding several variable case is left open

generalized functions and multiplication of distributions on c manifolds abstract an improvement in definition permits us to make invariant under the action of c diffeomorphisms a concept of generalized functions introduced by the first author to define arbitrary products of distributions

the generalized spectral radius and extremal norms the generalized spectral radius also known under the name of joint spectral radius or after taking logarithms maximal lyapunov exponent of a discrete inclusion is examined we present a new proof for a result of barabanov which states that for irreducible sets of matrices an extremal norm always exists this approach lends itself easily to the analysis of further properties of the generalized spectral radius we prove that the generalized spectral radius is locally lipschitz continuous on the space of compact irreducible sets of matrices and show a strict monotonicity property of the generalized spectral radius sufficient conditions for the existence of extremal norms are obtained

a global theory of algebras of generalized functions abstract we present a geometric approach to defining an algebra g m the colombeau algebra of generalized functions on a smooth manifold m containing the space d m of distributions on m based on differential calculus in convenient vector spaces we achieve an intrinsic construction of g m g m is a differential algebra its elements possessing lie derivatives with respect to arbitrary smooth vector fields moreover we construct a canonical linear embedding of d m into g m that renders c m a faithful subalgebra of g m finally it is shown that this embedding commutes with lie derivatives thus g m retains all the distinguishing properties of the local theory in a global context

geodesics and geodesic deviation for impulsive gravitational waves the geometry of impulsive pp waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric the geodesic equation involves formally ill defined products of distributions due to the nonlinearity of the equations and the presence of the dirac distribution in the space time metric thus strictly speaking it cannot be treated within schwartz s linear theory of distributions to cope with this difficulty we proceed by first regularizing the singularity then solving the regularized equation within classical smooth functions and finally obtaining a distributional limit as solution to the original problem furthermore it is shown that this limit is independent of the regularization without requiring any additional condition thereby confirming earlier results in a mathematically rigorous fashion we also treat the jacobi equation which despite being linear in the deviation vector field involves even more delicate singular expressions like the square of the dirac distribution again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization independent distributional limit hence it is concluded that the geometry of impulsive pp waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill defined products

nonlinear generalized sections of vector bundles abstract we present an extension of j f colombeau s theory of nonlinear generalized functions to spaces of generalized sections of vector bundles our construction builds on classical functional analytic notions which is the key to having a canonical geometric embedding of vector bundle valued distributions into spaces of generalized sections this permits to have tensor products invariance under diffeomorphisms covariant derivatives and the sheaf property while retaining as much compatibility to l schwartz theory of distributions as possible our theory provides the basis for a rigorous and general treatment of singular pseudo riemannian geometry in the setting of colombeau nonlinear generalized functions

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

on the accuracy of the ellipsoid norm approximation of the joint spectral radius the joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set this quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate we introduce in this paper an approximation hat rho that is based on ellipsoid norms that can be computed by convex optimization and that is such that the joint spectral radius belongs to the interval hat rho sqrt n hat rho where n is the dimension of the matrices we also provide a simple approximation for the special case where the entries of the matrices are non negative in this case the approximation is proved to be within a factor at most m m is the number of matrices of the exact value

overlap free words and spectra of matrices overlap free words are words over the binary alphabet a a b that do not contain factors of the form xvxvx where x a and v a we analyze the asymptotic growth of the number u n of overlap free words of length n as n we obtain explicit formulas for the minimal and maximal rates of growth of u n in terms of spectral characteristics the joint spectral subradius and the joint spectral radius of certain sets of matrices of dimension 20x20 using these descriptions we provide new estimates of the rates of growth that are within 0 4 and 0 03 of their exact values the best previously known bounds were within 11 and 3 respectively we then prove that the value of u n actually has the same rate of growth for almost all natural numbers n this average growth is distinct from the maximal and minimal rates and can also be expressed in terms of a spectral quantity the lyapunov exponent we use this expression to estimate it in order to obtain our estimates we introduce new algorithms to compute the spectral characteristics of sets of matrices these algorithms can be used in other contexts and are of independent interest

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

generating functions of switched linear systems analysis computation and stability applications in this paper a unified framework is proposed to study the exponential stability of discrete time switched linear systems and more generally the exponential growth rates of their trajectories under three types of switching rules arbitrary switching optimal switching and random switching it is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong the weak and the mean generating functions respectively in particular necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence various properties of the generating functions are established and their relations are discussed algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

on the stability of positive linear switched systems under arbitrary switching laws we consider n dimensional positive linear switched systems a necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is hurwitz several researchers conjectured that for positive linear switched systems this condition is also sufficient recently gurvits shorten and mason showed that this conjecture is true for the case n 2 but is not true in general their results imply that there exists some minimal integer np such that the conjecture is true for all n np but is not true for n np we show that np 3

third order nilpotency nice reachability and asymptotic stability we consider an affine control system whose vector fields span a third order nilpotent lie algebra we show that the reachable set at time t using measurable controls is equivalent to the reachable set at time t using piecewise constant controls with no more than four switches the bound on the number of switches is uniform over any final time t as a corollary we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching this provides a partial solution to an open problem posed in d liberzon lie algebras and stability of switched nonlinear systems in v blondel a megretski eds unsolved problems in mathematical systems and control theory princeton univ press 2004 pp 203 207