robustness of lambda tracking in the gap metric for m input m output finite dimensional linear systems satisfying the classical assumptions of adaptive control i e i minimum phase ii relative degree one and iii positive high frequency gain it is well known that the adaptive lambda tracker u k e dot k max 0 e lambda e achieves lambda tracking of the tracking error e if applied to such a system all states of the closed loop system are bounded and e is ultimately bounded by lambda where lambda 0 is prespecified and may be arbitrarily small invoking the conceptual framework of the nonlinear gap metric we show that the lambda tracker is robust in the present setup this means in particular that the lambda tracker copes with bounded input and output disturbances and more importantly it may even be applied to a system not satisfying any of the classical conditions i iii as long as the initial conditions and the disturbances are small and the system is close in terms of small gap to a system satisfying i iii

chow groups of projective varieties of very small degree see 12 5 and the references given there these facts together with various conjectures on the cohomology and chow groups of algebraic varieties suggest that the chow groups of x might satisfy chl x q chl pk q q for l 1 compare with remark 5 6 and corollary 5 7 this is explicitly formulated by v srinivas and k paranjape in 16 conjecture 1 8 the chain of reasoning goes roughly as follows suppose x is smooth one expects a good filtration 0 f j 1 f j f 0 ch x x q whose graded pieces f f l 1 are controlled by h2j l x x see 10 according to grothendieck s generalized conjecture 8 the groups h x should be generated by the image under the gysin morphism of the homology of a codimension subset together with the classes coming from p applying this to the diagonal in x x should then force the triviality of the chow groups in the desired range for zero cycles the conjecture follows from roitman s theorem see 17 and 18

arithmeticity of holonomy groups of lie foliations the point of this paper is to prove arithmeticity of a class of finitely generated groups that arise naturally in a purely geometric context namely we prove arithmeticity of the holonomy group of a lie foliation of a compact manifold assuming the existence of a metric for which the leaves are symmetric spaces of nonpositive curvature and rank at least 2 this provides a partial answer to the question raised a number of years ago by a haefliger 13 of determining the possible holonomy groups of lie foliations to indicate the nature of the questions we are considering let us first recall that if n is a riemannian manifold a fundamental question is to understand the relationship of the topology of n and the geometry of n or for many purposes the geometry of n there has been a wide variety of work on this type of question from a variety of viewpoints in particular we can ask how the geometry of n or n controls the nature of rc1 n one can ask a more general question namely to understand to what extent the geometry of n controls the actions of rc1 n on a manifold t a standard construction associates to every such action a foliation of a manifold m which has all leaves locally isometric to n in which t appears naturally as a submanifold of m transverse to the foliation and the 7 n action on t controls the way in which the leaves of this foliation are tied together we can thus reinterpret our last question and broaden its context as well by formulating the following question determine the extent to which the geometry of a simply connected riemannian manifold x controls the way in which the leaves of a foliation are tied together if we assume the leaves are all locally isometric to x in general it is of course difficult to understand actions of a group on a manifold and correspondingly difficult to understand foliations with leaves locally isometric to a given x it is natural to begin an investigation of such actions with the case in which the action on t preserves some geometric structure e g a riemannian metric in terms of the associated foliation this becomes a special case of the

systems of controlled functional differential equations and adaptive tracking an adaptive servomechanism is developed in the context of the problem of approximate or practical tracking with prescribed asymptotic accuracy by the system output of any admissible reference signal absolutely continuous and bounded with essentially bounded derivative for every member of a class of controlled dynamical systems modelled by functional differential equations

adaptive control and robustness in the gap metric we consider the construction of adaptive controllers for minimum phase linear systems that achieve nonzero robustness margins in the sense of the gap metric the gap perturbation margin may be more constrained for larger disturbances and for larger parametric uncertainties working in both and settings and within the framework of the nonlinear gap metric universal adaptive controllers are first given achieving stabilization for first order nominal plants and the results are then generalized to relative degree one nominal plants a notion of a semiuniversal control design is introduced which is the property that a bound on performance exists that is independent of the a priori known uncertainty level and a characterization is given for when semiuniversal designs outperform the class of memoryless controllers and the class of linear time invariant controllers robust semiuniversal adaptive control designs are given for nominal plants under the classical assumptions of adaptive control in both the and settings the results are applied throughout to explicit classes of unmodeled dynamics including the rohrs example

sheaves on artin stacks we develop a theory of quasi coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of laumon and moret bailly we study basic cohomological properties of such sheaves and prove stack theoretic versions of grothendieck s fundamental theorem for proper morphisms grothendieck s existence theorem zariski s connectedness theorem as well as finiteness theorems for proper pushforwards of coherent and constructible sheaves we also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to laumon and moret bailly 1 1 in the book lm b the lisse etale topos of an algebraic stack was introduced and a theory of quasi coherent and constructible sheaves in this topology was developed unfortunately it was since observed by gabber and behrend independently that the lisse etale topos is not functorial as asserted in loc cit and hence the development of the theory of sheaves in this book is not satisfactory as is in addition since the publication of the book lm b several new results have been obtained such as finiteness of coherent and etale cohomology fa ol and various other consequences of chow s lemma ol the purpose of this paper is to explain how one can modify the arguments of lm b to obtain good theories of quasi coherent and constructible sheaves on algebraic stacks and in addition we provide an account of the theory of sheaves which also includes the more recent results mentioned above 1 2 the paper is organized as follows in section 2 we recall some aspects of the theory of cohomological descent sga4 v which will be used in what follows in section 3 we review the basic definitions of the lisse etale site cartesian sheaves over a sheaf of algebras and verify some basic properties of such sheaves in section 4 we relate the derived category of cartesian sheaves over some sheaf of rings to various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space loosely speaking the main result states that the cohomology of a complex with cartesian cohomology sheaves can be computed by restricting to the simplicial space obtained from a covering and computing cohomology on this simplicial space using the etale topology in section 5 we generalize these results to comparisons between ext groups computed in the lisse etale topos and ext groups computed using the etale topology on a hypercovering in section 6 we specialize the discussion of sections 3 5 to quasi coherent sheaves we show that if x is an algebraic stack and oxlis et denotes the structure sheaf of the lisse etale topos then the triangulated category d qcoh x of bounded below complexes of oxlis et modules with quasi coherent cohomology sheaves satisfies all the basic properties that one would expect from the theory for schemes for example we show in this section that if f x y is a quasi compact morphism of algebraic stacks and m is a quasi coherent sheaf on x date november 2 2005 1

higher fano manifolds and rational surfaces let x be a fano manifold of pseudo index 3 such that c1 x 2c2 x is nef irreducibility of some spaces of rational curves on x implies a general point of x is contained in a rational surface

input constrained adaptive tracking with applications to exothermic chemical reaction models we consider input constrained adaptive output feedback control for a class of nonlinear systems which are prototype models for controlled exothermic chemical reactions our objective is set point control of the output i e the temperature of the reaction in the context of chemical reactions practical considerations lead us to work in the presence of input constraints we adopt an approach based on modified lambda tracking controllers whereby prespecified asymptotic tracking accuracy quantified by lambda 0 set by the designer is ensured the adaptive control strategy does not require any knowledge of the system s parameters and does not invoke an internal model only a feasibility assumption in terms of the reference temperature and the input constraints is assumed

robustness in the graph topology of a common adaptive controller for any m input m output finite dimensional linear minimum phase plant p with first markov parameter having spectrum in the open right half complex plane it is well known that the adaptive output feedback control c given by u ky dot k y 2 yields a closed loop system p c for which the state converges to zero the signal k converges to a finite limit and all other signals are of class l 2 it is first shown that these properties continue to hold in the presence of l 2 input and l 2 output disturbances working within the conceptual framework of the nonlinear gap metric approach to robust stability and by establishing gain function stability of an appropriate closed loop operator it is proved that these properties also persist when the plant p is replaced with a stabilizable and detectable linear plant p 1 within a sufficiently small neighborhood of p in the graph topology provided that the plant initial data and the l 2 magnitude of the disturbances are sufficiently small example 9 of georgiou and smith ieee trans automat control 42 1997 pp 1200 1221 is revisited unstable behavior for large initial conditions and or large l 2 disturbances is shown demonstrating that the bounds obtained from the l 2 theory are qualitatively tight this contrasts with the l infty robustness analysis of georgiou and smith which is insufficiently tight to predict the stable behavior for small initial conditions and zero disturbances

coprime fractional representations and stability of non linear feedback systems fractional representation methods for finite dimensional linear time invariant fdlti systems have been developed in an algebraic framework of principal ideal domains the concept of coprime fractional representations has been generalized to discrete time non linear systems by hammer 1985 1987 by viewing the bezout identity as the basis of the definition of coprimeness in this paper we view the role of right coprime fractional representation in representing the set of all stable input output pairs of a possibly unstable non linear system this notion of coprimeness is shown to be equivalent to that of fdlti systems using coprime fractional representations of non linear systems we obtain a characterization of the stability of non linear feedback systems this characterization is similar to that obtained in the case of fdlti systems we also obtain a parameterization of all non linear and time varying stabilizing controllers for linear possibly time varying plants which has the same form as t

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

robust stabilization by linear output delay feedback the main result of this paper establishes that if a controller c comprising of a linear feedback of the output and its derivatives globally stabilizes a nonlinear plant p then global stabilization of p can also be achieved by an output feedback controller c h where the output derivatives in c are replaced by an euler approximation with sufficiently small delay h 0 this is proved within the conceptual framework of the nonlinear gap metric approach to robust stability the main result is then applied to finite dimensional linear minimum phase systems with unknown coefficients but known relative degree and known sign of the high frequency gain results are also given for systems with nonzero initial conditions

a simple selftuning controller for stably invertible systems in this paper we construct a new trajectory following adaptive control algorithm for linear time invariant finite dimensional systems that possess a stable inverse the exponential stability of the control scheme is demonstrated

graph topologies gap metrics and robust stability for nonlinear systems graph topologies for nonlinear operators which admit coprime factorizations are defined w r t a gain function notion of stability in a general normed signal space setting several metrics are also defined and their relationship to the graph topologies are examined in particular relationships between nonlinear generalizations of the gap and graph metrics georgiou type formulae and the graph topologies are established closed loop robustness results are given w r t the graph topology where the role of a coercivity condition on the nominal plant is emphasized

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

2 cycles on higher fano hypersurfaces let x d be a smooth hypersurface of degree d in mathbb p mathbb c n suppose that the fano variety mathrm f x d of lines of x d is smooth we prove that the griffiths group mathrm griff 1 mathrm f mathrm x d of mathrm f x d is trivial if the hypersurface mathrm x d is of 2 fano type as a result we give a positive answer to a question of professor voisin about the first griffiths groups of fano varieties in some cases base on this result we prove that mathrm ch 2 mathrm x d mathbb z for a smooth 3 fano hypersurface mathrm x d subseteq mathbb p n mathbb c with smooth fano variety of lines

families of rationally connected varieties recall that a proper variety x is said to be rationally connected if two general points p q x are contained in the image of a map g p x this is clearly a birationally invariant property when x is smooth this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset x there is a map g p x whose image contains and such that gtx is an ample bundle

gap metrics representations and nonlinear robust stability various alternative definitions for the nonlinear h2 l2 and nu gap metrics are studied the concept of beta conjugacy and multiplicative homogeneity are introduced to relate the metrics to each other and to compare the stability margins of nonlinear feedback loops expressed in terms of the norms of complementary parallel projections left and right representations for the graph of a nonlinear system are studied a new definition of normalized is introduced for left representations formulas for the gap metrics as the norm of the product of left and right representations are derived the problem of controller synthesis for input affine nonlinear systems to achieve norm bounds on the parallel projection operators is studied for input affine nonlinear systems the duality between the optimization of the two parallel projections is highlighted state space realizations for the normalized left and right representations are derived using nonlinear h infty synthesis methods

the harmonic measures of lucy garnett abstract this paper presents an improved approach to the theory of harmonic measures for foliated spaces introduced by garnett this approach is based on a method for solving elliptic equations on foliated spaces and on the hille yosida theory the diffusion semigroup of a general laplacian and its infinitesimal generator are made explicit applications of the path space to the dynamical study of a foliated space are described in particular the final section studies cocycles on foliated spaces a formula for their asymptotic limit and some analytic and geometric consequences

brief paper robust stabilization of nonlinear systems via normalized coprime factor representations a reasonably complete theory for the synthesis of robust controllers for a broad class of nonlinear systems is now available we use this theory to generalize the linear theory of normalized coprime factor robustness optimization to the case of affine input nonlinear systems in particular we show that the equilibrium controller may be characterized in terms of the stabilizing and destabilizing solutions of the hamilton jacobi equation used to calculate the normalized right coprime factors of the plant we also show that the optimal robustness margin email protected 1 mn 2 hgeneralizes to the nonlinear case in preparation for the nonlinear analysis we review the linear case in a way which motivates our approach to the nonlinear case and highlights the parallels with it

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

robust stability of input output systems with initial conditions we consider the development of a general nonlinear input output theory which encompasses systems with initial conditions systems are defined in a set theoretic manner from input output pairs on a doubly infinite time axis and a general construction of the initial conditions is given in terms of an equivalence class of trajectories on the negative time axis input output operators are then defined for given initial conditions and a suitable notion of input output stability on the positive time axis with initial conditions is given this notion of stability is closely related to the iss ios concepts of sontag a fundamental robust stability theorem is derived which represents a generalization of the input output operator robust stability theorem of georgiou and smith to include the case of initial conditions this includes a suitable generalization of the nonlinear gap metric some applications are given to show the utility of the robust stability theorem

normalized coprime factorizations and balancing for unstable nonlinear systems in this paper we first study the normalized right and left coprime factorizations of a nonlinear system from a state space point of view in order to do so we introduce the notion of inner and co inner nonlinear systems secondly we deal with balancing for unstable nonlinear systems for linear systems there are several ways to approach this problem and we generalize the ideas of balancing the normalized coprime factorization and of lqg balancing to the nonlinear case lqg balancing can be seen as measuring the influence of a state component on a certain future and past energy function we extend this interpretation to the nonlinear case furthermore we use the normalized coprime factorization to give another way of balancing the unstable nonlinear system we give the relation between these two methods of balancing which is similar to the relation in the linear case

robust stabilization of normalized coprime factor plant descriptions with h sub infinity bounded uncertainty the problem of robustly stabilizing a family of linear systems is explicitly solved in the case where the family is characterized by h sub infinity bounded perturbations to the numerator and denominator of the normalized left coprime factorization of a nominal system this problem can be reduced to a nehari extension problem directly and gives an optimal stability margin all controllers satisfying a suboptimal stability margin are characterized and explicit state space formulas are given

a biased approach to nonlinear robust stability and performance with applications to adaptive control the nonlinear robust stability theory of georgiou and smith ieee trans automat control 42 1997 pp 1200 1229 is generalized to the case of notions of stability with bias terms an example from adaptive control illustrates nontrivial robust stability certificates for systems which the previous unbiased theory could not establish a nonzero robust stability margin this treatment also shows that the bounded input bounded output robust stability results for adaptive controllers in french ieee trans automat control 53 2008 pp 461 478 can be refined to show preservation of biased forms of stability under gap perturbations in the nonlinear setting it also is shown that in contrast to linear time invariant systems the problem of optimizing nominal performance is not equivalent to maximizing the robust stability margin