positive curvature local and global symmetry and fundamental groups a 1 invariant torus action on a manifold m is a t k action on the universal covering which extends to the action of a semi direct product 1 m t k in particular the t k action is the lift of a t k action on m if is the identity map the main result asserts that if a compact manifold m n of positive sectional curvature admits a 1 invariant isometric t k action then the fundamental group has a cyclic subgroup of index w n this refines the main result in ro1

adaptive regulation of high order lower triangular systems an adding a power integrator technique abstract in this paper a new adaptive control technique called adding a power integrator is proposed for a class of linearly parametrized high order nonlinear systems that are of a lower triangular structure but neither feedback linearizable nor affine in the control input using the adaptive adding a power integrator technique as a basic design tool we show how to explicitly construct in a recursive manner a smooth adaptive controller that solves the problem of adaptive regulation with global stability the development of the adding a power integrator technique leads to a significant relaxation on the system nonlinearities such as feedback linearizability and affineness which are known to be two standard hypotheses made in the existing nonlinear adaptive control schemes

saturated delayed controls for feedforward nonlinear systems the saturated delayed feedback is presented to globally stabilize a class of feedforward nonlinear systems whose nominal dynamics is the cascade of multiple oscillators and multiple integrators key strategies are stated as follows 1 to compensate the arbitrarily large input delay and to carry out the recursive analysis we transform the concerned system into a normal form 2 in the convergence analysis for multiple oscillators that are relatively difficult to treat we combine the saturated delayed terms define the composite lyapunov functions and use the cauchy formula 3 for ease of the analysis on nonlinear terms we employ saturation levels to restrict them as applications the usual feedforward nonlinear systems with multiple integrators as the nominal dynamics and the null controllable linear systems are dealt with in some new manners and a saturated delayed controller is presented for the well known tora system

frobenius schur indicators for subgroups and the drinfel d double of weyl groups if g is any finite group and k is a field there is a natural construction of a hopf algebra over k associated to g the drinfel d double d g we prove that if g is any finite real reflection group with drinfel d double d g over an algebraically closed field k of characteristic not 2 then every simple d g module has frobenius schur indicator 1 this generalizes the classical results for modules over the group itself we also prove some new results about weyl groups in particular we prove that any abelian subgroup is inverted by some involution also if e is any elementary abelian 2 subgroup of the weyl group w then all representations of cw e are defined over q

integrator backstepping for rounded controls and control rates we present a backstepping procedure for the design of globally stabilizing state feedback control laws such that the magnitudes of the control signals and their derivatives are bounded by constants which do not depend on the initial conditions we accomplish this by propagating such boundedness properties through each step of the recursive design

lyapunov design of stabilizing controllers for cascaded systems the design of a state feedback law for an affine nonlinear system to render a as small as possible compact neighborhood of the equilibrium of interest globally attractive is discussed following z artstein s theorem 1983 the problem can be solved by designing a so called control lyapunov function for systems which are in a cascade form a lyapunov function meeting artstein s conditions is designed assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem in particular for planar systems this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized

an explicit formula of bounded feedback stabilizers for feedforward systems abstract this work constitutes continuation of a recent result of the same authors which provides an explicit formula for bounded feedback stabilizers for a wide class of triangular nonlinear systems in the present paper the previous result is extended for nonlinear systems having feedforward structure and whose linearization at the equilibrium is in general uncontrollable

homogeneous lyapunov function for homogeneous continuous vector field the goal of this article is to provide a construction of a homogeneous lyapunov function p associated with a system of differential equations j f x x r n 1 under the hypotheses 1 f c r n vanishes at x 0 and is homogeneous 2 the zero solution of this system is locally asymptotically stable moreover the lyapunov function v x tends to infinity with 1 x i and belongs to c r 0 r n cp with p e as large as wanted as application to the theory of homogeneous systems we present two well known results of robustness in a slightly extended form and with simpler proofs

the almost cyclicity of the fundamental groups of positively curved manifolds recall that a pure f structure is a kind of generalized torus action the main result asserts that if a compact positively curved manifold mn admits an invariant pure f structure such that each orbit has positive dimension then the fundamental group has a finite cyclic subgroup with index less than wn a constant depending only on n as an application we conclude that for all 0 1 the fundamental group of a pinched n manifold either has a cyclic subgroup with index less than wn or has order less than w n a constant depending only on n and in particular this substantially improves the main result in ro1

stabilization of feedforward systems approximated by a non linear chain of integrators we prove that for any odd integer p and any strictly positive integer n feedforward systems which are approximated at the origin by a chain of integrators of degree p and length n can be globally asymptotically stabilized by bounded smooth time invariant state feedbacks our proof is based on the construction of a lyapunov function and the feedback laws we obtain are given by explicit formulas

asymptotically stabilizing feedback controls and the nonlinear regulator problem continuous asymptotically stabilizing feedback controls are constructed for two dimensional and certain three dimensional small time locally controllable affine systems the lie products which de

on the topology of positively curved 4 manifolds with symmetry soit m une variete compacte a courbure positive orientable a 4 dimensions si m a un champ de vecteurs de killing non trivial alors m est homeomorphe a s 4 ou cp 2

positively curved manifolds with maximal discrete symmetry rank let m be a closed simply connected n manifold of positive sectional curvature we determine its homeomorphism or homotopic type if m also admits an isometric elementary p group action of large rank our main results are there exists a constant p n 0 such that 1 if m 2 n admits an effective isometric inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i action for a prime p p n then k n and implies that m 2 n is homeomorphic to a sphere or a complex projective space 2 if m 2 n 1 admits an isometric s 1 inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i action for a prime p p n then k n and implies that m is homeomorphic to a sphere 3 for m in 1 or 2 if n 7 and k 3 n 4 2 then m is homeomorphic to a sphere or homotopic to a complex projective space

simple saturated designs for ancbc systems and extension to feedforward nonlinear systems this article re examines the robust stabilisation of the asymptotically null controllable with bounded controls ancbc systems and extends the established algorithm to a wide class of feedforward nonlinear systems whose nominal dynamics contains both multiple integrators and multiple oscillators based on the notion of converging input bounded state cibs rather than small input small state siss the computation burden in sussmann et al sussmann h j sontag e d and yang y 1994 a general result on the stabilization of linear systems using bounded controls ieee transactions on automatic control 39 2411 2425 is reduced and a class of simple saturated control laws is presented for the cibs stabilisation of ancbc systems then by combining the technique of dealing with higher order terms the algorithm for ancbc systems is extended to feedforward nonlinear systems

adding an integrator for the stabilization problem abstract we study the relationship between the following two properties p1 the system x f x y y v is locally asymptotically stabilizable and p2 the system x f x u is locally asymptotically stabilizable where x r n y r dayawansa martin and knowles have proved that these properties are equivalent if the dimension n 1 here using the so called control lyapunov function approach a we propose another more constructive and somewhat simpler proof of dayawansa martin and knowles s result b we show that in general p1 does not imply p2 for dimensions n larger than 1 c we prove that p2 implies p1 if some extra assumptions are added like homogeneity of the system by using the latter result recursively we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form

explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization for a class of nonlinear systems having triangular structure we provide explicit formulas for feedback controllers that exhibit global stabilization the global part of our analysis is mainly based on a version of sontag s input to state stability property and constitutes a generalization of previous works of the second author

obstructions to positive curvature and symmetry we discuss new obstructions to positive sectional curvature and symmetry the main result asserts that the index of the dirac operator twisted with the tangent bundle vanishes on a 2 connected manifold of dimension 8 if the manifold admits a metric of positive sectional curvature and isometric effective s1 action the proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds

collapsing riemannian manifolds while keeping their curvature bounded i soit y n une variete de riemann connexe complete et p y n le rayon d injectivite i p de l application exponentielle en p est defini comme etant le plus petit r tel que exp p b r p n est pas un diffeomorphisme sur son image on etudie la situation dans laquelle la taille du rayon d injectivite est petite relativement a la courbure

constructive nonsmooth stabilization of triangular systems the problem of local asymptotic continuous feedback stabilization of single input nonlinear systems is considered here the aim is to explicitly construct a continuous asymptotically stabilizing feedback for a class of nonlinear systems that is known to be continuous feedback asymptotically stabilizable but for which the known results do not provide an effective method to compute the stabilizer computer simulations are included to show practical applicability of our approach

nilpotent structures and invariant metrics on collapsed manifolds let mn be a complete riemannian manifold of bounded curvature say iki 0 we put mn w n c u fn c where 1n e consists of those points at which the injectivity radius of the exponential map is e the complementary set n c is called the e collapsed part of mn if x e 4 n r e then the metric ball bx r is quasi isometric with small distortion to the flat ball b0 r in the euclidean space rn after slightly