locally f g invariant distributions conditions are derived for the existence of f g invariant distributions in nonlinear systems and a procedure for the construction of the corresponding feedback control law is given

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

classical dirichlet forms on topological vector spaces the construction of the associated diffusion process given a minimal classical dirichlet form onl 2 e we construct the associated diffusion process heree is a locally convex topological vector space and is a not necessarily quasi invariant probability measure one the construction is carried out under certain assumptions one and which can be easily verified in many examples in particular we explicitly apply our results to time zero and space time quantum fields with or with out cut off

interface fluctuations for the d 1 stochastic ginzburg landau equation with nonsymmetric reaction term we consider a ginzburg landau equation in the interval e 1 e 1 e 0 with neumann boundary conditions perturbed by an additive white noise of strength sqrt varepsilon and reaction term being the derivative of a function which has two equal depth wells at 1 but is not symmetric when e 0 the equation has equilibrium solutions that are increasing and connect 1 with 1 we call them instantons and we study the evolution of the solutions of the perturbed equation in the limit e 0 when the initial datum is close to an instanton we prove that for times that may be of the order of e 1 the solution stays close to some instanton whose center suitably normalized converges to a brownian motion plus a drift this drift is known to be zero in the symmetric case and using a perturbative analysis we show that if the nonsymmetric part of the reaction term is sufficiently small it determines the sign of the drift

on cohomology isomorphisms of groups abstract we show how mislin s theorem on group homomorphisms that induce an isomorphism in cohomology can be proved based on ideas of alperin and the use of lannes s t functor this enables us to extend the class of groups to include groups of finite virtual cohomological dimension and profinite groups

optimal gaussian solutions of nonlinear stochastic partial differential equations we present a linearization procedure of a stochastic partial differential equation for a vector field x i t x t 0 x r d i l n t x i t x b i x t x d x i t x i f i t x here is the laplace beltrami operator inr d and f i t x is a gaussian random field with f i t x f j t x ij t t x x the procedure is a natural extension of the equivalent linearization for stochastic ordinary differential equations the linearized solution is optimal in the sense that the distance between true and approximate solutions is minimal when it is measured by the kullback leibler entropy the procedure is applied to the scalar valued ginzburg landau model in r1 withb 1 z z vz 3 stationary values of mean variance and correlation length are calculated they almost agree with exact ones if 1 24 2 1 4 d 1 1 3 c when c there appear quasistationary states fluctuating around one of the bottoms of the potentialu z b 1 z dz the second moment at the quasistationary states almost agrees with the exact one transient phenomena are also discussed half width at half maximum of a structure function decays liket 1 2 for small t the diffusion term 2 x accelerates the relaxation from the neighborhood of an unstable initial statex 0 x 0

the early days of geometric nonlinear control around 1970 the study of nonlinear control systems took a sharp turn in part this was driven by the hope for a more inclusive theory which would be applicable to various newly emerging aerospace problems lying outside the scope of linear theory and also by the gradual realization that tools from differential geometry and lie theory in particular could be seen as providing a remarkably nice fit with what seemed to be needed for the wholesale extension of linear control theory into a nonlinear setting this paper discusses an initial phase of the development of geometric nonlinear control including material on the broader context from which it emerged we limit our account to developments occurring up to the early 1980s not because the field stopped developing at that point but rather to limit the scope of the project to something manageable even so because of the volume and diversity of the literature we have had to be selective even within the given time frame

lie poisson structures dual spin spacecraft and asymptotic stability on traite la dynamique d un astronef rigide portant trois rotors a partir des equations dynamiques on deduit les structures de lie poisson

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

the brown comenetz dual of the k 2 local sphere at the prime 3 abstract we calculate the homotopy type of the brown comenetz dual i 2 of the k 2 local sphere at the prime 3 and show that there is an equivalence in the k 2 local category between i 2 and a smash product of the determinant twisted sphere and an exotic element p in the picard group we give a characterization of p as well a secondary aim of the paper is to extend our library of calculations in the k 2 local category

a resolution of the k 2 local sphere at the prime 3 we develop a framework for displaying the stable homotopy theory of the sphere at least after localization at the second morava k theory k 2 at the prime 3 we write the spectrum lk 2 s 0 as the inverse limit of a tower of fibrations with four layers the successive fibers are of the form e hf where f is a finite subgroup of the morava stabilizer group and e2 is the second morava or lubin tate homology theory we give explicit calculation of the homotopy groups of these fibers the case n 2a tp 3 represents the edge of our current knowledge n 1 is classical and at n 2 the prime 3 is the largest prime where the morava stabilizer group has a p torsion subgroup so that the homotopy theory is not entirely algebraic the problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics a period of calculation beginning with serre s computation of the cohomology of eilenberg maclane spaces and the advent of the adams spectral sequence culminated in the late 1970s with the work of miller ravenel and wilson on periodic phenomena in the homotopy groups of spheres and ravenel s nilpotence conjectures the solutions to most of these conjectures by devinatz hopkins and smith in the middle 1980s established the primacy of the chromatic point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory computations passed from center stage to some extent although there has been steady work in the wings most notably by shimomura and his coworkers and ravenel and more lately by hopkins and

linearization by output injection and nonlinear observers abstract observers can easily be constructed for those nonlinear systems which can be transformed into a linear system by change of state variables and output injection necessary and sufficient conditions for the existence of such a transformation are given

application of a new functional expansion to the cubic anharmonic oscillator a new representation of causal functionals is introduced which makes use of noncommutative generating power series and iterated integrals this technique allows the solutions of nonlinear differential equations with forcing terms to be obtained in a simple and natural way it generalizes some properties of fourier and laplace transforms to nonlinear systems and leads to effective computations of various perturbative expansions illustrations by means of the cubic anharmonic oscillator are given in both the deterministic and the stochastic cases

the homotopy of the k 2 local moore spectrum at the prime 3 revisited in this paper we use the approach introduced in goerss et al ann math 162 2 777 822 2005 in order to analyze the homotopy groups of l k 2 v 0 the mod 3 moore spectrum v 0 localized with respect to morava k theory k 2 these homotopy groups have already been calculated by shimomura j math soc japan 52 1 65 90 2000 the results are very complicated so that an independent verification via an alternative approach is of interest in fact we end up with a result which is more precise and also differs in some of its details from that of shimomura j math soc japan 52 1 65 90 2000 an additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the k 2 localization of the spectrum tmf of topological modular forms and related spectra even more the adams novikov differentials for l k 2 v 0 can be read off from those for tmf

coordinate independent adaptive attitude tracking control design for spacecraft robust to time varying system uncertainties abstractin this paper we consider the adaptive attitude control of spacecraft with time varying inertial properties we use a coordinate independent approach for the purpose of designing the control and estimation laws in terms of the rotation matrices representing the spacecraft body frame and reference tracking signals this method helps to overcome the difficulties regarding the attitude representation on so 3 such as ambiguities associated with quaternion representation and inherent singularities inside euler parameters we model the time variations in inertial parameters in two different ways and design adaptive control schemes for each case as the first uncertain dynamic model we consider a setting of spacecraft with multiple moving appendages and based on this model design an adaptive control scheme with three different versions where frobenious norm is used in measuring the deviation of the estimated inertia tensors from their actual values the proposed adaptive control scheme is later ex

nonlinear parabolic stochastic differential equations with additive colored noise on bf r d times bf r _ a regulated stochastic quantization we prove the existence of solutions to the nonlinear parabolic stochastic differential equation partial mathord left vphantom partial partial t right kern nulldelimiterspace partial t delta varphi v varphi eta c for polynomialsv of even degree with positive leading coefficient and c a gaussian colored noise process onr d r when c is colored enough that the gaussian solution to the linear problem has holder continuous covariance the nongaussian processes are almost surely realized by continuous functions uniqueness regularity properties asymptotic perturbation expansions and nonperturbative fluctuation bounds are obtained for the infinite volume processes these equations are a cutoff version of the parisi wu stochastic quantization procedure forp d models and the results of this paper rigorously establish the nonperturbative nature of regularization via modification of the noise process in the limit c gaussian white noise we find that the asymptotic expansion and the rigorous bounds agree for processes corresponding to the regulated stochastic quantization of super renormalizable and small coupling strictly renormalizable scalar field theories and disagree for nonrenormalizable models

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

existence and uniqueness of realizations of nonlinear systems necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input output maps are given it is also shown that two minimal realizations are diffeomorphic

centralizers of elementary abelian p subgroups and mod p cohomology of profinite groups 1 1 let g be a profinite group and p be a fixed prime in this paper we will be concerned with h c g fp the continuous cohomology of g with coefficients in the trivial module fp we will abbreviate h c g fp by h g fp or simply by h g if p is understood from the context we recall that if g is the inverse limit of finite groups gi then h g colim h gi throughout this paper we will assume that h g is finitely generated as fp algebra by work of lazard la it is known that this holds for many interesting groups for example for profinite p analytic groups like gl n zp the general linear groups over the p adic integers in case h g is finitely generated as fp algebra quillen has shown q1 that there are only finitely many conjugacy classes of elementary abelian p subgroups of g i e groups isomorphic to z p for some natural number n in other words the following category a g is equivalent to a finite category objects of a g are all elementary abelian p subgroups of g and if e1 and e2 are elementary abelian p subgroups of g then the set of morphisms from e1 to e2 in a g consists precisely of those homomorphisms e1 e2 of abelian groups for which there exists an element g g with e geg 1 e e1 the category a g plays an important role both in quillen s results and in the work presented here this category entered into quillen s work as follows the assignment e 7 h e extends to a functor from the opposite category a g op to graded fp algebras and the restriction homomorphisms h g h e for e running through the elementary abelian p subgroups of g induce a canonical map of algebras q h g lima g op h e

the image of j in the ehp sequence the ehp sequence is based on a result of james j who showed that there is h a map h such that s q sn s2n 1 is a fibration when localized at 2 the map sn sn is usually labeled e the boundary homomorphism in the homotopy sequence is usually labeled p for our purposes it will be most convenient to combine all the ehp sequences into one system this gives the following filtration of o201 s q s

the homotopy groups of the localized toda smith spectrum 1 at the prime 3 in this paper we try to compute the homotopy groups of the l2 localized toda smith spectrum v 1 at the prime 3 by using the adamsnovikov spectral sequence and have almost done so this computation involves non trivial differentials d5 and dg of the adams novikov spectral sequence different from the case p 3 we also determine the homotopy groups of some l2 localized finite spectra relating to v 1 we further show some of the non trivial differentials on elements relating so called 3 elements in the adams novikov spectral sequence for 7r s0

on the controllability of quantum mechanical systems the systems theoretic concept of controllability is elaborated for quantum mechanical systems sufficient conditions being sought under which the state vector can be guided in time to a chosen point in the hilbert space h of the system the schrodinger equation for a quantum object influenced by adjustable external fields provides a state evolution equation which is linear in and linear in the external controls thus a bilinear control system for such systems the existence of a dense analytic domain d in the sense of nelson together with the assumption that the lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension permits the adaptation of the geometric approach developed for finite dimensional bilinear and nonlinear control systems conditions are derived for global controllability on the intersection of d with a suitably defined finite dimensional submanifold of the unit sphere sh in h several soluble examples are presented to illuminate th

a mathcal b invariant distributions and disturbance decoupling of nonlinear systems the concept of a b invariant subspaces has resulted in a unified approach to many of the basic structural properties of time invariant linear systems w m wonham lecture notes in economics and mathematical systems vol 101 springer verlag new york 1974 the purpose of this paper is to introduce the more general notion of a mathcal b invariant distributions on differentiable manifolds and to use this idea to study the disturbance decoupling problem for a class of nonlinear systems which evolve on real analytic manifolds

frequency analysis and norms of distributed spatially periodic systems we investigate several fundamental aspects of the theory of linear distributed systems with spatially periodic coefficients we develop a spatial frequency domain representation analogous to the lifted or frequency response operator representation for linear time periodic systems using this representation we introduce the notion of the h2 norm for this class of systems and provide algorithms for its computation a stochastic interpretation of the h2 norm is given in terms of spatially cyclostationary random fields and spectral correlation density operators when the periodic coefficients are viewed as feedback modifications of spatially invariant systems we show how they can stabilize or destabilize the original systems in a manner analogous to vibrational control or parametric resonance in time periodic systems two examples from physics are provided to illustrate the main results

a two space dimensional semilinear heat equation perturbed by gaussian white noise a two space dimensional heat equation perturbed by a white noise in a bounded volume is considered the equation is perturbed by a non linearity of the type f au where means wick re ordering with respect to the free solution a are small parameters u denotes a solution f is the fourier transform of a complex measure with compact support