adaptive control design based on adaptive optimization principles recently we introduced an adaptive control design for linearly parameterized multi input nonlinear systems admitting a known control lyapunov function clf that depends on the unknown system parameters the main advantage of that design is that it overcomes the problem where the estimation model becomes uncontrollable at regions of the state space where the actual system is controllable however the resulted adaptive control design is quite complicated and moreover it exhibited poor transient behavior in various applications in this technical note we propose and analyze a new computationally efficient adaptive control design that overcomes the aforementioned shortcomings the proposed design is based on an adaptive optimization algorithm introduced recently by the author which makes sure that the parameters to be optimized which correspond to the controller parameters in this technical note are modified so as to both lead to a decrease of the function to be minimized and satisfy a persistence of excitation condition the main advantage of the proposed adaptive control design is that it can produce arbitrarily good transient performance outside a the regions of the state space where the actual system becomes uncontrollable and b a region of the parameter estimates space which shrinks exponentially fast it is also worth noting that the class of systems where the proposed algorithm is applicable is more general than that of our previous work however it has to be emphasized that due to the fact that the proposed algorithm involves the use of random sequences all the established stability and convergence results are guaranteed to hold with probability one

stable localized solutions of arbitrary length for the quintic swift hohenberg equation abstract we show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic swift hohenberg equation with a destabilizing cubic term this equation is thought to model a weakly hysteretic transition to stationary patterns we argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell these results are critically compared with other known mechanisms to stabilize localized states in various situations we also discuss for which experimental systems the states predicted here could be detected including e g the stationary onset of binary fluid convection

a variant of k theory and topological t duality for real circle bundles for a space with involutive action there is a variant of k theory motivated by t duality in type ii orbifold string theory we establish that a twisted version of the variant enjoys a topological t duality for real circle bundles i e circle bundles with real structure

efficient rigorous numerics for higher dimensional pdes via one dimensional estimates we present an efficient rigorous computational method which is an extension of the work analytic estimates and rigorous continuation for equilibria of higher dimensional pdes m gameiro and j p lessard j differential equations 249 2010 pp 2237 2268 the idea is to generate sharp one dimensional estimates using interval arithmetic which are then used to produce high dimensional estimates these estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable computing the equilibria using a finite dimensional projection the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the pde a new construction of the polynomials is presented where the nonlinearities are bounded by products of one dimensional estimates as opposed to using fft with large inputs it is demonstrated that with this approac

heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible hamiltonian hopft bifurcation this paper considers an unfolding of a degenerate reversible 1 1 resonance or hamiltonian hopf bifurcation for four dimensional systems of time reversible ordinary differential equations odes this bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs the degeneracy occurs via the vanishing of a normal form coefficient q2 0 that determines whether the bifurcation is super or subcritical of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits a partial unfolding of such solutions already occurs in the work of dias and iooss eur j mech b fluids 15 1996 367 393 given a sign of the coefficient of a higher order term q4 0 and 1 q2 0 the normal form then shows a small amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit a genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies this is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies at each limit point the homoclinic orbit generates another bump close to the periodic orbit as q2 is further decreased from zero the snake widens until at a critical value a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium a kink for q2 less than this value the kinks heteroclinic tangencies and snake like curves no longer occur

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

brief paper an adaptive optimization scheme with satisfactory transient performance adaptive optimization ao schemes based on stochastic approximation principles such as the random directions kiefer wolfowitz rdkw the simultaneous perturbation stochastic approximation spsa and the adaptive fine tuning aft algorithms possess the serious disadvantage of not guaranteeing satisfactory transient behavior due to their requirement for using random or random like perturbations of the parameter vector the use of random or random like perturbations may lead to particularly large values of the objective function which may result to severe poor performance or stability problems when these methods are applied to closed loop controller optimization applications in this paper we introduce and analyze a new algorithm for alleviating this problem mathematical analysis establishes satisfactory transient performance and convergence of the proposed scheme under a general set of assumptions application of the proposed scheme to the adaptive optimization of a large scale complex control system demonstrates the efficiency of the proposed scheme

degree of approximation results for feedforward networks approximating unknown mappings and their derivatives recently barron 1993 has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition these rates do not depend on the dimension of the input space we extend barron s results to feedforward networks with possibly nonsigmoid activation functions approximating mappings and their derivatives simultaneously our conditions are similar but not identical to barron s but we obtain the same rates of approximation showing that the approximation error decreases at rates as fast as n 1 2 where n is the number of hidden units the dimension of the input space appears only in the constants of our bounds

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

adaptive neural control of mimo nonlinear state time varying delay systems with unknown dead zones and gain signs in this paper adaptive neural control is proposed for a class of uncertain multi input multi output mimo nonlinear state time varying delay systems in a triangular control structure with unknown nonlinear dead zones and gain signs the design is based on the principle of sliding mode control and the use of nussbaum type functions in solving the problem of the completely unknown control directions the unknown time varying delays are compensated for using appropriate lyapunov krasovskii functionals in the design the approach removes the assumption of linear functions outside the deadband as an added contribution by utilizing the integral lyapunov function and introducing an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead zone disturbance the closed loop control system is proved to be semi globally uniformly ultimately bounded simulation results demonstrate the effectiveness of the approach

neural networks for optimal approximation of smooth and analytic functions we prove that neural networks with a single hidden layer are capable of providing an optimal order of approximation for functions assumed to possess a given number of derivatives if the activation function evaluated by each principal element satisfies certain technical conditions under these conditions it is also possible to construct networks that provide a geometric order of approximation for analytic target functions the permissible activation functions include the squashing function 1 e x 1 as well as a variety of radial basis functions our proofs are constructive the weights and thresholds of our networks are chosen independently of the target function we give explicit formulas for the coefficients as simple continuous linear functionals of the target function

perturbed homoclinic solutions in reversible 1 1 resonance vector fields we consider a smooth reversible vector field in r 4 such that the origin is a fixed point the differential at the origin has two double pure imaginary eigenvalues iq for the critical value 0 of the parameter we show by a normal form analysis that the vector field can be approximated by an integrable field in r 4 for which we know all solutions specially interesting ones are the homoclinics to 0 and homoclinics to periodic solutions depending on the sign of a leading nonlinear coefficient we prove in particular the persistence of these homoclinics for the full vector field

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

approximation by network operators with logistic activation functions this paper aims to study the construction and multivariate approximation of a class of network operators with logistic sigmoidal functions first a class of even and bell shaped function with support on r is constructed by using appropriate translation and combination of the logistic function then the constructed function is employed as activation function to construct a kind of so called cardaliaguet euvrard type network operators finally these network operators are used to approximate bivariate functions in c 1 1 2 and a jackson type theorem for the approximation errors is established

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

approximation of conditional densities by smooth mixtures of regressions this paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the kullback leibler distance by different specifications of finite mixtures of normal regressions in which normal means and variances and mixing probabilities can depend on variables in the conditioning set covariates these models are a special case of models known as mixtures of experts in statistics and computer science literature flexible specifications include models in which only mixing probabilities modeled by multinomial logit depend on the covariates and in the univariate case models in which only means of the mixed normals depend flexibly on the covariates modeling the variance of the mixed normals by flexible functions of the covariates can weaken restrictions on the class of the approximable densities obtained results can be generalized to mixtures of general location scale densities rates of convergence and easy to interpret bounds are also obtained for different model specifications these approximation results can be useful for proving consistency of bayesian and maximum likelihood density estimators based on these models the results also have interesting implications for applied researchers

classification of quaternionic bloch bundles we provide a classification of type aii topological quantum systems in dimension d 1 2 3 4 our analysis is based on the construction of a topological invariant the fkmm invariant which completely classifies quaternionic vector bundles a k a symplectic vector bundles in dimension d leqslant 3 this invariant takes value in a proper equivariant cohomology theory and in the case of examples of physical interest it reproduces the familiar fu kane mele index in the case d 4 the classification requires a combined use of the fkmm invariant and the second chern class among the other things we prove that the fkmm invariant is a bona fide characteristic class for the category of quaternionic vector bundles in the sense that it can be realized as the pullback of a universal topological invariant

locally optimal controllers and globally inverse optimal controllers in this paper we consider the problem of global asymptotic stabilization with prescribed local behavior we show that this problem can be formulated in terms of control lyapunov functions moreover we show that if the local control law has been synthesized employing an lq approach then the associated lyapunov function can be seen as the value function of an optimal problem with some specific local properties we illustrate these results on two specific classes of systems backstepping and feedforward systems finally we show how this framework can be employed when considering the orbital transfer problem

chiral vector bundles this paper focuses on the study of a new category of vector bundles the objects of this category called chiral vector bundles are pairs given by a complex vector bundle along with one of its automorphisms we provide a classification for the homotopy equivalence classes of these objects based on the construction of a suitable classifying space the computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes some of those just reproduce the ordinary chern classes but there are also new odd dimensional classes which take care of the extra topological information introduced by the chiral structure chiral vector bundles provide a geometric model for topological quantum systems in class aiii namely for systems endowed with a pseudo symmetry of chiral type the classification of the chiral vector bundles over spheres and tori explicitly computable up to dimension 4 recovers the commonly accepted classification for topological insulators of class aiii which is usually based on the k group k 1 however our classification turns out to be even richer since it takes care also for possible non trivial chern classes

clf based control design for unknown multiinput nonlinear systems with good transient performance this technical note tackles the problem of constructing state feedback stabilizers that guarantee good transient closed loop performance when applied to general unknown nonlinear multiinput state feedback stabilizable systems an adaptive control lyapunov function based control scheme is proposed in order to address this problem mathematical analysis establishes that the proposed control scheme guarantees good closed loop transient performance outside the regions where the system is uncontrollable provided the controlled system admits a controllability like property

approximation by neural networks with scattered data in this paper the approximation capability of feed forward neural networks fnns formed as k 1 nc k f x x k is considered where f is an even periodic continuous function and x k s are scattered data the best approximation error of trigonometric polynomials and the continuous modulus of the activation function f are introduced to describe the approximation capability of fnns

error bounds for functional approximation and estimation using mixtures of experts we examine some mathematical aspects of learning unknown mappings with the mixture of experts model mem specifically we observe that the mem is at least as powerful as a class of neural networks in a sense that will be made precise upper bounds on the approximation error are established for a wide class of target functions the general theorem states that spl par f f sub n spl par sub p spl les c n sup r d for f spl isin w sub p sup r l a sobolev class over 1 1 sup d and f sub n belongs to an n dimensional manifold of normalized ridge functions the same bound holds for the mem as a special case of the above the stochastic error in the context of learning from independent and identically distributed i i d examples is also examined an asymptotic analysis establishes the limiting behavior of this error in terms of certain pseudo information matrices these results substantiate the intuition behind the mem and motivate applications

analysis of sdc matrices for successfully implementing the sdre scheme abstract the state dependent riccati equation sdre approach for stabilization of nonlinear affine systems was recently reported to be effective in many practical applications however there is no guideline on the construction of state dependent coefficient sdc matrix when the sdre solvability condition is violated which may result in the sdre scheme being terminated in this study we present several easy checking conditions so that the sdre scheme can be successfully implemented additionally when the presented checking conditions are satisfied the sets of all feasible sdc matrices and their structures are explicitly depicted for the planar system

accuracy analysis for wavelet approximations constructive wavelet networks are investigated as a universal tool for function approximation the parameters of such networks are obtained via some direct monte carlo procedures approximation bounds are given typically it is shown that such networks with one layer of wavelons achieve an l sub 2 error of order o n sup spl rhod where n is the number of nodes d is the problem dimension and spl rho is the number of summable derivatives of the approximated function an algorithm is also proposed to estimate this approximation based on noisy input output data observed from the function under consideration unlike neural network training this estimation procedure does not rely on stochastic gradient type techniques such as the celebrated backpropagation and it completely avoids the problem of poor convergence or undesirable local minima

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

sampled data feedback practical semi global controllability and stabilization for time varying systems we deal with the problems of the closed loop controllability and stabilization for time varying systems whose dynamics are in general unbounded with respect to time it is shown that a general notion of global asymptotic controllability being in general non uniform with respect to initial values of time implies existence of a family of continuous control lyapunov functions and this enables the establishment of practical semi global closed loop controllability by means of a sampled data time varying feedback moreover the usual concept of global asymptotic controllability being uniform with respect to the initial values of time implies practical semi global stabilization by sampled data feedback the work is a continuation of a recent paper of the author in the same journal and extends krichman s 2000 result relying practical stabilization of time varying systems with bounded in time dynamics

robust switching adaptive control of multi input nonlinear systems during the last decade a considerable progress has been made in the design of stabilizing controllers for nonlinear systems with known and unknown constant parameters new design tools such as adaptive feedback linearization adaptive back stepping control lyapunov functions clfs and robust control lyapunov functions rclfs nonlinear damping and switching adaptive control have been introduced most of the results developed are applicable to single input feedback linearizable systems and parametric strict feedback systems these results however cannot be applied to multi input feedback linearizable systems parametric pure feedback systems and systems that admit a linear in the parameters clf in this paper we develop a general procedure for designing robust adaptive controllers for a large class of multi input nonlinear systems this class of nonlinear systems includes as a special case multi input feedback linearizable systems parametric pure feedback systems and systems that admit a linear in the parameters clf the proposed approach uses tools from the theory of rclf and the switching adaptive controllers proposed by the authors for overcoming the problem of computing the feedback control law when the estimation model becomes uncontrollable the proposed control approach has also been shown to be robust with respect to exogenous bounded input disturbances

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

the errors of simultaneous approximation of multivariate functions by neural networks there have been many studies on the simultaneous approximation capability of feed forward neural networks fnns most of these however are only concerned with the density or feasibility of performing simultaneous approximations this paper considers the simultaneous approximation of algebraic polynomials employing taylor expansion and an algebraic constructive approach to construct a class of fnns which realize the simultaneous approximation of any smooth multivariate function and all of its derivatives we also present an upper bound on the approximation accuracy of the fnns expressed in terms of the modulus of continuity of the functions to be approximated

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

approximation bounds for smooth functions in c r sup d by neural and mixture networks we consider the approximation of smooth multivariate functions in c r sup d by feedforward neural networks with a single hidden layer of nonlinear ridge functions under certain assumptions on the smoothness of the functions being approximated and on the activation functions in the neural network we present upper bounds on the degree of approximation achieved over the domain r sup d thereby generalizing available results for compact domains we extend the approximation results to the so called mixture of expert architecture which has received considerable attention in recent years showing that the same type of approximation bound may be achieved

an efficient estimate based on fft in topological verification method in this paper localized patterns of the quintic swift hohenberg equation are studied a numerical verification method with the conley index theory developed in zgliczynski and mischaikow rigorous numerics for partial differential equations the kuramoto sivashinsky equation found comput math 1 2001 255 288 is used in order to prove these patterns a new technique to efficiently obtain estimates for nonlinear terms is presented the key idea is based on the pseudo spectral method it is shown that this technique is inevitable for the verification of the localized patterns

asymptotic stabilization of two dimensional real analytic systems abstract we investigate the local asymptotic stabilizability of a real analytic control system around an equilibrium point and give necessary and sufficient conditions for local asymptotic stabilization using continuous feedback

non commutative odd chern numbers and topological phases of disordered chiral systems abstract an index theorem for higher chern characters of odd fredholm modules over crossed product algebras is proved together with a local formula for the associated cyclic cocycle the result generalizes the classic noether gohberg krein index theorem which in its simplest form states that the winding number of a complex valued function over the circle is equal to the index of the associated toeplitz operator when applied to the non commutative brillouin zone this generalization allows to define topological invariants for all condensed matter phases from the chiral unitary or aiii symmetry class in the presence of strong disorder and magnetic fields whenever the fermi level lies in a region of anderson localized spectrum

hysteresis switching adaptive control of linear multivariable systems this paper presents a model reference adaptive control scheme for deterministic continuous time multivariable systems represented by square strictly proper minimum phase transfer function matrices a typical requirement of existing algorithms is to assume that the zero structure at infinity and the high frequency gain matrix are fully or at least partially known it is well known that these requirements may be very restrictive since in general both the zero structure at infinity and the high frequency gain matrix depend on plant parameters in this paper we show that these restrictive assumptions may be considerably weakened using morse et al s hysteresis switching control strategy 1992 the strategy entails running a finite number of parameter estimators in parallel and using a switching algorithm to select between candidate estimators based on their associated prediction errors hysteresis in the switching algorithm precludes switching arbitrarily rapidly between estimators and all switching ceases within a finite time the results represent a significant step forward in understanding the minimal amount of prior knowledge necessary to design a stabilizing controller for a linear multivariable system

convex design control for practical nonlinear systems this paper describes a new control scheme for approximately optimal control aoc of nonlinear systems convex control design convcd the key idea of convcd is to transform the approximate optimal control problem into a convex semi definite programming sdp problem contrary to the majority of existing aoc designs where the problem that is addressed is to provide a control design which approximates the performance of the optimal controller by increasing the controller complexity the proposed approach addresses a different problem given a controller of fixed complexity it provides a control design that renders the controller as close to the optimal as possible and moreover the resulted closed loop system stable two numerical examples are used to show the effectiveness of the method

asymptotics of cellular buckling close to the maxwell load we study the deformation of an elastic strut on a nonlinear winkler foundation subjected to an axial compressive load p using multi scale analysis and numerical methods we describe the localized cellular post buckled state of the system when p is removed from the critical load p 2 the solutions and their modulation frequencies differ significantly from those predicted by weakly nonlinear analysis very close to p 2 in particular when p approaches the maxwell load pm the localized solutions approach a large amplitude heteroclinic connection between an unbuckled solution and a periodic solution an asymptotic description of pm in terms of the system parameters is given the agreement between the numerical calculations and the asymptotic approximations is striking

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

the approximation operators with sigmoidal functions the aim of this paper is to investigate the error which results from the method of approximation operators with logarithmic sigmoidal function by means of the method of extending functions a class of feed forward neural network operators is introduced using these operators as approximation tools the upper bounds of errors in uniform norm approximating continuous functions are estimated also a class of quasi interpolation operators with logarithmic sigmoidal function is constructed for approximating continuous functions defined on the total real axis

differential geometric invariants for time reversal symmetric bloch bundles the real case topological quantum systems subjected to an even resp odd time reversal symmetry can be classified by looking at the related real resp quaternionic bloch bundles if from one side the topological classification of these time reversal vector bundle theories has been completely described in de nittis and gomi j geom phys 86 303 338 2014 for the real case and in de nittis and gomi commun math phys 339 1 55 2015 for the quaternionic case from the other side it seems that a classification in terms of differential geometric invariants is still missing in the literature with this article and its companion g de nittis and k gomi unpublished we want to cover this gap more precisely we extend in an equivariant way the theory of connections on principal bundles and vector bundles endowed with a time reversal symmetry in the real case we generalize the chern weil theory and we show that the assignment of a real connection along with the related differential chern class and its holonomy suffices for the classification of real vector bundles in low dimensions