multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

computational plasticity for composite structures based on mathematical homogenization theory and practice abstract this paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases the overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro constituent and history data is updated at two points reinforcement and matrix in the microstructure one for each phase macroscopic history data is stored in the database and then subjected in the post processing stage onto the unit cell in the critical locations for numerical examples considered the cpu time obtained by means of the two point averaging scheme with variational micro history recovery with 30 seconds on sparc 10 51 as opposed to 7 hours using classical mathematical homogenization theory at the same time the maximum error in the microstress field in the critical unit cell was only 3 5 in comparison with the classical mathematical homogenization theory

almost sure convergence of extremum seeking algorithm using stochastic perturbation abstract this paper presents an extremum seeking es algorithm where the perturbation signal is a martingale difference sequence m d s with a vanishing variance the measurement noise at the plant output is modeled by a superposition of deterministic component and a non stationary colored noise signal the optimizing set point of the uncertain reference to output equilibrium map is estimated by a stochastic approximation sa type algorithm the algorithm has a vanishing gain sequence dependent on the set point estimates by utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point in spite of the presence of a measurement noise this result is derived without requiring boundedness or any prior condition on the set point estimates

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

newton based stochastic extremum seeking in this paper we introduce a newton based approach to stochastic extremum seeking and prove local stability of newton based stochastic extremum seeking algorithm in the sense of both almost sure convergence and convergence in probability the convergence of the newton algorithm is proved to be independent of the hessian matrix and can be arbitrarily assigned which is an advantage over the standard gradient based stochastic extremum seeking simulation shows the effectiveness and advantage of the proposed algorithm over gradient based stochastic extremum seeking

extremum seeking with bounded update rates a b s t r a c t in this work we present a form of extremum seeking es in which the unknown function being mini mized enters the system s dynamics as the argument of a cosine or sine term thereby guaranteeing known bounds on update rates and control efforts we present general n dimensional optimization and stabi lization results as well as 2d vehicle control with bounded velocity and control efforts for application to autonomous vehicles tracking a source in a gps denied environment with unknown orientation this es approach allows for smooth heading angle actuation with constant velocity and in application to a unicycle type vehicle results in control ability as if the vehicle is fully actuated our stability analysis is made possible by the classic results of kurzweil jarnik sussmann and liu regarding systems with highly oscillatory terms in our stability analysis we combine the averaging results with a semiglobal practical stability result under small parametric perturbations developed by moreau and aeyels published by elsevier b v

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

extremum seeking loops with quadratic functions estimation and control extremum seeking also peak seeking controllers are designed to operate at an a priori unknown set point that extremizes the value of a performance function traditional approaches to the problem assume a time scale separation between the gradient computation and function minimization and the system dynamics the work here in contrast assumes that the performance function can be approximated by a quadratic function with a finite number of parameters these parameters are estimated on line and the extremum seeking controller operates based on these estimated values a significant advantage of a quadratic function is that it allows the peak seeking control loop to be reduced to a linear system for such a loop the wealth of linear system analysis and synthesis tools can be employed first the control loop is analysed assuming that the parameters in the function are known full information case and then when the parameters are estimated on line the partial information case

multivariable newton based extremum seeking we present a newton based extremum seeking algorithm for the multivariable case the design extends the recent newton based extremum seeking algorithms for the scalar case and introduces a dynamic estimator of the inverse of the hessian matrix that removes the difficulty with the possible singularity of a possible direct estimate of the hessian matrix the estimator of the inverse of the hessian has the form of a differential riccati equation we prove local stability of the new algorithm for general nonlinear dynamic systems using averaging and singular perturbations in comparison with the standard gradient based multivariable extremum seeking the proposed algorithm removes the dependence of the convergence rate on the unknown hessian matrix and makes the convergence rate of both the parameter estimates and of the estimates of the hessian inverse user assignable in particular the new algorithm allows all the parameters to converge with the same speed yielding straight trajectories to the extremum even with maps that have highly elongated level sets in contrast to curved steepest descent trajectories of the gradient algorithm simulation results show the advantage of the proposed approach over gradient based extremum seeking by assigning equal desired convergence rates to all the parameters using newton s approach

observer based perturbation extremum seeking control with input constraints for direct contact membrane distillation process abstractan observer based perturbation extremum seeking control is proposed for a direct contact membrane distillation dcmd process the process is described with a dynamic model that is based on a 2d advection diffusion equation model which has pump flow rates as process inputs the objective of the controller is to optimise the trade off between the permeate mass flux and the energy consumption by the pumps inside the process cases of single and multiple control inputs are considered through the use of only the feed pump flow rate or both the feed and the permeate pump flow rates a nonlinear lyapunov based observer is designed to provide an estimation for the temperature distribution all over the designated domain of the dcmd process moreover control inputs are constrained with an anti windup technique to be within feasible and physical ranges performance of the proposed structure is analysed and simulations based on real dcmd process parameters for each control input are provided

stochastic averaging in continuous time and its applications to extremum seeking we investigate stochastic averaging theory in continuous time for locally lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms first we establish a general stochastic averaging principle and some related stability theorems for a class of continuous time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results global lipschitzness of the nonlinear vector field equilibrium preservation under the stochastic perturbation global exponential stability of the average system and compactness of the state space of the perturbation process then we propose a continuous time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals we analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems

batch to batch finite horizon lq control for unknown discrete time linear systems via stochastic extremum seeking we employ our recent discrete time stochastic averaging theorems and stochastic extremum seeking to iteratively batch to batch optimize open loop control sequences for unknown but reachable discrete time linear systems with a scalar input and without known system dimension for a cost that is quadratic in the measurable output and the input first for a multivariable gradient based stochastic extremum seeking algorithm we prove local exponential convergence to the optimal open loop control sequence second to remove the convergence rate s dependence on the hessian matrix of the cost function which is unknown since the system s model the system matrices a b c is unknown we develop a multivariable discrete time newton based stochastic extremum seeking method design the newton based algorithm for the iteration of the input sequence and prove local exponential convergence to the optimal input sequence finally two simulation examples are given to illustrate the effectiveness of the two methods

proper generalized decomposition of time multiscale models summary models encountered in computational mechanics could involve many time scales when these time scales cannot be separated one must solve the evolution model in the entire time interval by using the finest time step that the model implies in some cases the solution procedure becomes cumbersome because of the extremely large number of time steps needed for integrating the evolution model in the whole time interval in this paper we considered an alternative approach that lies in separating the time axis one dimensional in nature in a multidimensional time space then for circumventing the resulting curse of dimensionality the proper generalized decomposition was applied allowing a fast solution with significant computing time savings with respect to a standard incremental integration copyright 2011 john wiley sons ltd

newton like extremum seeking for the control of thermoacoustic instability in practice the convergence rate and stability of perturbation based extremum seeking schemes can be very sensitive to the curvature of the plant map an example of this can be seen in the use of extremum seeking to reduce the amplitude of thermoacoustic oscillations in premixed gas turbine combustors this sensitivity to the plant map curvature arises from the use of a gradient descent adaptation algorithm such extremum seeking schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions resulting in slower optimization than could be achieved for a fixed operating condition this can severely reduce the effectiveness of perturbation based extremum seeking schemes in some applications in this paper a sinusoidally perturbed extremum seeking scheme using a newton like step is developed non local stability results for the scheme are formulated using a lyapunov analysis a local analysis of the scheme is given to investigate the influence of plant dynamics and to show that the local rate of convergence is independent of the plant map curvature the benefit of this plant map curvature independence is then experimentally demonstrated in minimizing the thermoacoustic oscillations in a model premixed combustor

extremum seeking under stochastic noise and applications to mobile sensors in this paper the extremum seeking algorithm with sinusoidal perturbations has been extended and modified in two ways a the output of the system is corrupted with measurement noise b the amplitudes of the perturbation signals as well as the gain of the integrator block are time varying and tend to zero at a pre specified rate convergence to the extremal point with probability one has been proved also as a consequence of being able to cope with a stochastic environment it has been shown how the proposed algorithm can be applied to mobile sensors as a tool for achieving the optimal observation positions the proposed algorithm has been illustrated through several simulations

on energy based coupled elastoplastic damage theories constitutive modeling and computational aspects abstract novel energy based coupled elastoplastic damage theories are presented in this paper the proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials at variance with lemaitre s work on damage elastoplasticity the present formulation renders rational thermodynamic potential and damage energy release rate in contrast to previous work by simo and ju featuring an additive split of the stress tensor current formulation assumes an additive split of the strain tensor it is shown that the strain split damage elastoplasticity formulation leads to more robust tangent moduli than the stress split formulation the plastic flow rule and hardening law are characterized in terms of the effective quantities viz the effective stress space plasticity this mechanism is both physically well motivated and computationally efficient further a fourth order anisotropic damage mechanism is proposed for brittle materials rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain rate dependency of microcrack growth efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the operator splitting methodology in particular new three step operator split algorithms are presented application is made to a class of inviseid and rate dependent cap damage models for concrete and mortar experimental validations are also given to illustrate the applicability of the proposed damage models

extremum seeking with stochastic perturbations extremum seeking es using deterministic periodic perturbations has been an effective method for non model based real time optimization when only limited plant knowledge is available however periodicity can naturally lead to predictability which is undesirable in some tracking applications and unrepresentative of biological optimization processes such as bacterial chemotaxis with this in mind it is useful to investigate employing stochastic perturbations in the context of a typical es architecture and to compare the approach with existing stochastic optimization techniques in this work we show that convergence towards the extremum of a static map can be guaranteed with a stochastic es algorithm and quantify the behavior of a system with gaussian distributed perturbations at the extremum in terms of the es constants and map parameters we then examine the closed loop system when actuator dynamics are included as the separation of time scales between the perturbation signal and plant dynamics recommended in periodic es schemes cannot be guaranteed with stochastic perturbations consequently we investigate how actuator dynamics influence the allowable range of es parameters and necessitate changes in the closed loop structure finally simulation results are presented to demonstrate convergence and to validate predicted behavior about the extremum for the sake of analogy with the classical methods of stochastic approximation stochastic es in this technical note is pursued in discrete time

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

extremum seeking control with an adaptive dither signal we present an output feedback law that steers a fully actuated control affine system with general drift vector field to a minimum of the output function the proposed control law is based on a suitable approximation of descent directions by lie brackets in the high frequency limit a novelty of our approach is an adaptive choice of the frequency parameter in this way the problem of determining a sufficiently large frequency parameter becomes obsolete the adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift the proposed control law does not only lead to convergence into a neighborhood of the set of minima but leads to exact convergence for the case of an output function with a global minimum and no other stationary points we prove global convergence

multi scale domain decomposition method for large scale structural analysis with a zooming technique application to plate assembly this article is concerned with a multi scale domain decomposition method ddm based on the feti dp solver for large scale structural elastic analysis and suited to problems that exhibit structural heterogeneities such as plate assemblies in the presence of structural details in this approach once a partition of the global fine mesh into subdomains has been performed all subdomains possess a fine mesh and to optimize the computational time the fine mesh is preserved only in the zones of interest with local phenomena due to discontinuity hole etc while the remaining subdomains are replaced by numerical homogenized coarse elements indeed the multi scale aspect is introduced by the description of subdomains with either a fine or a coarse scale mesh as a result an extension of the feti dp ddm is proposed in this article called herein feti dp micro macro that allows the simultaneous usage of different discretizations fine microscopic mesh for subdomains in zones of interest and coarse macroscopic or homogenized mesh for the complementary part of the structure using this strategy raises the problem of the determination of the stiffness of coarse subdomains and of the incompatible finite element connection between fine and coarse subdomains two approaches collocation and mortar are presented and compared the article ends with patch tests and some numerical examples in 2d and 3d the obtained numerical results exemplify the efficiency and capability of the feti dp micro macro approach and reveal that the mortar approach is more accurate at constant cost than the collocation approach copyright 2009 john wiley sons ltd

numerical solution of a fast diffusion equation in this paper the authors consider the first boundary value problem for the nonlinear reaction diffusion equation u t u m u p 1 in a smooth bounded domain in r d d 1 with the zero lateral boundary condition and with a positive initial condition m e 0 1 fast diffusion problem 0 and p1 m sufficient conditions on the initial data are obtained for the solution to vanish or become infinite in a finite time a scheme for the discretization in time of this problem is proposed the numerical scheme preserves the essential properties of the initial problem namely existence of an extinction or a blow up time for which estimates have been obtained the convergence of the method is also proved

recent advances on the use of separated representations separated representations based on finite sum decompositions constitute an appealing strategy for reducing the computer resources and the calculation costs by reducing drastically the number of degrees of freedom that the functional approximations involve the number of degrees of freedom scale linearly with the dimension of the space in which the model is defined instead of the exponential growing characteristic of mesh based discretization strategies in our knowledge the use of separated representations is the only possibility for circumventing the terrific curse of dimensionality related to some highly multidimensional models involving hundreds of dimensions as we proved in some of our former works its application is not restricted to multidimensional models obviously separated representation can also be applied in standard 2d or 3d models allowing for high resolution computations because its early life numerous issues persist many of them attracting the curiosity of many research groups within the computational mechanics community in this paper we are focusing in two issues never until now addressed i the imposition of non homogenous essential boundary conditions and ii the consideration of complex geometries copyright 2009 john wiley sons ltd

linking microstructure and properties through a predictive multiresolution continuum abstract under the sponsorship of the nsf sandia life cycle science based engineering program the sandia predictive science program and the onr digital 3d d3d program a multiresolution continuum theory c mcveigh f vernerey w k liu l c brinson comput methods appl mech engrg 195 2006 5053 is developed to predict material response when spatial and temporal microstructure evolution gives rise to severely inhomogeneous deformation at multiple scales the proposed theory is applied by concurrently homogenizing the microstructure at each characteristic length scale associated with the inhomogeneous response a continuum microstructure work rate equivalence approach is used to develop a set of continuum partial differential governing equations in terms of multiresolution microstresses and couple microstresses constitutive models relating to each microstress are determined from numerical microstructure models the multiresolution governing equations can be solved with a conventional finite element approach hence numerical and modeling errors analyses probabilistic and reliability analyses and petaflop computing can be considered using existing approaches or with transparent modifications when only a single scale of inhomogeneous deformation at the scale of the rve is considered the multiresolution theory decomposes to the strain gradient theory of fleck and hutchinson the theory is applied to i an alloy with two scales of statistically embedded particles ii a cemented carbide and iii adiabatic shear banding in steel alloys only the mean constitutive behavior is considered at each scale the full probabilistic analysis will be presented in a separate paper

an extended stochastic finite element method for solving stochastic partial differential equations on random domains abstract recently a new strategy was proposed to solve stochastic partial differential equations on random domains it is based on the extension to the stochastic framework of the extended finite element method x fem this method leads by a direct calculus to an explicit solution in terms of the variables describing the randomness on the geometry it relies on two major points the implicit representation of complex geometries using random level set functions and the use of a galerkin approximation at both stochastic and deterministic levels in this article we detail the basis of this technique from theoretical and technical points of view several numerical examples illustrate the efficiency of this method and compare it to other approaches

on a mixed and multiscale domain decomposition method this paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems focusing on its suitability for parallel computing spatially this strategy can be viewed as a mixed multilevel domain decomposition method or more accurately as a structure decomposition method regarding time a parallel property is also described we also draw bridges between this and other current approaches

stochastic averaging in discrete time and its applications to extremum seeking we investigate stochastic averaging theory for locally lipschitz discrete time nonlinear systems with stochastic perturbation and its applications to convergence analysis of discrete time stochastic extremum seeking algorithms firstly by defining two average systems one is continuous time the other is discrete time we develop discrete time stochastic averaging theorem for locally lipschitz nonlinear systems with stochastic perturbation our results only need some simple and applicable conditions which are easy to verify and remove a significant restriction present in existing results global lipschitzness of the nonlinear vector field secondly we provide a discrete time stochastic extremum seeking algorithm for a static map in which measurement noise is considered and an ergodic discrete time stochastic process is used as the excitation signal finally for discrete time nonlinear dynamical systems in which the output equilibrium map has an extremum we present a discrete time stochastic extremum seeking scheme and with a singular perturbation reduction we prove the stability of the reduced system compared with classical stochastic approximation methods while the convergence that we prove is in a weaker sense the conditions of the algorithm are easy to verify and no requirements e g boundedness are imposed on the algorithm itself

a perturbation based proportional integral extremum seeking control approach this paper proposes a proportional integral extremum seeking control technique the technique is a generalization of the standard perturbation based techniques which provides fast transient performance of the closed loop system to the optimum equilibrium of a measured cost function under the assumption that the equations describing the dynamics of the nonlinear system and the cost function to be minimized are unknown it is shown that for a class of minimum phase nonlinear system with a cost function of relative degree one the closed loop esc can practically stabilize the unknown equilibrium optimum the stabilization result provides an alternative approach to output feedback control of nonlinear systems the effectiveness of the proposed approach is demonstrated using simulation examples

a study of strain localization in a multiple scale framework the one dimensional problem abstract this work approaches strain localization by recognizing the multiple scales inherent in the problem a component associated with the region of localized strain typically one with high gradient is referred to as the fine scale it represents the microstructure the field obtained by removing the fine scale from the total solution is referred to as the coarse scale the aim of the multiple scale method advanced here is to derive a model for the coarse scale field that accounts for the finite scale this process eliminates the fine scale from the problem yet retains its effect in applying this framework to the nonlinear problems with which localized strains are associated a crucial step is a first order approximation of the relevant relations the fully nonlinear problem is solved by an iterative scheme by accounting directly for the microstructure the multiple scale model recovers the regularizing effects of various alternative formulations for softening strain localization it thus presents itself as a unifying framework for such models numerical solutions are shown to be invariant with respect to the discretization furthermore for cases in which the displacements assume a distinct profile within the localization band the multiple scale model provides a resolution that can be made as accurate as desired even with the coarsest mesh possible the model is applied to strain localization problems that arise in inviscid and viscoplastic solids numerical simulations are presented that demonstrate the efficacy of the approach

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

a pgd based multiscale formulation for non linear solid mechanics under small deformations abstract model reduction techniques have became an attractive and a promising field to be applied in multiscale methods the main objective of this work is to formulate a multiscale procedure for non linear problems based on parametrized microscale models the novelty of this work relies in the implementation of the model reduction technique known as proper generalized decomposition for solving the high dimensional parametrized problem resulting from the microscale model the multiscale framework here proposed is formulated to non linear problems specifically to material non linearities where material response is governed by a strain dependent evolution law two strategies to deal with this kind of problem under small deformations are detailed in this work both strategies based on parametrized microscale models solved by pgd have been applied to a problem with a rate dependent isotropic damage model first a procedure where the problem is solved by uncoupling the equilibrium equation to the state variable expression has been explored in order to alleviate the parametrized microscale problem a second strategy for problems with material non linearity has been proposed incorporating a parametrized microscale problem at each macroscale increment fe pgd the basis of those procedures are described and compared highlighting the solution accuracy and computer time consumption in comparison to a traditional finite element analysis

nash equilibrium seeking in noncooperative games we introduce a non model based approach for locally stable convergence to nash equilibria in static noncooperative games with n players in classical game theory algorithms each player employs the knowledge of the functional form of his payoff and the knowledge of the other players actions whereas in the proposed algorithm the players need to measure only their own payoff values this strategy is based on the extremum seeking approach which has previously been developed for standard optimization problems and employs sinusoidal perturbations to estimate the gradient we consider static games with quadratic payoff functions before generalizing our results to games with non quadratic payoff functions that are the output of a dynamic system specifically we consider general nonlinear differential equations with n inputs and n outputs where in the steady state the output signals represent the payoff functions of a noncooperative game played by the steady state values of the input signals we employ the standard local averaging theory and obtain local convergence results for both quadratic payoffs where the actual convergence is semi global and non quadratic payoffs where the potential existence of multiple nash equilibria precludes semi global convergence our convergence conditions coincide with conditions that arise in model based nash equilibrium seeking however in our framework the user is not meant to check these conditions because the payoff functions are presumed to be unknown for non quadratic payoffs convergence to a nash equilibrium is not perfect but is biased in proportion to the perturbation amplitudes and the higher derivatives of the payoff functions we quantify the size of these residual biases and confirm their existence numerically in an example noncooperative game in this example we present the first application of extremum seeking with projection to ensure that the players actions remain in a given closed and bounded action set

extremum seeking control for discrete time systems we present an extremum seeking control algorithm for discrete time systems applied to a class of plants that are represented as a series combination of a linear input dynamics a static nonlinearity with an extremum and a linear output dynamics by using the two time scale averaging theory we derive a mild sufficient condition under which the plant output exponentially converges to an o spl alpha sup 2 neighborhood of the extremum value where spl alpha is the magnitude of modulation signal the sufficient condition is related to positive realness of linear parts of the plant but only at the modulation frequency the algorithm is illustrated with a brief simulation study

distributed seeking of nash equilibria with applications to mobile sensor networks we consider the problem of distributed convergence to a nash equilibrium in a noncooperative game where the players generate their actions based only on online measurements of their individual cost functions corrupted with additive measurement noise exact analytical forms and or parameters of the cost functions as well as the current actions of the players may be unknown additionally the players actions are subject to linear dynamic constraints we propose an algorithm based on discrete time stochastic extremum seeking using sinusoidal perturbations and prove its almost sure convergence to a nash equilibrium we show how the proposed algorithm can be applied to solving coordination problems in mobile sensor networks where motion dynamics of the players can be modeled as 1 single integrators velocity actuated vehicles 2 double integrators force actuated vehicles and 3 unicycles a kinematic model with nonholonomic constraints examples are given in which the cost functions are selected such that the problems of connectivity control formation control rendezvous and coverage control are solved in an adaptive and distributed way the methodology is illustrated through simulations

a multilevel finite element method fe2 to describe the response of highly non linear structures using generalized continua abstract a general method called fe2 has been introduced which consists in describing the behavior of heterogeneous structures using a multiscale finite element model instead of trying to build differential systems to establish a stress strain relation at the macroscale a finite element computation of the representative volume element is carried out simultaneously doing so does not require any constitutive equations to be written at the macroscopic scale all non linearities come directly from the microscale in this paper we describe how this method can be used in the context of generalized continua for such continua constitutive equations are very difficult to write and a new set of material is difficult to fit to experimental data the use of fe2 models bypasses this problem because no analytical equation is needed at the macroscale an academic application is presented to show that generalized continua are necessary when the size of the heterogeneities increases and that fe2 models behave well compared to a reference solution