output integral sliding mode for min max optimization of multi plant linear uncertain systems we consider the application of a min max optimal control based on the lq index for a set of systems where only the output information is available here every system is affected by matched uncertainties and we propose to use an output integral sliding mode to compensate the matched uncertainties right after the beginning of the process for the case when the extended system is free of invariant zeros a hierarchical sliding mode observer is applied the error of realization of the proposed control algorithm is estimated in terms of the sampling step and actuator time constant an example illustrates the suggested method of design

nature of singularities for analyticity domains of invariant curves we present theoretical arguments based on infinite dimensional bifurcation theory and numerical evidence based on nonperturbative methods that the boundaries of analyticity of invariant curves can be described as an accumulation of branch points which are typically of order 2 we show how this fact would explain previous numerical results of several authors and how it suggests more efficient numerical algorithms which we implement

application of shifted jacobi pseudospectral method for solving in finite horizon min max optimal control problems with uncertainty abstractthe difficulty of solving the min max optimal control problems m mocps with uncertainty using generalised euler lagrange equations is caused by the combination of split boundary conditions nonlinear differential equations and the manner in which the final time is treated in this investigation the shifted jacobi pseudospectral method sjpm as a numerical technique for solving two point boundary value problems tpbvps in m mocps for several boundary states is proposed at first a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented then by applying the generalised euler lagrange equations and expanding the required approximate solutions as elements of shifted jacobi polynomials finding a solution of tpbvps in nonlinear m mocps with uncertainty is reduced to the solution of a system of algebraic equations moreover the jacobi polynomials are particularly useful for boundary value problems in unbounded

the dynamic programming approach to multi model robust optimization abstract the aim of this paper is to extend the dynamic programming dp approach to multi model optimal control problems ocps we deal with robust optimization of multi model control systems and are particularly interested in the hamilton jacobi bellman hjb equation for the above class of problems in this paper we study a variant of the hjb for multi model ocps and examine the natural relationship between the bellman dp techniques and the robust maximum principle mp moreover we describe how to carry out the practical calculations in the context of multi model lq problems and derive the associated riccati type equation

mini max integral sliding mode control for multimodel linear uncertain systems an original linear time varying system with matched and unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external unmatched bounded disturbances such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system containing only matched uncertainties leads to a linear multi model system with known unmatched bounded disturbances and unknown matched disturbances as well each model from a given finite set is characterized by a quadratic performance index the developed minimax integral sliding mode control strategy gives an optimal minimax linear quadratic lq control with additional integral sliding mode term the design of this controller is reduced to a solution of an equivalent mini max lq problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex the additional integral sliding mode controller part completely dismisses the influence of matched uncertainties from the initial time instant two numerical examples illustrate this study

measurement of body segment parameters using dual energy x ray absorptiometry and three dimensional geometry an application in gait analysis abstract body segment parameters bsp are essential input for the computations in kinetics of motion applied in the field of biomechanics these data are usually obtained from population specific predictive equations which present limitations in being representative of the population under study more recently medical imaging techniques have been adopted but are limited to two dimensional 2 d measurements or required extensive tomographic images for three dimensional 3 d reconstruction we proposed an in vivo method to measure 3 d bsp using x ray imaging and 3 d exterior geometry criterion values of the bsp were determined using magnetic resonance imaging mri which has previously been validated errors for all bsp values were less than 2 when values derived from our method were compared to the criterion values we found no significant difference between our method and four selected bsp models in both stance and swing phase significant phase effects were observed for our method and other bsp models between stance and swing phase significant differences p

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

a criterion for non existence of invariant tori for hamiltonian systems abstract a particularly interesting class of invariant tori for a hamiltonian system satisfying the legendre condition is the class of invariant lagrangian graphs a classical result of the calculus of variations is that for such systems every orbit on an invariant lagrangian graph has minimal action this provides a simple criterion for non existence of invariant lagrangian graphs namely if an orbit has a variation which decreases the action or it possesses conjugate points then there is no invariant lagrangian graph containing it as an example the criterion is applied to the motion of a particle in the field of two waves to prove that there are no invariant tori with initial velocity between those of the two waves if the amplitudes of the waves exceed certain values

mri derived body segment parameters of children differ from age based estimates derived using photogrammetry abstract body segment parameters are required when researching joint kinetics using inverse dynamics models however the only regression equations for estimating pediatric body segment parameters across a wide age range were developed using photogrammetry based on 12 boys and have not been validated to date jensen r k 1986 body segment mass radius and radius of gyration proportions of children journal of biomechanics 19 359 368 to assess whether these equations could validly be applied to girls we asked whether body segment parameters estimated by the equations differ from parameters measured using a validated magnetic resonance imaging mri method if so do the differences cause significant differences in joint kinetics during normal gait body segment parameters were estimated from axial mris of the left thigh and shank of 10 healthy girls 9 6 0 9 years and compared to those from jensen s equations kinematics and kinetics were collected for 10 walking trials extrema in hip and knee moments and powers were compared between the two sets of body segment parameters with the exception of the shank mass center and radius of gyration body segment parameters measured using mri were significantly different from those estimated using regression equations these systematic differences in body segment parameters resulted in significant differences in sagittal plane joint moments and powers during gait nevertheless it is doubtful that even the greatest differences in kinetics are practically meaningful 0 3 bw ht and 0 7 bw ht s for moments and power at the hip respectively therefore body segment parameters estimated using jensen s regression equations are a suitable substitute for more detailed anatomical imaging of 8 10 year old girls when quantifying joint kinetics during gait

determination of the threshold of the break up of invariant tori in a class of three frequency hamiltonian systems we consider a class of hamiltonians with three degrees of freedom that can be mapped into quasi periodically driven pendulums the purpose of this paper is to determine the threshold of the break up of invariant tori with a specific frequency vector we apply two techniques the frequency map analysis and renormalization group methods the renormalization transformation acting on a hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the hamiltonian and a rescaling of phase space around the considered torus we give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break up of invariant tori furthermore we obtain by these methods numerical values of the threshold of the break up of the last invariant torus

minimax optimal control this paper provides a framework for deriving necessary conditions in the form of a maximum principle for minimax optimal control problems the distinguishing feature of these problems is that the data depends on a vector alpha of unknown parameters and optimality is defined on a worst case basis as alpha ranges over the parameter set cal a the centerpiece a minimax maximum principle is a set of optimality conditions for such problems here the parameter set cal a is taken to be an arbitrary compact metric space and the hypotheses imposed on the dynamics and endpoint constraints are of an unrestrictive nature the minimax maximum principle captures as special cases necessary conditions for optimal control problems with minimax costs for problems involving semi infinite endpoint constraints and also a maximum principle for state constrained optimal control problems

a remark on the kam theorem applied to a four vortex system we consider a planar four vortex system with unit intensities and apply the kam theorem for two dimensional tori with fixed frequency we obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation number 5 1 2 and compare our result with numerical experiments

bounds on the trajectories of a system of weakly coupled rotators in this paper we study a classical mechanical system of weakly coupled rotators on a one dimensional lattice such systems are of interest in statistical mechanics we prove that for any site in the system there is a large set of initial conditions for which there exists a canonical change of variables such that the trajectory of that site in the transformed system is essentially indistinguishable from that of an integrable system for a long but finite time alternatively the trajectory of this site lies very close to a torus in the phase space of the system for times very long in comparison with the typical period of the unperturbed rotators all the estimates in this theory areindependent of the number of degrees of freedom in the system we propose this mechanism as an explanation of certain numerical experiments

critical kam circles and the brjuno function the authors consider a complex hamiltonian map and show how to approximate the critical value of the perturbation parameter at which a given kam circle disappears by means of a function the brjuno function which only depends on the continued fraction expansion of the rotation number

a posteriori kam theory with optimal estimates for partially integrable systems abstract in this paper we present a posteriori kam results for existence of d dimensional isotropic invariant tori for n dof hamiltonian systems with additional n d independent first integrals in involution we carry out a covariant formulation that does not require the use of action angle variables nor symplectic reduction techniques the main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates which may cause difficulties and underperformance when quantifying the hypotheses of the kam theorem in such reduced coordinates the results include ordinary and generalized iso energetic kam theorems the approach is suitable to perform numerical computations and computer assisted proofs

robust stochastic maximum principle for multi model worst case optimization this paper develops a version of the robust maximum principle applied to the minimax mayer problem formulated for stochastic differential equations with a control dependent diffusion term the parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each possible value of the parameter the cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term a terminal condition given by a vector function is also considered the optimal control strategies adapted for available information for the wide class of multi model systems given by a stochastic differential equation with parameters from a given finite set are constructed this problem belongs to the class of minimax stochastic optimization problems the p

natural boundaries for area preserving twist maps we consider kam invariant curves for generalizations of the standard map of the form x y x y y f x wheref x is an odd trigonometric polynomial we study numerically their analytic properties by a pade approximant method applied to the function which conjugates the dynamics to a rotation in the complex plane natural boundaries of different shapes are found in the complex plane the analyticity region appears to be a strip bounded by a natural boundary whose width tends linearly to 0 as tends to the critical value

a 3d reconstruction method of the body envelope from biplanar x rays evaluation of its accuracy and reliability the aim of this study was to propose a novel method for reconstructing the external body envelope from the low dose biplanar x rays of a person the 3d body envelope was obtained by deforming a template to match the surface profiles in two x rays images in three successive steps global morphing to adopt the position of a person and scale the template s body segments followed by a gross deformation and a fine deformation using two sets of pre defined control points to evaluate the method a biplanar x ray acquisition was obtained from head to foot for 12 volunteers in a standing posture up to 172 radio opaque skin markers were attached to the body surface and used as reference positions each envelope was reconstructed three times by three operators results showed a bias lower than 7 mm and a confidence interval 95 of reproducibility lower than 6 mm for all body parts comparable to other existing methods matching a template onto stereographic photographs the proposed method offers the possibility of reconstructing body shape in addition to the skeleton using a low dose biplanar x rays system

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

accurate strategies for small divisor problems many interesting problems in mechanics are close to systems that can be solved exactly in such systems it is natural to consider perturbative expansions about the known system frequently it is possible to compute all of the terms in the expansion recursively which can lead to expressions involving denominators that become arbitrarily small

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

robust optimal control for minimax stochastic linear quadratic problem the robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control dependent diffusion term the parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding hamiltonian formalism the hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic hamiltonians corresponding to each value of the uncertain parameter from a given finite set the cost function is considered on a finite horizon contains the mathematical expectation of both an integral and a terminal term and on an infinite one a time averaged losses function these problems belong to the class of minimax stochastic optimization problems it is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of ric

robust maximum principle for multi model lq problem this paper presents the version of the robust maximum principle in the context of multi model control formulated as the minimax bolza problem the cost function contains a terminal term as well as an integral one a fixed horizon and terminal set are considered the necessary conditions of the optimality are derived for the class of uncertain systems given by an ordinary differential equation with parameters from a given finite set this problem consists in the control design providing a good behaviour for a given class of multi model system it is shown that the design of the minimax optimal controller is reduced to a finite dimensional optimization problem given at the corresponding simplex set containing the weight parameters to be found the robust optimal control may be interpreted as a mixture with the optimal weights of the controls which are optimal for each fixed parameter value the proof is based on the recent results obtained for minimax mayer problem boltyanski and poznyak 1999a the mini

computer assisted proofs in analysis and programming in logic a case study in this paper we present a computer assisted proof of the existence of a solution for the feigenbaum equation varphi x frac 1 lambda varphi varphi lambda x there exist by now various such proofs in the literature although the one presented here is new the main purpose of this paper is not to provide yet another version but to give an easy to read and self contained introduction to the technique of computer assisted proofs in analysis our proof is written in prolog programming in logic a programming language which we found to be well suited for this purpose in this paper we also give an introduction to prolog so that even a reader without prior exposure to programming should be able to verify the correctness of the proof

on the numerical treatment of a small divisor problem quasiperiodic solutions of perturbed integrable hamiltonian equations such as weakly coupled harmonic oscillators can be found by constructing an appropriate coordinate transformation which leads to a small divisor problem however the numerical difficulties are not merely caused by the small divisors but rather by the appearence of ghost solutions which appear in any reasonable discretization of the problem our numerical treatment based on a newton type iteration guarantees an approximation of the relevant solution of the nonlinear problem numerical solutions are found up to a critical value of the coupling constant which is much larger than the coupling constants allowed by the existence theory available so far

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

high order control for symplectic maps abstract we revisit the problem of introducing an a priori control for devices that can be modeled via a symplectic map in a neighborhood of an elliptic equilibrium using a technique based on lie transform methods we produce a normal form algorithm that avoids the usual step of interpolating the map with a flow the formal algorithm is completed with quantitative estimates that bring into evidence the asymptotic character of the normal form transformation then we perform an heuristic analysis of the dynamical behavior of the map using the invariant function for the normalized map finally we discuss how control terms of different orders may be introduced so as to increase the size of the stable domain of the map the numerical examples are worked out on a two dimensional map of henon type

a method for accurate stability bounds in a small denominator problem the author considers the problem of obtaining realistic lower bounds for the siegel radius recent advances of the analysis of siegel disks allows one to give a very accurate numerical algorithm based on rigorous results he finds that for non quadratic polynomial maps the maximal siegel radius might correspond to rotation numbers different from the golden mean

robust maximum principle in minimax control this paper presents a new version of the maximum principle dealing with construction of optimal control strategies for the class of uncertain systems given by an ordinary differential equation with unknown parameters from a given finite set the problem under consideration belongs to the class of optimization problems of the minimax type the proof is based on the tent method illustrative examples conclude this study

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

invariant curves for area preserving twist maps far from integrable one parameter families of area preserving twist maps of the formf e x y x y ef x y ef x are considered various invariant curves for the maps corresponding tof x sin andf x sinx 1 50 sin 5x are rigorously constructed forlarge values of the nonlinearity parameter for larger values of close to critical some numerical experiments are briefly discussed

min max sliding mode control for multimodel linear time varying systems an original linear time varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system with a complete knowledge leads to a linear multi model system with known bounded disturbances each model from a given finite set is characterized by a quadratic performance index the developed min max sliding mode control strategy gives an optimal robust sliding surface design algorithm which is reduced to a solution of an equivalent linear quadratic problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex an illustrative numerical example is presented

the kam theory of systems with short range interactions i the existence of quasiperiodic trajectories for hamiltonian systems consisting of long chains of nearly identical subsystems with interactions which decay rapidly with increasing distance between the interacting components is studied such models are of interest in statistical mechanics it is shown that nonergodic motions persist for much larger perturbations than prior work indicated if the number of degrees of freedom of the system isn the allowed perturbation decreases only as an inverse power ofn as the number of degrees of freedom increases rather than the inverse power ofn which previous estimates yielded

a rigorous partial justification of greene s criterion we prove several theorems that lend support to greene s criterion for the existence or not of invariant circles in twist maps in particular we show that some of the implications of the criterion are correct when the aubry mather sets are smooth invariant circles or uniformly hyperbolic we also suggest a simple modification that can work in the case that the aubry mather sets have nonzero lyapunov exponents the latter is based on a closing lemma for sets with nonzero lyapunov exponents which may have several other applications

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

min max optimal control of linear systems with uncertainty and terminal state constraints in this paper a class of min max optimal control problems with continuous dynamical systems and quadratic terminal constraints is studied the main contribution is that the original terminal state constraint in which the disturbance is involved is transformed into an equivalent linear matrix inequality without disturbance under certain conditions then the original min max optimal control problem is solved via solving a sequence of semi definite programming problems an example is presented to illustrate the proposed method

compensations in small divisor problems a general direct method alternative to kam theory apt to deal with small divisor problems in the real analytic category is presented and tested on several small divisor problems including the construction of maximal quasi periodic solutions for nearly integrable non degenerate hamiltonian or lagrangian systems and the construction of lower dimensional resonant tori for nearly integrable hamiltonian systems the method is based on an explicit graph theoretical representation of the formal power series solutions which allows to prove compensations among the monomials forming such representation

determination of a new uniform thorax density representative of the living population from 3d external body shape modeling body segment parameters bsp for each body s segment are needed for biomechanical analysis to provide population specific bsp precise estimation of body s segments volume and density are needed widely used uniform densities provided by cadavers studies did not consider the air present in the lungs when determining the thorax density the purpose of this study was to propose a new uniform thorax density representative of the living population from 3d external body shape modeling bi planar x ray radiographies were acquired on 58 participants allowing 3d reconstructions of the spine rib cage and human body shape three methods of computing the thorax mass were compared for 48 subjects 1 the dempster uniform density method currently in use for bsps calculation using dempster density data 2 the personalized method using full description of the thorax based on 3d reconstruction of the rib cage and spine and 3 the improved uniform density method using a uniform thorax density resulting from the personalized method for 10 participants comparison was made between the body mass obtained from a force plate and the body mass computed with each of the three methods the dempster uniform density method presented a mean error of 4 8 in the total body mass compared to the force plate vs 0 2 for the personalized method and 0 4 for the improved uniform density method the adjusted thorax density found from the 3d reconstruction was 0 74g cm 3 for men and 0 73g cm 3 for women instead of the one provided by dempster 0 92g cm 3 leading to a better estimate of the thorax mass and body mass

the simulation of aerial movement ii a mathematical inertia model of the human body a mathematical inertia model which permits the determination of personalized segmental inertia parameter values from anthropometric measurements is described the human body is modelled using 40 geometric solids which are specified by 95 anthropometric measurements a stadium solid is introduced for modelling the torso segments using perimeter and width measurements this procedure is more accurate than the use of elliptical discs of given width and depth and permits a smaller number of such solids to be used inertia parameter values may be obtained for body models of up to 20 segments errors in total body mass estimates from this and other models are discussed with reference to the unknown lung volumes

rigorous estimates for a computer assisted kam theory nonautonomous hamiltonian systems of one degree of freedom close to integrable ones are considered let e be a positive parameter measuring the strength of the perturbation and denote by e c the critical value at which a given kam kolmogorov arnold moser torus breaks down a computer assisted method that allows one to give rigorous lower bounds for e c is presented this method has been applied in celletti falcolini porzio to be published in ann inst h poincare to the escande and doveil pendulum yielding a bound which is within a factor 40 2 of the value indicated by numerical experiments