the extended euler lagrange condition for nonconvex variational problems this paper provides necessary conditions of optimality for a general variational problem for which the dynamic constraint is a differential inclusion with a possibly nonconvex right side they take the form of an euler lagrange inclusion involving convexification in only one coordinate supplemented by the transversality and weierstrass conditions it is also shown that for time invariant free time problems the adjoint arc can be chosen so that the hamiltonian function is constant along the minimizing state arc the methods used here based on simple finite dimensional nonsmooth calculus clarke decoupling and a rudimentary version of the maximum principle offer an alternative and somewhat simpler derivation of such results to those used by ioffe and rockafellar in concurrent research

necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

the hamiltonian inclusion for nonconvex velocity sets since clarke s 1973 proof of the hamiltonian inclusion for optimal control problems with convex velocity sets there has been speculation and more recently speculation relating to a stronger partially convexified version of the hamiltonian inclusion as to whether these necessary conditions are valid in the absence of the convexity hypothesis the issue was in part resolved by clarke himself when in 2005 he showed that l infty local minimizers satisfy the hamiltonian inclusion in this paper it is shown by counterexample that the hamiltonian inclusion and so also the stronger partially convexified hamiltonian inclusion are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a w 1 1 local minimizer not an l infty local minimizer the counterexample demonstrates that the need to consider l infty local minimizers not w 1 1 local minimizers in the proof of the hamiltonian inclusion for nonconvex velocity sets is fundamental not j

localisation in a cosserat continuum under static and dynamic loading conditions abstract localisation studies have been carried out for an unconstrained elasto plastic strain softening cosserat continuum because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement a finite constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions because of the existence of rotational degrees of freedom in a cosserat continuum additional wave types arise and wave propagation becomes dispersive this has been investigated analytically and numerically for an elastic cosserat continuum and an excellent agreement has been found between both solutions

a maximum principle for nonsmooth optimal control problems with state constraints abstract optimality conditions are derived in the form of a maximum principle governing solutions to an optimal control problem which involves state constraints the conditions which apply in the absence of differentiability assumptions on the data are stated in terms of clarke s generalized jacobians although not the most general available the conditions are derived by a novel method this involves removal of the state constraints by introduction of a penalty term and application of ekeland s variational principle

dynamic finite element formulation for cosserat elastic plates a displacement and rotation based dynamic finite element formulation for cosserat plates is discussed in detail in this paper special attention is given to the validation of the element through adequate benchmarks and patch tests the choice of the interpolation functions and the order of integration of the stiffness and the mass matrices are extensively argued the possibility of local system deficiencies is explored and a shear locking investigation specifically elaborated for cosserat plates is carried out it is shown how the present formulation has interesting computational properties as compared to a classical continuum based formulation and how it can provide suitable results despite the use of reduced integration an example of application of the finite element is given in which the natural frequencies of a masonry panel modelled by means of discrete elements are computed and compared with the finite elements solution

asymptotic analysis of heterogeneous cosserat media abstract the present work deals with the development of homogenization procedures for periodic heterogeneous linear elastic cosserat media it is resorted to asymptotic methods classically used in periodic homogenization it is shown that the nature of the homogeneous equivalent medium depends on the hierarchy of three characteristic lengths the size l of the heterogeneities the cosserat intrinsic lengths l c of the constituents and the typical size l of the considered structure when l and l c are comparable and much smaller than l the effective medium is proved to be a cauchy continuum with volume couples whereas the case l c l leads to a cosserat effective medium finite element simulations are provided in the case of a fiber matrix composite for a large range of characteristic lengths l c and for two different volume fractions reference calculations involving every heterogeneity are compared to the response obtained using a homogeneous equivalent medium the results confirm the predicted hierarchy of models and also show that a cosserat effective medium still provide a good estimation when all characteristic lengths have the same order of magnitude

the adjoint arc in nonsmooth optimization we extend the theory of necessary conditions for nonsmooth problems of bolza in three ways first we incorporate state constraints of the intrinsic type x t e x t for all t second we make no assumption of calmness or normality and third we show that a single adjoint function of bounded variation simultaneously satisfies the hamiltonian inclusion the euler lagrange inclusion and the weierstrass pontryagin maximum condition along with the usual transversality relations

a new finite element with drilling d o f abstract the aim of this paper is to study a new finite element with drilling degrees of freedom for linear elasticity problems and to show its convergence properties we will compare this element on a test problem with the corresponding drill element introduced by hughes and brezzi and we will analyze the behaviour of both with respect to the stiffening effect

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

some reductions of the self dual yang mills equations to integrable systems in 2 1 dimensions a reduction of the self dual yang mills sdym equations is studied by imposing two space time symmetries and by requiring that the connection one form belongs to a lie algebra of formal matrix valued differential operators in an auxiliary variable in this article the scalar case and the canonical cases for 2 2 matrices are examined in the scalar case it is shown that the field equations can be reduced to the forced burgers equation in the matrix case several well known 2 1 integrable equations are obtained also examined are certain transformation properties between the solutions of some of these 2 1 equations

computational aspects of a new multi scale dispersive gradient elasticity model with micro inertia computational aspects of a recently developed gradient elasticity model are discussed in this paper this model includes the aifantis strain gradient term along with two higher order acceleration terms micro inertia contributions it has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy in this paper the discretisation details of this model are thoroughly investigated including both discretisation in time and in space firstly the critical time step is derived that is relevant for conditionally stable time integrators secondly recommendations on how to choose the numerical parameters primarily the element size and time step are given by comparing the dispersion behaviour of the original higher order continuum with that of the discretised medium in so doing the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length scales a set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy these guidelines are then verified numerically by examining the wave propagation in a one dimensional bar as well as in a two dimensional example

homogenization of interlocking masonry structures using a generalized differential expansion technique in this paper a micromorphic continuum is derived for the homogenization of masonry structures with interlocking blocks this is done by constructing a continuum which maps exactly the kinematics of the corresponding discrete masonry structure and has the same internal and kinetic energy for any virtual translational and rotational field the obtained continuum is an anisotropic micromorphic continuum of second order the enriched kinematics of micromorphic continua allows to model microelement systems undergoing both translations and rotations the homogenization technique applied here excludes averaging and keeps all the necessary information of the discrete structure therefore all the dispersion curves of the discrete system are reproduced in the continuum model

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

two 2 1 dimensional hierarchies of evolution equations and their hamiltonian structures two kinds of appropriate isospectral problems are introduced by using a lie algebra with the help of the tah scheme we generate two new 2 1 dimensional hierarchies of evolution equations whose hamiltonian structures are derived from the trace identity proposed by tu guizhang andrushkiw r i and huang x c finally we propose some problems worth thinking about

darboux transformation and solutions for an equation in 2 1 dimensions painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2 1 dimensions this procedure has allowed us to obtain the lax pair darboux transformation and functions in such a way that a plethora of different solutions with solitonic behavior can be constructed iteratively

backlund transformations for the anti self dual yang mills equations beginning from any given local solution of the gl n c anti self dual yang mills asdym equations on minkowski space a simple technique for the generation of large classes of solutions perhaps in some sense all is given the origin of this technique is described in terms of two versions of the ward construction the resulting description of backlund transformations is sufficiently simple that it is then possible to identify the group generated by the collection of all such backlund transformations and the space on which it acts in terms of concrete functions

on an algorithmic construction of lump solutions in a 2 1 integrable equation the singular manifold method is used to generate lump solutions of a generalized integrable nonlinear schrodinger equation in 2 1 dimensions we present several essentially different types of lump solutions the connection between this method and the ablowitz villarroel scheme is also analysed

determination of hourglass coefficients in the theory of a cosserat point for nonlinear elastic beams abstract the theory of a cosserat point has recently been used int j solids struct 38 2001 4395 to formulate the numerical solution of problems of nonlinear elastic beams in that theory the constitutive equations for inhomogeneous elastic deformations included undetermined constants associated with hourglass modes which can occur due to nonuniform cross sectional extension and nonuniform torsion the objective of this paper is to determine these hourglass coefficients by matching exact solutions of pure bending and pure torsion applied in different directions on each of the surfaces of the element it is shown that the resulting constitutive equations in the cosserat theory do not exhibit unphysical stiffness increases due to thinness of the beam mesh refinement or incompressibility that are present in the associated bubnov galerkin formulation also example problems of a bar hanging under its own weight and a bar attached to a spinning rigid hub are analyzed

dualization of the euler and hamiltonian inclusions optimal control emerged as a distinct eld of research in the late 1950s with the derivation of the pontryagin maximum principle a fundamental set of necessary conditions on minimizing arcs which takes account of di erential constraints the pontryagin maximum principle directly generates the euler equation and other classical necessary conditions from the calculus of variations when specialised to problems without dynamic constraints however deeper connections between necessary conditions in optimal control and the calculus of variations were only subsequently revealed when clarke developed new analytic machinery now referred to as nonsmooth analysis a term introduced by clarke himself for extending classical necessary conditions to apply to the generalized

exotic coherent structures in the 2 1 dimensional long dispersive wave equation in this paper we investigate the integrability aspects of the 2 1 dimensional coupled long dispersive wave 2ldw equation introduced recently by chakravarty kent and newman and establish its painleve p property we then deduce its bilinear form from the p analysis and use it to construct wave type solutions for the field variables we then identify line solitons for the composite field variable qr which eventually helps to bring out the peculiar localization behavior of the system by generating localized structures dromions for the composite field from out of only one ghost soliton driving the boundary we have then extended this analysis to multidromion solutions

miura transformation between two non linear equations in 2 1 dimensions a dispersive wave equation in 2 1 dimensions 2ldw widely discussed by different authors is shown to be nothing but the modified version of the generalized dispersive wave equation gldw using singularity analysis and techniques based upon the painleve property leading to the double singular manifold expansion we shall find the miura transformation which converts the 2ldw equation into the gldw equation through this miura transformation we shall also present the lax pair of the 2ldw equation as well as some interesting reductions to several already known integrable systems in 1 1 dimensions as the 2ldw equation arises from a miura transformation we propose that it should be treated conventionally as a modified equation in this case we propose its designation as the mgldw equation

a connection between the einstein and yang mills equations it is our purpose here to show an unusual relationship between the einstein equations and the yang mills equations we give a correspondence between solutions of the self dual einstein vacuum equations and the self dual yang mills equations with a special choice of gauge group the extension of the argument to the full yang mills equations yields einstein s unifield equations we try to incorporate the full einstein vacuum equations but the approach is incomplete we first consider yang mills theory for an arbitrary lie algebra with the condition that the connection 1 form and curvature are constant on minkowski space this leads to a set of algebraic equations on the connection components we then specialize the lie algebra to be the infinite dimensional lie algebra of a group of diffeomorphisms of some manifold the algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act in the self dual case if we choose the connection components from the lie algebra of the volume preserving 4 dimensional diffeomorphism group the resulting equations are the same as those obtained by ashtekar jacobsen and smolin in their remarkable simplification of the self dual einstein vacuum equations an alternative derivation of the same equations begins with the self dual yang mills connection now depending only on the time then choosing the lie algebra as that of the volume preserving 3 dimensional diffeomorphisms when the reduced full yang mills equations are used in the same context we get einstein s equations for his unified theory based on absolute parallelism to incorporate the full einsteinvacuum equations we use as the lie group the semi direct product of the diffeomorphism group of a 4 dimensional manifold with the group of frame rotations of anso 1 3 bundle over the 4 manifold this last approach however yields equations more general than the vacuum equations

a self dual yang mills hierarchy and its reductions to integrable systems in 1 1 and 2 1 dimensions the self dual yang mills equations play a central role in the study of integrable systems in this paper we develop a formalism for deriving a four dimensional integrable hierarchy of commuting nonlinear flows containing the self dual yang mills flow as the first member we show that upon appropriate reduction and suitable choice of gauge group it produces virtually all well known hierarchies of soliton equations in 1 1 and 2 1 dimensions and can be considered as a universal integrable hierarchy prototypical examples of reductions to classical soliton equations are presented and related issues such as recursion operators symmetries and conservation laws are discussed

bicomplex formulation and moyal deformation of 2 1 dimensional fordy kulish systems using bicomplex formalism we construct generalizations of fordy kulish systems of matrix nonlinear schrodinger equations on two dimensional space time in two respects firstly we obtain corresponding equations in three space time dimensions secondly a moyal deformation is applied to the space time coordinates and the ordinary product of functions replaced by the moyal product in a suitable way both generalizations preserve the existence of an infinite set of conservation laws

equivalent subgradient versions of hamiltonian and euler llagrange equations in variational analysis much effort in recent years has gone into generalizing the classical hamiltonian and euler lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints these generalizations in which nonsmoothness abounds and gradients are systematically replaced by subgradients have succeeded in furnishing necessary conditions for optimality that reduce to the classical ones in the classical setting but important issues have remained unsettled especially concerning the exact relationship of the subgradient versions of the hamiltonian equations versus those of the euler lagrange equations here it is shown that new tighter subgradient versions of these equations are actually equivalent to each other the theory of epi convergence of convex functions provides the technical basis for this development

hamiltonian analysis of the generalized problem of bolza on obtient des conditions necessaires en forme hamiltonienne sous des hypotheses moins exigeantes qu anterieurement en particulier sans qualification sur les contraintes on developpe le lien avec les problemes de commande optimale ainsi que l apport de ces conditions a la theorie de la regularite de la solution

finite elements for elasticity with microstructure and gradient elasticity we present a general finite element discretisation of mindlin s elasticity with microstructure a total of twelve isoparametric elements are developed and presented six for plane strain conditions and six for the general case of three dimensional deformation all elements interpolate both the displacement and microdeformation fields the minimum order of integration is determined for each element and they are all shown to pass the single element test and the patch test numerical results for the benchmark problem of one dimensional deformation show good convergence to the closed form solution the behaviour of all elements is also examined at the limiting case of vanishing relative deformation where elasticity with microstructure degenerates to gradient elasticity an appropriate parameter selection that enforces this degeneration in an approximate manner is presented and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid

diffeomorphism algebras and the nahm and ward equations the question is investigated as to what equations arise from the reduction of the anti self dual yang mills equations by the imposition of three space time translational symmetries and by the choice of the connection coefficients having values in the infinite dimensional lie algebras associated with one and two dimensional diffeomorphism groups on one and two dimensional auxiliary manifolds special cases of this reduction yield the incompressible euler equations for fluid flow in two spatial dimensions the membrane equations in 4 1 dimensions the continuous version of the heisenberg spin chain equations for antiferromagnets as well as the toda lattice and liouville equations