morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

removal of boundary singularities of pseudo holomorphic curves with lagrangian boundary conditions we give a proof of the theorem of removing isolated singularities of pseudo holomorphic curves with lagrangian boundary conditions and bounded symplectic area the proof is a combination of some lp type estimates standard techniques of geometric p d e and some ideas from symplectic geometry and calibration theory

robust partial pole assignment problem for high order control systems in this paper we consider the partial pole assignment problem ppap for high order control systems it is shown that solving the ppap is essentially solving a pole assignment for a linear system of a much lower order and the robust ppap is then concerning the robust pole assignment problem for this linear system based on this theory a rather simple algorithm for solving the robust ppap is proposed and numerical examples show that this algorithm does lead to comparable results with earlier algorithms but at much lower computational cost

existence of weak solutions for steady viscous fluids with general slip boundary conditions abstract we study the existence of weak solutions for stationary viscous fluids with general slip boundary conditions in this paper applying monotone operator theory we first establish the existence result of weak solutions for an approximation problem then using the compactness methods and the point wise convergence property of velocity gradients we get the desired results

on the global regularity of shear thinning flows in smooth domains abstract in some recent papers we have been pursuing regularity results up to the boundary in w 2 l spaces for the velocity and in w 1 l spaces for the pressure for fluid flows with shear dependent viscosity to fix ideas we assume the classical non slip boundary condition from the mathematical point of view it is appropriate to distinguish between the shear thickening case p 2 and the shear thinning case p 2 and between flat boundaries and smooth arbitrary boundaries the p 2 non flat boundary case is still open the aim of this work is to extend to smooth boundaries the results proved in reference h beirao da veiga on non newtonian p fluids the pseudo plastic case j math anal appl 344 1 2008 175 185 this is done here by appealing to a quite general method introduced in reference h beirao da veiga on the ladyzhenskaya smagorinsky turbulence model of the navier stokes equations in smooth domains the regularity problem j eur math soc in press suitable for considering non flat boundaries

lagrangian floer theory on compact toric manifolds i we introduced the notion of weakly unobstructed lagrangian submanifolds and constructed their potential function po purely in terms of a model data in fooo3 in this article we carry out explicit calculations involving po on toric manifolds and study the relationship between this class of lagrangian submanifolds with the earlier work of givental g1 which advocates that the quantum cohomology ring is isomorphic to the jacobian ring of a certain function called the landau ginzburg superpotential combining this study with the results from fooo3 we also apply the study to various examples to illustrate its implications to symplectic topology of lagrangian fibers of toric manifolds in particular we relate it to the hamiltonian displacement property of lagrangian fibers and to entov polterovich s symplectic quasi states

multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied we prove an infinitely many existence result via z 2 index theory this is a multiplicity result about the arnold chord in some sense

on the existence of euler lagrange orbits satisfying the conormal boundary conditions abstract let m g be a closed connected riemannian manifold l t m r be a tonelli lagrangian given two non empty closed submanifolds q 0 q 1 m and a real number k we study the existence of euler lagrange orbits with energy k connecting q 0 to q 1 and satisfying suitable boundary conditions known as conormal boundary conditions we introduce the mane critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies we also provide counterexamples showing that all the results are sharp

hamiltonian pseudo rotations of projective spaces the main theme of the paper is the dynamics of hamiltonian diffeomorphisms of mathbb c mathbb p n with the minimal possible number of periodic points equal to n 1 by arnold s conjecture called here hamiltonian pseudo rotations we prove several results on the dynamics of pseudo rotations going beyond periodic orbits using floer theoretical methods one of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points partially extending a theorem of le calvez and yoccoz and of franks and misiurewicz to higher dimensions the other is a strong variant of the lagrangian poincare recurrence conjecture for pseudo rotations we also prove the c 0 rigidity of pseudo rotations with exponentially liouville mean index vector this is a higher dimensional counterpart of a theorem of bramham establishing such rigidity for pseudo rotations of the disk

balancing and model reduction for second order form linear systems model reduction of second order form linear systems is considered where a second order form reduced model is desired the focus is on reduction methods that employ or mimic moore s balance and truncate 1981 first we examine second order form model reduction by conversion to first order form and obtain a complete solution for this problem then new gramians and input output i o invariants for second order systems are motivated and defined based on these direct second order balancing methods are developed this leads naturally to direct second order form analogs for the well known first order form balance and truncate model reduction method explicit algorithms are given throughout the paper

an optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures the partial quadratic eigenvalue assignment problem pqevap concerns the reassignment of a small number of undesirable eigenvalues of a quadratic matrix pencil while leaving the remaining large number of eigenvalues and the corresponding eigenvectors unchanged the problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems the solution of this problem requires computations of a pair of feedback matrices for practical effectiveness these feedback matrices must be computed in such a way that their norms and the condition number of the closed loop eigenvector matrix are as small as possible these considerations give rise to the minimum norm partial quadratic eigenvalue assignment problem mnpqevap and the robust partial quadratic eigenvalue assignment problem rpqevap respectively in this paper we propose new optimization based algorithms for solving these problems the problems are solved directly in a second order setting without resorting to a standard first order formulation so as to avoid the inversion of a possibly ill conditioned matrix and the loss of exploitable structures of the original model the algorithms require the knowledge of only the open loop eigenvalues to be replaced and their corresponding eigenvectors the remaining open loop eigenvalues and their corresponding eigenvectors are kept unchanged the invariance of the large number of eigenvalues and eigenvectors under feedback is guaranteed by a proven mathematical result furthermore the gradient formulas needed to solve the problems by using the quasi newton optimization technique employed are computed in terms of the known quantities only above all the proposed methods do not require the reduction of the model order or the order of the controller even when the underlying finite element model has a very large degree of freedom these attractive features coupled with minimal computational requirements such as solutions of small diagonal sylvester equations make the proposed algorithms ideally suited for application to large real life structures numerical results show significant improvement in feedback norms and in the condition number of the closed loop system also the closed loop eigenvalues have acceptable accuracy

quantitative measures of modal controllability and observability in vibration control of defective and near defective systems in this paper we focus on the control problem of defective systems with repeated eigenvalues using the singular value decomposition of input matrix b and output matrix c in the modal subspace the quantitative measures of modal controllability and observability of defective systems are discussed for a near defective system with close eigenvalues we first transform it into a defective one and then apply the same method to deal with near defective systems numerical examples for a general damping system and an airfoil are given to illustrate the applications of the present method

dissipative lqg control systems an indefinite lyapunov test matrix for closed loop dissipativeness using lqg linear quadratic gaussian design is presented the application of this test to relaxation systems lossless systems hamiltonian systems positive real systems and second order systems with collocated rate sensors and actuators is explored the lyapunov test also helps clarify the role of collocated sensors and actuators in lqg design a set of duality theorems based on the indefinite lyapunov test matrix is derived

on the regularity of flows with ladyzhenskaya shear dependent viscosity and slip or non slip boundary conditions navier stokes equations with shear dependent viscosity under the classical nonslip boundary condition were introduced and studied in the 1960s by o a ladyzhenskaya and in the case of gradient dependent viscosity by j l lions a particular case is the well known smagorinsky turbulence model this is nowadays a central subject of investigation on the other hand boundary conditions of slip type seems to be more realistic in some situations in particular in numerical applications they are a main research subject the existence of weak solutions u to the above problems with slip or nonslip type boundary conditions is well known in many cases however regularity up to the boundary still presents many open questions in what follows we present some regularity results in the stationary case for weak solutions to this kind of problem see theorem 3 1 and theorem 3 2 the evolution problem is studied in a forthcoming paper 6 a cornerstone in our proof is the classical nirenberg translation method see 38

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

controllability and observability of second order descriptor systems we analyze controllability and observability conditions for second order descriptor systems and show how the classical conditions for first order systems can be generalized to this case we show that performing a classical transformation to first order form may destroy some controllability and observability properties as an example we demonstrate that the loss of impulse controllability in constrained multibody systems is due to the representation as a first order system to avoid this problem we will derive a canonical form and new first order formulations that do not destroy the controllability and observability properties

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

controllability and observability criteria for multivariable linear second order models criteria are discussed for the determination of controllability stabilizability observability or detectability of linear second order multivariable models of for example large space structures an initial modal transformation is not required and the criteria are thus applicable to models with arbitrary damping coefficients moreover the criteria are modal in the sense that some or all of the modes may be tested for controllability or observability this aspect has advantages if not all the modes are known or easily computable the criteria are further illustrated for a number of important special cases in a series of corollaries

orthogonality and partial pole assignment for the symmetric definite quadratic pencil abstract the eigenvectors of a symmetric matrix can be chosen to form an orthogonal set with respect to the identity and to the matrix itself similarly the eigenvectors of a symmetric definite linear pencil can be chosen to be orthogonal with respect to the pair this paper presents the three sets of matrix weights with respect to which the eigenvectors of the symmetric definite quadratic pencil are orthogonal one of these is used to derive an explicit solution of the partial pole assignment problem by state feedback control for a control system modeled by a system of second order differential equations the solution may be of particular interest in the stabilization and control of flexible large space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum is required to remain unchanged

parametric eigenstructure assignment in second order descriptor linear systems this note considers eigenstructure assignment in second order descriptor linear systems via proportional plus derivative feedback it is shown that the problem is closely related with a type of so called second order sylvester matrix equations through establishing two general parametric solutions to this type of matrix equations two complete parametric methods for the proposed eigenstructure assignment problem are presented both methods give simple complete parametric expressions for the feedback gains and the closed loop eigenvector matrices the first one mainly depends on a series of singular value decompositions and is thus numerically simple and reliable the second one utilizes the right factorization of the system and allows the closed loop eigenvalues to be set undetermined and sought via certain optimization procedures an example shows the effect of the proposed approaches

natural second order observers for second order distributed parameter systems the aim of this manuscript is to present an alternative method for designing state observers for second order distributed parameter systems without resorting to a first order formulation this method has the advantage of utilizing the algebraic structure that second order systems enjoy with the obvious computational savings in observer gain calculations the proposed scheme ensures that the derivative of the estimated position is indeed the estimate of the velocity component and to achieve such a result a parameter dependent lyapunov function was utilized to ensure the asymptotic convergence of the state estimation error

cup length estimate for lagrangian intersections abstract in this paper we consider the arnold conjecture on the lagrangian intersections of some closed lagrangian submanifold of a closed symplectic manifold with its image of a hamiltonian diffeomorphism we prove that if the hofer s symplectic energy of the hamiltonian diffeomorphism is less than a topology number defined by the lagrangian submanifold then the arnold conjecture is true in the degenerated nontransversal case

euler maclaurin with remainder for a simple integral polytope we give an euler maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope we prove this formula by elementary methods

on the topological properties of symplectic maps in this paper we show that symplectic maps have surprising topological properties in particular we construct an interesting metric for the symplectic diffeomorphism groups which is related but not obviously to the topological properties of symplectic maps and phase space geometry we also prove a certain number of generalised symplectic fixed point theorems and give an application to a hamiltonian system

orthogonal basis selection method for robust partial eigenvalue assignment problem in second order control systems in this paper we study the robust partial eigenvalue assignment problem using state feedback control in second order control systems which means that the feedback control matrices not only assign specific eigenvalues to the second order closed loop system but also that the system is robust or insensitive to perturbations in the coefficient matrices some measures of robustness of the closed loop system are discussed and a numerical method is proposed such that it aims to minimize one measure of the conditioning of the closed loop system numerical examples show that the method is convergent and often leads to better conditioned closed loop system

controllability and observability at infinity of multivariable linear second order models in this note we give in the spirit of 4 numerically attractive tests of controllability and observability of linear multivariable second order systems in which the leading matrix is singular

neurodynamics based robust pole assignment for high order descriptor systems in this paper a neurodynamic optimization approach is proposed for synthesizing high order descriptor linear systems with state feedback control via robust pole assignment with a new robustness measure serving as the objective function the robust eigenstructure assignment problem is formulated as a pseudoconvex optimization problem a neurodynamic optimization approach is applied and shown to be capable of maximizing the robust stability margin for high order singular systems with guaranteed optimality and exact pole assignment two numerical examples and vehicle vibration control application are discussed to substantiate the efficacy of the proposed approach

coisotropic displacement and small subsets of a symplectic manifold we prove a coisotropic intersection result and deduce the following a lower bounds on the displacement energy of a subset of a symplectic manifold in particular a sharp stable energy gromov width inequality b a stable non squeezing result for neighborhoods of products of unit spheres c existence of a badly squeezable set in mathbb r 2n of hausdorff dimension at most d for every n 2 and d n d existence of a stably exotic symplectic form on mathbb r 2n for every n 2 e non triviality of a new capacity which is based on the minimal action of a regular coisotropic submanifold of dimension d

the geometry of symplectic energy non squeezing theorem which says that it is impossible to embed a large ball symplectically into a thin cylinder of the form r2 x b2 where b2 is a 2 disc this led to hofer s discovery of symplectic capacities which give a way of measuring the size of subsets in symplectic manifolds recently hofer found a way to measure the size or energy of symplectic diffeomorphisms by looking at the total variation of their generating hamiltonians this gives rise to a bi invariant pseudo norm on the group ham m of compactly supported hamiltonian symplectomorphisms of the manifold m the deep fact is that this pseudo norm is a norm in other words the only symplectomorphism on m with zero energy is the identity map up to now this had been proved only for sufficiently nice symplectic manifolds and by rather complicated analytic arguments in this paper we consider a more geometric version of this energy which was first considered by eliashberg and hofer in connection with their study of the extent to which the interior of a region in a symplectic manifold determines its boundary we prove by a simple geometric argument that both versions of energy give rise to genuine norms on all symplectic manifolds roughly speaking we show that if there were a symplectomorphism of m which had too little energy one could embed a large ball into a thin cylinder m x

observability of linear discrete time systems of algebraic and difference equations abstractthe notion of observability for higher order discrete time systems of algebraic and difference equations is studied such systems are also known as polynomial matrix descriptions attention is first given to a special form of descriptor systems with a state lead in the output this system is transformed into its causal and noncausal subsystems and observability criteria are given in terms of the subsystem s matrices and the fundamental matrix sequence of the matrix pencil e a afterwards the higher order system is studied by transforming it into a first order descriptor system of the above form an observability criterion is provided for the higher order system in terms of the laurent expansion at infinity of the system s polynomial matrix in addition observability is connected with the coprimeness of the polynomial matrices of the higher order system and the coprimeness of the matrix pencils of the descriptor system

some quantitative results in mathcal c 0 c 0 symplectic geometry this paper proceeds with the study of the mathcal c 0 symplectic geometry of smooth submanifolds as initiated in humiliere et al duke math j 164 4 767 799 2015 and opshtein ann sci ec norm super 42 5 857 864 2009 with the main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities our main result is that a symplectic homeomorphism may preserve and squeeze codimension 4 symplectic submanifolds mathcal c 0 flexibility while this is impossible for codimension 2 symplectic submanifolds mathcal c 0 rigidity we also discuss mathcal c 0 invariants of coistropic and lagrangian submanifolds proving some rigidity results and formulating some conjectures we finally formulate an eliashberg gromov mathcal c 0 rigidity type question for submanifolds which we solve in many cases our main technical tool is a quantitative h principle result in symplectic geometry

metric and isoperimetric problems in symplectic geometry introduction 411 1 some basic results in symplectic topology 413 2 capacity and symplectic reduction 414 3 volume estimates for lagrange submanifolds 416 3 1 the case of r 417 3 2 deformations of the zero section in cotangent bundles 419 3 3 generalization to the case of cp 422 4 an application to billiards 423 5 geometry of convex sets and periodic orbits 425 6 compensated compactness and closure of the symplectic group 426 appendix a a summary of symplectic geometry through generating functions 427 appendix b john s ellipsoid 428 references 430

punctured holomorphic curves and lagrangian embeddings we use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and maslov class with boundary on lagrangian submanifolds of nonpositive curvature applications include the proof of audin s conjecture on the maslov class of lagrangian tori in linear symplectic space the construction of a new symplectic capacity obstructions to lagrangian embeddings into uniruled symplectic manifolds a quantitative version of arnold s chord conjecture and estimates on the size of weinstein neighbourhoods the main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

taubes s proof of the weinstein conjecture in dimension three does every smooth vector field on a closed three manifold for example the three sphere have a closed orbit no according to counterexamples by k kuperberg and others on the other hand there is a special class of vector fields called reeb vector fields which are associated to contact forms the three dimensional case of the weinstein conjecture asserts that every reeb vector field on a closed oriented three manifold has a closed orbit this conjecture was recently proved by taubes using seiberg witten theory we give an introduction to the weinstein conjecture the main ideas in taubes s proof and the bigger picture into which it fits taubes s proof of the weinstein conjecture is the culmination of a large body of work both by taubes and by others in an attempt to make this story accessible to nonspecialists much of the present article is devoted to background and context and taubes s proof itself is only partially explained hopefully this article will help prepare the reader to learn the full story from taubes s paper 62 more exposition of this subject which was invaluable in the preparation of this article can be found in the online video archive from the june 2008 msri hot topics workshop 44 and in the article by auroux 5 below in 1 3 we introduce the statement of the weinstein conjecture and discuss some examples in 4 6 we discuss a natural strategy for approaching the weinstein conjecture which proves it in many but not all cases and provides background for taubes s work in 7 we give an overview of the big picture surrounding taubes s proof of the weinstein conjecture readers who already have some familiarity with the weinstein conjecture may wish to start here in 8 9 we recall necessary material from seiberg witten theory in 10 we give an outline of taubes s proof and in 11 we explain some more details of it to conclude in 12 we discuss some further results and open problems related to the weinstein conjecture 1 statement of the weinstein conjecture the weinstein conjecture asserts that certain vector fields must have closed orbits before stating the conjecture at the end of this section we first outline its origins this is discussion is only semi historical because only a sample of the relevant works will be cited and not always in chronological order 1 1 closed orbits of vector fields let y be a closed manifold in this article all manifolds and all objects defined on them are smooth unless otherwise stated 2000 mathematics subject classification 57r17 57r57 53d40 partially supported by nsf grant dms 0806037

seifert conjecture in the even convex case in this paper we prove that there exist at least n geometrically distinct brake orbits on every c2 compact convex symmetric hypersurface in 2n satisfying the reversible condition n with n diag in in as a consequence we show that if the hamiltonian function is convex and even then seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integern 2014 wiley periodicals inc

multiplicity of characteristics with lagrangian boundary values on symmetric star shaped hypersurfaces abstract in this paper the multiplicity of lagrangian orbits on c 2 smooth compact symmetric star shaped hypersurfaces with respect to the origin in r 2 n is studied these lagrangian orbits begin from one lagrangian subspace and end on another an infinitely many existence result is proved via z 2 index theory this is a multiplicity result about the arnold chord conjecture in some sense and is a generalization of the problem about the multiplicity of lagrangian orbits beginning from and ending on the same lagrangian subspace which was considered in the authors previous paper f guo c liu multiplicity of lagrangian orbits on symmetric star shaped hypersurfaces nonlinear anal 69 4 2008 1425 1436

lagrangian non squeezing and a geometric inequality we prove that if the unit codisc bundle of a closed riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest non trivial closed geodesic is at most half the area of the unit disc

displacement energy of unit disk cotangent bundles we give an upper bound of a hamiltonian displacement energy of a unit disk cotangent bundle d m in a cotangent bundle t m when the base manifold m is an open riemannian manifold our main result is that the displacement energy is not greater than c r m where r m is the inner radius of m and c is a dimensional constant as an immediate application we study symplectic embedding problems of unit disk cotangent bundles moreover combined with results in symplectic geometry our main result shows the existence of short periodic billiard trajectories and short geodesic loops

a variational integrators approach to second order modeling and identification of linear mechanical systems the theory of variational integration provides a systematic procedure to discretize the equations of motion of a mechanical system preserving key properties of the continuous time flow the discrete time model obtained by variational integration theory inherits structural conditions which in general are not guaranteed under general discretization procedures we discuss a simple class of variational integrators for linear second order mechanical systems and propose a constrained identification technique which employs simple linear transformation formulas to recover the continuous time parameters of the system from the discrete time identified model we test this approach on a simulated eight degrees of freedom system and show that the new procedure leads to an accurate identification of the continuous time parameters of second order mechanical systems starting from discrete measured data

existence and uniqueness of the flow of second grade fluids with slip boundary conditions this paper treats the solvability of the equations of motion for an incompressible fluid of grade two subject to nonlinear partial slip boundary conditions in a bounded simply connected domain the existence of a unique classical solution global in time is proved under suitable regularity and growth restrictions on the initial data the slip law and the body and surface forces the method is based on a fixed point formulation of the problem