a new topological technique for characterizing homoclinic tangles we have developed a topological approach called homotopic lobe dynamics for describing the qualitative structure of homoclinic tangles this approach begins from an efficient and accurate description of the initial development of a tangle up to some finite number of iterates j where the value of j indicates the amount of information that one puts into the theory our approach can then compute the topologically forced structure of the tangle at all iterates after j this allows one for example to predict a minimal set of homoclinic intersections this technique places few assumptions on the homoclinic tangles considered in fact one main advantage is its ability to describe the wide variety of behaviour seen in physically significant tangles

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

transmutation techniques and observability for time discrete approximation schemes of conservative systems in this article we consider abstract linear conservative systems and their time discrete counterparts our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time discrete one as an application we show how observability properties for the time continuous model yield uniform with respect to the time step observability results for its time discrete approximation counterparts provided the initial data are suitably filtered the main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time discrete models besides using a reverse representation formula we also prove that this estimate on the time of uniform observability for the time discrete models is sharp we then conclude with some general comments and open problems

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

controllability cost of conservative systems resolvent condition and transmutation abstract this article concerns the exact controllability of unitary groups on hilbert spaces with unbounded control operator it provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality by the transmutation of controls in some time l for the corresponding second order conservative system it is proved that the cost of controls in time t for the unitary group grows at most like exp l 2 t as t tends to 0 in the application to the cost of fast controls for the schrodinger equation l is the length of the longest ray of geometric optics which does not intersect the control region this article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system

closed form sums for some perturbation series involving associated laguerre polynomials infinite series n 1 2 n n 1 n 1f1 n x2 where 1f1 n x2 n n ln 1 x2 appear in the first order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator hamiltonian h d2 dx2 bx2 a x2 x 0 x 0 a 0 it is proved that the series is convergent for all x 0 and 2 where 1 1 4a closed form sums are presented for these series for the cases 2 4 and 6 a general formula for finding the sum for 2 2 m m 0 1 2 in terms of associated laguerre polynomials is also provided

controllability and observability for non autonomous evolution equations the averaged hautus test abstract we consider the observability problem for non autonomous evolution systems i e the operators governing the system depend on time we introduce an averaged hautus condition and prove that for skew adjoint operators it characterizes exact observability next we extend this to more general class of operators under a growth condition on the associated evolution family we give an application to the schrodinger equation with time dependent potential and the damped wave equation with a time dependent damping coefficient

control of three dimensional water waves we study the exact controllability for spatially periodic water waves with surface tension by localized exterior pressures applied to free surfaces we prove that in any dimension the exact controllability holds within an arbitrarily short time for sufficiently small and regular data provided that the region of control satisfies the geometric control condition this result was previously obtained by alazard et al j eur math soc 20 3 657 745 2018 for 2 d water waves our proof combines an iterative scheme that reduces the controllability of the original quasi linear equation to that of a sequence of linear equations with a semiclassical approach for the linear control problems

decay estimates of functions through singular extensions of vector valued laplace transforms abstract let x be a banach space and let f l r x whose laplace transform extends analytically to some region containing i r 0 possibly having a pole at the origin in this paper we give estimates of the decay of certain slight suitable modification of f in terms of the growth of its laplace transform along the imaginary axis this technique is applied to obtain decay estimates of smooth orbits of bounded c 0 semigroups whose infinitesimal generators have an arbitrary finite boundary spectrum these results are close to those given recently by c j k batty and t duyckaerts

scaling properties of a scattering system with an incomplete horseshoe we investigate a parameter dependent map describing a chaotic scattering system in parameter ranges leading to an incomplete horseshoe we construct an approximate symbolic dynamics which describes quite well the hyperbolic component of the invariant set its grammatical rules are correlated with the convergence properties of the thermodynamical formalism for the measures characterizing the invariant set

the topology of nested homoclinic and heteroclinic tangles abstract homoclinic and heteroclinic tangles are fundamental phase space structures that help in organizing the transport defined by a chaotic map on a two dimensional phase space previous work introduced the technique of homotopic lobe dynamics to study the topology of a homoclinic tangle the present work generalizes homotopic lobe dynamics to describe the topology of a network of nested homoclinic and or heteroclinic tangles using this theory we demonstrate that the topological analysis of a network of nested tangles can lead to an efficient symbolic description of a chaotic phase space even in the vicinity of islands of stability by increasing the amount of topological information used e g by introducing additional tangles at finer scales this technique suggests an efficient route for defining accurate symbolic dynamics for phase spaces that exhibit a mixture of chaos and regularity

resolvent conditions for the control of parabolic equations abstract since the seminal work of russell and weiss in 1994 resolvent conditions for various notions of admissibility observability and controllability and for various notions of linear evolution equations have been investigated intensively sometimes under the name of infinite dimensional hautus test this paper sets out resolvent conditions for null controllability in arbitrary time necessary for general semigroups sufficient for analytic normal semigroups for a positive self adjoint operator a it gives a sufficient condition for the null controllability of the semigroup generated by a which is only logarithmically stronger than the usual condition for the unitary group generated by ia this condition is sharp when the observation operator is bounded the proof combines the so called control transmutation method and a new version of the direct lebeau robbiano strategy the improvement of this strategy also yields interior null controllability of new logarithmic anomalous diffusions

entropy and bifurcations in a chaotic laser we compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection we consider the poincare return map to a fixed plane and are able to compute the stable and unstable manifolds of periodic points globally even though it is impossible to find a plane on which the poincare map is globally smoothly defined in this way we obtain the information that forms the input of the entropy calculations and characterize the boundary crisis in which the chaotic attractor is destroyed this boundary crisis involves a periodic point with negative eigenvalues and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation

nonperturbative treatment of singular potentials a nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of hamiltonian with a singular potential of the form h d2 dx2 x2 x in the domain 0 which obey dirichlet boundary conditons the formal structure of the algorithm is identical to that of the lanczos algorithm when it is formally extended to self adjoint operators

tauberian theorems and stability of one parameter semigroups the main result is the following stability theorem let t t t tzo be a bounded co semigroup on a reflexive space x denote by a the generator of t and by a a the spectrum of a if a a n ir is countable and no eigenvalue of a lies on the imaginary axis then limt boo t t x o for all x e x

on the observability of time discrete conservative linear systems abstract we consider various time discretization schemes of abstract conservative evolution equations of the form z a z where a is a skew adjoint operator we analyze the problem of observability through an operator b more precisely we assume that the pair a b is exactly observable for the continuous model and we derive uniform observability inequalities for suitable time discretization schemes within the class of conveniently filtered initial data the method we use is mainly based on the resolvent estimate given by burq and zworski in n burq m zworski geometric control in the presence of a black box j amer math soc 17 2 2004 443 471 electronic we present some applications of our results to time discrete schemes for wave schrodinger and kdv equations and fully discrete approximation schemes for wave equations

observability and stabilization of magnetic schrodinger equations abstract we are mainly interested in extending the known results on observability inequalities and stabilization for the schrodinger equation to the magnetic schrodinger equation that is in presence of a magnetic potential we establish observability inequalities and exponential stabilization by extending the usual multiplier method under the same geometric condition that needed for the schrodinger equation we also prove with the help of elliptic carleman inequalities logarithmic stabilization results through a resolvent estimate although the approach is classical these results on logarithmic stabilization seem to be new even for the schrodinger equation

new blow up rates for fast controls of schrodinger and heat equations abstract we consider the null controllability problem for the schrodinger and heat equations with boundary control we concentrate on short time or fast controls we improve recent estimates see l miller geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 202 226 l miller how violent are fast controls for schrodinger and plate vibrations arch ration mech anal 172 2004 429 456 l miller controllability cost of conservative systems resolvent condition and transmutation j funct anal 218 2005 425 444 l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 on the norm of the operator associating to any initial state the minimal norm control driving the system to zero our main results concern the schrodinger and heat equations in one space dimension they yield new estimates concerning window problems for series of exponentials as described in t i seidman the coefficient map for certain exponential sums nederl akad wetensch indag math 48 1986 463 478 and in t i seidman s a avdonin s a ivanov the window problem for series of complex exponentials j fourier anal appl 6 2000 233 254 these results are used following l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 to deal with the case of several space dimensions

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

eigenvalues of schrodinger operators with definite and indefinite weights in this paper computational algorithms are presented to compute the eigenvalues of schrodinger operators with definite and indefinite weights the algorithms are based on titchmarsh weyl s theory and can be applied to solve a wide class of problems in quantum mechanics explicit examples are presented some of which recover the known exact solutions finally some analytical bounds have been reviewed for definite and indefinite singular problems weyl s spectral bisection algorithm has been proposed to solve definite problems as algebraic problems open problems have been presented based on the results of the presented algorithms

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

guidelines for the construction of a generating partition in the standard map abstract the validity of a recently developed procedure to construct a symbolic dynamics for the standard map is accurately investigated the implications of all the stages that involve the determination of homoclinic tangencies and the identification of appropriate symmetry lines are discussed moreover an extension of the method from the strongly to the weakly chaotic regime is presented as a result the general validity of the approach is strengthened although many intermediate steps still have no rigorous justification

how violent are fast controls iii abstract for coupled systems of partial differential equations for which nullcontrol is possible in arbitrarily short time the standard blowup rate for the control cost is exponential in 1 t it is shown how to derive this rate for a variety of systems including the thermoelastic system with control restricted to a small patch in the domain and to a single component thermal displacement or velocity

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

hautus condition on the exact controllability of conservative systems we consider the exact controllability of a linear conservative system a b associated with hilbert spaces h and u we get a necessary and sufficient controllability condition this condition is related to the hautus lemma from the finite dimensional systems theory it is an estimate in terms of operators a and b alone

a degenerate bifurcation to chaotic scattering in a multicentre potential many scattering systems can be described as scattering of a point particle off a multicentre potential in this paper we present a two centre system which shows either regular or chaotic scattering depending on the kinetic energy i e the velocity of the incoming particle the transition points to chaotic scattering can be derived analytically by linearization of the poincare map at one of these transition velocities there is a degenerate bifurcation where the invariant set contains a parabolic surface and where the time delay statistics is algebraic with power sigma 2

on the exact controllability of hyperbolic magnetic schrodinger equations abstract in this paper we address the exact controllability problem for the hyperbolic magnetic schrodinger equation which plays an important role in the research of electromagnetics typical techniques such as hamiltonian induced hilbert spaces and pseudodifferential operators are introduced by choosing an appropriate multiplier we proved the observability inequality with sharp constants in particular a genuine compactness uniqueness argument is applied to obtain the minimal time in the final analysis a suitable boundary control is constructed by the systematic hilbert uniqueness method introduced by j l lions compared with the micro local discussion in bardos et al 1992 we do not require the coefficients belong to c actually c 1 is already sufficient for the vector potential of the hyperbolic electromagnetic equation

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

spectral conditions for admissibility and observability of wave systems applications to finite element schemes in this article we derive uniform admissibility and observability properties for the finite element space semi discretizations of ddot u a 0 u 0 where a 0 is an unbounded self adjoint positive definite operator with compact resolvent to address this problem we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems our approach provides very general admissibility and observability results for finite element approximation schemes of ddot u a 0 u 0 which stand in any dimension and for any regular mesh in the sense of finite elements our results can be combined with previous works to derive admissibility and observability properties for full discretizations of ddot u a 0 u 0 we also present applications of our results to controllability and stabilization problems

closed form sums for some perturbation series involving hypergeometric functions infinite series of the type are investigated closed form sums are obtained for a positive integer 1 2 3 the limiting case of b after y is replaced with x2 b leads to this type of series appears in the first order perturbation correction for the wavefunction of the generalized spiked harmonic oscillator hamiltonian these results have immediate applications to perturbation series for the energy and wavefunction of the spiked harmonic oscillator hamiltonian

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

simplified conditions of initial observability for infinite dimensional second order damped dynamical systems abstract in the paper the initial observability of abstract second order infinite dimensional dynamical systems is considered two cases of observation operator are explored first the observation operator is assumed to be bounded then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation using the frequency domain method it is proved that initial observability of second order system under discussion can be verified by the initial observability conditions for the corresponding simplified first order system general results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures some special cases are also discussed and practical comments and remarks are given the paper extends earlier results on initial observability for a class of infinite dimensional second order abstract dynamical systems

trellises formed by stable and unstable manifolds in the plane a trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diseomorphism of a 2 manifold this paper describes and classifies some trellises the set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold each homoclinic point is assigned a type number which is constant on its orbit combinatorial properties of trellises are studied using type numbers and the pair of linear orderings trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones introduction in chapter 33 of new methods of celestial mechanics poincare describes the figure formed by the stable and unstable manifolds of a hyperbolic fixed point of a transformation of the upper half plane as follows when we try to represent the figure formed by these two curves and their intersections each of which corresponds to a doubly asymptotic solution these intersections form a type of trellis tissue or grid with infinitely serrated mesh neither of the two curves must ever 6ut across itself again but must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times the complexity of this figure will be striking and i shall not even try to draw it nothing is more suitable for providing us with an idea of the complex nature of the three body problem and of all the problems of dynamics in general the aim of this paper is to understand figures of the type poincare described so well if p is a hyperbolic periodic point of a diffeomorphism of a 2 manifold then the figure formed by its stable and unstable manifolds will be called the trellis of p whenever these manifolds have nonempty intersection figure 1 illustrates the beginning development of such a trellis similarly the figure formed by the stable and unstable manifolds of a cycle of hyperbolic periodic points is called the trellis of the cycle a cycle of hyperbolic periodic points of a diffeomorphism is a finite collection p of periodic points having a cyclic permutation p p such that the unstable manifold of p intersects the stable manifold of p for each p g p it is natural to conjecture that in some cases ergodic zones are the closures of trellises anosov transformations of the torus provide examples where this is the case 1 one may also conjecture that strange attractors are sometimes the closures of the unstable manifolds of cycles of hyperbolic periodic points for certain values of the parameters a and b the henon map of the plane defined by t x y 1 y ax by may provide examples received by the editors may 7 1985 1980 mathematics subject classification primary 58f13 secondary 58f05 58f11 1986 american mathematical society 0002 9947 86 1 00 25 per page

nonsingular spiked harmonic oscillator a perturbative study of a class of nonsingular spiked harmonic oscillators defined by the hamiltonian h d2 dr 2 r 2 r in the domain 0 is carried out in the two extremes of a weak coupling and a strong coupling regimes a path has been found to connect both expansions for near 2

on stability equivalence between dynamic output feedback and static output feedback for a class of second order infinite dimensional systems we consider stabilization for a class of abstract second order infinite dimensional systems with collocated control and observation we show that the closed loop system under a proportional direct output feedback control is asymptotically stable if and only if the closed loop system under some dynamic output feedback control is asymptotically stable a hautus test is developed to ensure the asymptotic stability two types of dynamic output feedback controls are investigated the results are applied to some coupled wave heat equations where the heat system is considered as a controller of the wave system this study provides a different view in the study of the coupled systems described by partial differential equations

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

energy bounds for a class of singular potentials and some related series perturbation expansions up to third order for the generalized spiked harmonic oscillator hamiltonians h d2 dx2 x2 a x2 x a 0 2 1 1 2 1 4a and small values of the coupling 0 are developed upper and lower bounds for the eigenvalues are computed by the procedure of burrows et al 1987 j phys a math gen 20 889 97 for assessing the accuracy of a truncated perturbation expansion closed form sums for some related perturbation double infinite series then immediately follow as a result of this investigation

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

necessary conditions for the exact observability of systems on hilbert spaces abstract in this paper we introduce the zero class of admissible observation operators for which the norm of the output map tends to zero as the time tends to zero we show that for such an observation operator exact observability implies that the semigroup of the system is bounded from below in addition if the semigroup generator has no residual spectrum then the semigroup can be extended to a c 0 group

how do supersingular perturbations generate non taylor series we show that a suitable choice of the zero order problem can lead directly to the emergence of non taylor expansions in for the ground state energies of the hamiltonian family h 2 r2 r in regions 3 the discussion includes the role of dimension and the right order parameter the effect of choosing a more general potential of the form r r is also considered

locally distributed control for a model of fluid structure interaction we consider the equations modeling the coupled vibrations of a fluid solid system the control acts in a subset of a domain occupied by the fluid our main result asserts that we have exact controllability and exponential stabilizability provided that the support of the control contains a neighborhood of the solid and a neighborhood of the exterior boundary this improves the existing exact controllability results which require a control which is active in the whole fluid domain the proof is based on a frequency domain approach combined with the use of appropriate multipliers moreover we show that the strong stabilizability property holds for any open control region

on the cost of fast controls for some families of dispersive or parabolic equations in one space dimension in this paper we consider the cost of fast controls for a large class of linear equations of parabolic or dispersive type in one space dimension in small time by extending the work of tenenbaum and tucsnak j differential equations 243 2007 pp 70 100 we are able to give precise upper bounds on the time dependance of the cost of fast controls when the time of control t tends to 0 we also give a lower bound of the cost of fast controls for the same class of equations which proves the optimality of the power of t involved in the cost of the control these general results are then applied to treat notably the case of linear kdv equations and fractional heat or schrodinger equations

spiked oscillators exact solution a procedure to obtain the eigenenergies and eigenfunctions of a quantum spiked oscillator is presented the originality of the method lies in an adequate use of asymptotic expansions of wronskians of algebraic solutions of the schrodinger equation the procedure is applied to three familiar examples of spiked oscillators

eigenvalue bounds for a class of singular potentials we study smooth transformations of the solvable potentials eigenvalue approximation formulae are obtained which provide lower or upper energy bounds for all the discrete energy eigenvalues accordingly as the transformation functions g and f are both convex or both concave detailed results are presented for the special case of two term singular potentials of the form and also for the potentials and for

riccati equations and normalized coprime factorizations for strongly stabilizable infinite dimensional systems the first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic riccati equation for a class of infinite dimensional systems of the form a b s 1 2b d where a is dissipative and all the other operators are bounded these systems are not exponentially stabilizable and so the standard theory is not applicable the second part uses the riccati equation results to give formulas for normalized coprime factorizations over h for positive real transfer functions of the form d s 1 2b author a 1 b

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

stabilization and control for the nonlinear schrodinger equation on a compact surface in this paper we study the stabilization property and the exact controllability for the nonlinear schrodinger equation on a two dimensional compact riemannian manifold without boundary we use a pseudo differential dissipation the proofs are based on a result of propagation of singularities and on recent dispersion estimates strichartz type inequalities due to n burq p gerard and n tzvetkov

exact observability square functions and spectral theory in the first part of this article we introduce the notion of a backward forward conditioning bfc system that generalises the notion of zero class admissibility introduced in xu et al 2008 30 we can show that unless the spectrum contains a half plane the bfc property occurs only in situations where the underlying semigroup extends to a group in a second part we present a sufficient condition for exact and final state observability in banach spaces that is designed for infinite dimensional output spaces and general strongly continuous semigroups to obtain this we make use of certain weighted square function estimates specialising to the hilbert space situation we obtain a result for contraction semigroups without an analyticity condition on the semigroup

transport rates of a class of two dimensional maps and flows abstract a method is developed for estimating the transport rates of phase space areas for a class of two dimensional diffeomorphisms and flows the class of diffeomorphisms we considered are defined by the topological structure of their stable and unstable manifolds and hence are universal we show how to estimate the transport rates for a class of diffeomorphisms found by easton and for an extension of this class of diffeomorphisms which is found via a perturbation of the topology of the stable and unstable manifolds this is done by introducing symbolic dynamics and transfer matrices which in turn relate transport phenomena in phase space to markov processes in a precise manner in addition to the transport rates we use the transfer matrices to obtain estimates for the topological entropy averaged stretching rates and the elongation rate of the unstable manifold the flows we consider are two dimensional time periodic flows which can be reduced via poincare section to the extended family of maps we develop an analytical method based on chirikov s whisker map to classify a given flow according to the structure of its manifolds in its poincare section this allows the techniques developed here for maps to be directly applied to time periodic flows

escape function conditions for the observation control and stabilization of the wave equation for the linear wave equation with time invariant coefficients on a connected compact riemannian manifold omega g with c 3 boundary the geodesics condition of bardos lebeau and rauch siam j control optim 30 1992 pp 1024 1065 is characterized in terms of escape functions which are some lyapunov functions on the phase space s bar omega the unit sphere cotangent bundle differentiable escape functions yield a sufficient condition which is slightly less sharp but does not refer to geodesics the escape function condition yields a straightforward geometric proof that the geodesics condition holds in the situations where first order differential multiplier methods apply using microlocal control results it allows us to generalize some control results that were obtained by multiplier methods to variable coefficients and lower order terms it also allows us to prove in some class of simple situations e g in mathbb r 2 with constant coefficients that no first order differential multiplier method can reach the optimal control time or control regions

observability of polynomially stable systems abstract for finite dimensional systems the hautus test is a well known and easily checkable condition for observability russell and weiss a general necessary condition for exact observability siam j control optim 32 1994 1 23 suggested an infinite dimensional version of the hautus test which is necessary for exact infinite time observability and sufficient for approximate infinite time observability of exponentially stable systems in this paper the notion of observability is studied for polynomially stable systems several known results for exponentially stable systems are extended to the setting of polynomially stable systems by means of an example the obtained results are illustrated

locally distributed control and damping for the conservative systems in this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control this enables us to establish a frequency domain characterization for the exact controllability uniform exponential decay property of second order elastic systems such as the wave equation and the petrovsky equation with locally distributed control damping a piecewise multiplier method for frequency domain is introduced for several classes of pdes on regions which are not necessarily smooth we obtain a sufficient condition for the subregion on which the application of control damping will yield the exact controllability uniform exponential decay property this result provides useful information for designing the location of controllers dampers for distributed systems with a law of conservation

on power bounded operators abstract the main result asserts that for any contraction t on an arbitrary banach space x t n t n 1 0 as n if and only if the spectrum of t has no points on the unit circle except perhaps z 1 this theorem is extended for t t n where is a function of spectral synthesis on the unit circle as an application we generalize the so called zero two law of ornstein and sucheston and zaharopol to positive contraction on a very large class of banach lattices

the minimal time function in infinite dimensions the goal of this paper is to study regularity properties of the minimal time function t cdot associated with linear control systems in infinite dimensions this paper proves that various regularity properties of t cdot hold on the whole reachable set provided they hold around the target some methods to obtain estimates on t cdot around the target are given and applications to both distributed and boundary control problems are presented

exponential stabilization of well posed systems by colocated feedback we consider well posed linear systems whose state trajectories satisfy dot x ax bu where u is the input and a is an essentially skew adjoint and dissipative operator on the hilbert space x this means that the domains of a and a are equal and a a q where q geq 0 is bounded on x the control operator b is possibly unbounded but admissible and the observation operator of the system is b such a description fits many wave and beam equations with colocated sensors and actuators and it has been shown for many particular cases that the feedback u ka y v with ka 0 stabilizes the system strongly or even exponentially here y is the output of the system and v is the new input we show by means of a counterexample that if b is sufficiently unbounded then such a feedback may be unsuitable the closed loop semigroup may even grow exponentially our counterexample is a simple regular system with feedthrough operator zero however we prove that if the original system is exactly controllable and observable and if ka is sufficiently small then the closed loop system is exponentially stable

non structural controllability of linear elastic systems with structural damping this paper proves that any initial condition in the energy space for the plate equation with square root damping 2 u on a smooth bounded domain with hinged boundary conditions 0 can be steered to zero by a square integrable input function u supported in arbitrarily small time interval 0 t and subdomain as t tends to zero for initial states with unit energy norm the norm of this u grows at most like exp cp tp for any real p 1 and some cp 0 indeed this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non structural controls satisfying a spectral observability condition moreover under some geometric optics condition on the subdomain allowing to apply the control transmutation method this estimate is improved into p 1 and the dependence of cp on the subdomain is made explicit these results are analogous to the optimal ones known for the heat flow

almost periodic solutions of first and second order cauchy problems abstract almost periodicity of solutions of first and second order cauchy problems on the real line is proved under the assumption that the imaginary resp real spectrum of the underlying operator is countable related results have been obtained by ruess v and basit our proof uses a new idea it is based on a factorisation method which also gives a short proof of the vector valued version of loomis classical theorem saying that a bounded uniformly continuous function from r into a banach spacexwith countable spectrum is almost periodic ifc0 x our method can also be used for solutions on the half line this is done in a separate paper

partitioning two dimensional mixed phase spaces abstract hamiltonian systems typically exhibit a mixture of chaos and regularity complicating any scheme to partition phase space and extract a symbolic description of the dynamics in particular the dynamics in the vicinity of stable islands can exhibit complicated topology that is qualitatively distinct from that away from the islands we develop an approach to partition the chaotic phase space of a general dynamical system represented by a two dimensional map homeomorphism this approach can accommodate mixed phase space structure with an arbitrarily high degree of accuracy the partitioning scheme is built around networks of nested heteroclinic tangles fundamental geometric objects that organize phase space transport these tangles can be used to progressively approximate the dynamics in the vicinity of stable island chains the net result is a symbolic approximation to the dynamics in the chaotic sea and an associated phase space partition which includes the influence of stable islands and which is approximately markov

what symbolic dynamics do we get with a misplaced partition on the validity of threshold crossings analysis of chaotic time series abstract an increasingly popular method of encoding chaotic time series from physical experiments is the so called threshold crossings technique where one simply replaces the real valued data with symbolic data of relative positions to an arbitrary partition at discrete times the implication has been that this symbolic encoding describes the original dynamical system on the other hand the literature on generating partitions of non hyperbolic dynamical systems has shown that a good partition is non trivial to find it is believed that the generating partition of non uniformly hyperbolic dynamical system connects primary tangencies which are generally not simple lines as used by a threshold crossings therefore we investigate consequences of using itineraries generated by a non generating partition we do most of our rigorous analysis using the tent map as a benchmark example but show numerically that our results likely generalize in summary we find the misrepresentation of the dynamical system by sample path symbolic dynamics of an arbitrary partition can be severe including sometimes extremely diminished topological entropy and a high degree of non uniqueness interestingly we find topological entropy as a function of misplacement to be devil s staircase like but surprisingly non monotone

a spectral approach for the exact observability of infinite dimensional systems with skew adjoint generator abstract let a be a possibly unbounded skew adjoint operator on the hilbert space x with compact resolvent let c be a bounded operator from d a to another hilbert space y we consider the system governed by the state equation z t az t with the output y t cz t we characterize the exact observability of this system only in terms of c and of the spectral elements of the operator a the starting point in the proof of this result is a hautus type test recently obtained in burq and zworski j amer soc 17 2004 443 471 and miller j funct anal 218 2 2005 425 444 we then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain the schrodinger equation the bernoulli euler plate equation and the wave equation in a square are considered for the plate and schrodinger equations the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary this is done by combining our spectral observability test to a theorem of beurling on nonharmonic fourier series and to a new number theoretic result on shifted squares

generating partition for the standard map a procedure to obtain the symbolic dynamics for conservative dynamical systems is introduced with reference to the standard map in a strongly chaotic regime the method extends an approach previously developed for highly dissipative systems it is based on the construction of a generating partition from homoclinic tangencies and fibers of invariant manifolds it is found that some arbitrariness in the construction of the partition is unavoidable

admissibility and exact observability of observation operators for micro beam model time and frequency domain approaches this study focuses on the exact observability of a non classical euler bernoulli micro beam equation this non classical model was derived based on the strain gradient elasticity theory which is intended to explain the phenomenon of size effect at the micron scale spectral properties of the corresponding state operator are studied an asymptotic expression for eigenvalues is calculated and eigenfunctions are analyzed in order to check the necessary conditions for the exact observability of the system by examining the eigenfunctions it is shown that among non collocated boundary outputs only measurement of the non classical moment at the root of the beam yields an admissible observation operator and also defines an exactly observable system an alternative proof based on the multiplier method which is commonly employed in the literature on the observability and controllability of infinite dimensional dynamical systems is presented to provide a comparison between the time and frequency domain approaches

internal null controllability of a linear schrodinger kdv system on a bounded interval abstract the control of a linear dispersive system coupling a schrodinger and a linear korteweg de vries equation is studied in this paper the system can be viewed as three coupled real valued equations by taking real and imaginary parts in the schrodinger equation the internal null controllability is proven by using either one complex valued control on the schrodinger equation or two real valued controls one on each equation notice that the single schrodinger equation is not known to be controllable with a real valued control the standard duality method is used to reduce the controllability property to an observability inequality which is obtained by means of a carleman estimates approach