simultaneous exact controllability and some applications we study the exact controllability of two systems by means of a common finite dimensional input function a property called simultaneous exact controllability most of the time we consider one system to be infinite dimensional and the other finite dimensional in this case we show that if both systems are exactly controllable in time t 0 and the generators have no common eigenvalues then they are simultaneously exactly controllable in any time t t0 moreover we show that similar results hold for approximate controllability for exactly controllable systems we characterize the reachable subspaces corresponding to input functions of class h1 and h2 we apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end finally we consider an example of two infinite dimensional systems we characterize the simultaneously reachable subspace for two strings controlled from a common end the result is obtained using a recent generalization of a classical inequality of ingham

averaged controllability of parameter dependent conservative semigroups abstract we consider the problem of averaged controllability for parameter depending either in a discrete or continuous fashion control systems the aim being to find a control independent of the unknown parameters so that the average of the states is controlled we do it in the context of conservative models both in an abstract setting and also analysing the specific examples of the wave and schrodinger equations our first result is of perturbative nature assuming the averaging probability measure to be a small parameter dependent perturbation in a sense that we make precise of an atomic measure given by a dirac mass corresponding to a specific realisation of the system we show that the averaged controllability property is achieved whenever the system corresponding to the support of the dirac is controllable similar tools can be employed to obtain averaged versions of the so called ingham inequalities particular attention is devoted to the 1d wave equation in which the time periodicity of solutions can be exploited to obtain more precise results provided the parameters involved satisfy diophantine conditions ensuring the lack of resonances

analytic smoothing for equipercentile equating under the common item nonequivalent populations design a cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method degree of smoothing these include a asymptotic significance tests that compare no equating and linear equating to equipercentile equating b a scheme for estimating total equating error and for dividing total estimated error into systematic and random components the smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test

a smooth nonparametric estimator of a quantile function abstract a smooth alternative to the conventional sample quantile function as a nonparametric estimator of a population quantile function is proposed the proposed estimator is essentially a kernel type of estimator and has the same asymptotic distribution as the conventional sample quantile the mean squared convergence rate of the proposed estimator is also estimated monte carlo studies are conducted to compare the proposed estimator with the sample quantile and the estimator proposed by kaigh and lachenbruch the feasibility of using bootstrap techniques to estimate the optimal window width for the proposed estimator is also considered

ensemble control of bloch equations in this article we study a class of control problems which involves controlling a continuum of dynamical systems with different values of parameters characterizing the system dynamics by using the same control signal we call such problems control of ensembles the motivation for looking into these problems comes from the manipulation of an ensemble of nuclear spins in nuclear magnetic resonance nmr spectroscopy and imaging mri with dispersions in natural frequencies and the strengths of the applied radio frequency rf field from the standpoint of mathematical control theory the challenge is to simultaneously steer a continuum of systems between points of interest with the same control input this raises some new and unexplored questions about controllability of such systems we show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the noncommuting vector fields that govern the system dynamics a systematic study of these systems has immediate applications to broad areas of control of ensembles of quantum systems as arising in coherent spectroscopy and quantum information processing the new mathematical structures appearing in such problems are excellent motivation for new developments in control theory

a variational characterization and geometric integration for bertrand curves in this paper we introduce a class of functionals in the three dimensional semi euclidean space rq3 having an energy density that depends only on curvature and whose moduli space of trajectories consists of lw curves i e curves with curvature and torsion for which there are three real constants such that with 2 2 0 this family of curves includes plane curves helices curves of constant curvature curves of constant torsion lancret curves also called generalized helices and bertrand curves we present an algorithm to construct bertrand curves in rq3 by using an arclength parametrized curve in a totally umbilical surface s2 s12 or h2 and prove that every bertrand curve in rq3 can be obtained in this way a second algorithm is presented for the construction of lw curves by using a curve of constant slope in the ruled surface s whose directrix is a certain curve with non zero curvature and whose rulings are generated by its modified darboux vector field

observation and control of vibrations in tree shaped networks of strings in this paper we study the controllability problem for a system that models the vibrations of a controlled tree shaped network of vibrating elastic strings the control acts through one of the exterior nodes of the network with the help of the dalembert representation formula for the solutions of the one dimensional wave equation we find certain linear relations between the traces of the solutions at the nodes of the network these relations allow us to prove a weighted observability inequality with weights that may be explicitly computed in terms of the eigenvalues of the associated elliptic problem we characterize the class of trees for which all those weights are different from zero which leads to the spectral controllability of the system additionally we consider the same one node control problem for several networks that are controlled simultaneously

boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass we consider a hybrid system consisting of two flexible beams connected by a point mass the constant of rotational inertia is assumed to be nonzero in a previous paper we have proved that in the presence of the point mass the system is well posed in asymmetric spaces in which solutions have one more degree of regularity to one side of the mass we are interested in the problem of controllability when the control acts on the free extreme of one of the beams we prove that when the control time is large enough the system is exactly controllable in an asymmetric space this result is sharp the proofs combine classical techniques from asymptotic analysis and the theory of nonharmonic fourier series

bootstrapping regression models the regression and correlation models are considered it is shown that the bootstrap approximation to the distribution of the least squares estimates is valid and some error bounds are given

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets abstract we consider a restricted dirichlet to neumann map s r t associated with the operator t 2 g a q where g is the laplace beltrami operator of a riemannian manifold m g and a and q are a vector field and a function on m the restriction s r t corresponds to the case where the dirichlet traces are supported on 0 t s and the neumann traces are restricted on 0 t r here s and r are open sets which may be disjoint on the boundary of m we show that s r t determines uniquely up the natural gauge invariance the lower order terms a and q in a neighborhood of the set r assuming that r is strictly convex and that the wave equation is exactly controllable from s in time t 2 we give also a global result under a convex foliation condition the main novelty is the recovery of a and q when the sets r and s are disjoint we allow a and q to be non self adjoint and in particular the corresponding physical system may have dissipation of energy

the dirichlet neumann boundary control problem associated with maxwell s equations in a cylindrical region in a cylindrical region we consider electromagnetic fields independent of the axial coordinate controlling the time evolution of such fields by means of boundary currents likewise independent of the axial direction is equivalent to controlling simultaneously two wave equations one with boundary control of dirichlet type the other of neumann type in this paper we provide a preliminary study of control problems of this type and indicate what is necessary for extensions of our work

discrete ingham type inequalities and simultaneous observability of strings or beams abstract harmonic analysis has been an efficient tool in control theory for a long time see e g russell d l russell controllability and stabilizability theory for linear partial differential equations recent progress and open questions siam rev 20 1978 639 739 and its numerous references here we establish discrete ingham type and haraux type inequalities for exponential sums satisfying a weakened gap condition they enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams

output controllability in a long time horizon abstract in this article we consider a linear finite dimensional system our aim is to design a control such that the output of the system reach a given target at a final time t 0 this notion is known as output controllability we extend this notion to the one of long time output controllability more precisely we consider the question is it possible to steer the output of the system to some prescribed value in time t 0 and then keep the output of the system at this prescribed value for all times t t we provide a necessary and sufficient condition for this property to hold once the condition is satisfied one can apply a feedback control that keeps the average fixed during a given time period we also address the l 2 norm optimality of such controls we apply our results to long time averaged control problems

optimal control of averaged state of a parabolic equation with missing boundary condition we consider the optimal control of general heat governed by an operator depending on an unknown parameter and with missing boundary condition using the notion of no regret and low regret control

nonparametric estimates of standard error the jackknife the bootstrap and other methods summary we discuss several nonparametric methods for attaching a standard error to a point estimate the jackknife the bootstrap half sampling subsampling balanced repeated replications the infinitesimal jackknife influence function techniques and the delta method the last three methods are shown to be identical all the methods derive from the same basic idea which is also the idea underlying the common parametric methods extended numerical comparisons are made for the special case of the correlation coefficient

on controllability of a rotating string we consider the problem of controllability of a rotating string we say that a set of initial data of the string is controllable if for any initial data of this set by suitable manipulation of the exterior force the string goes to rest the main result of the paper is a description of controllable sets of initial data the equation is not strongly hyperbolic to get exact controllability of such equation in the sharp time interval we use control functions from sobolev spaces with noninteger indices to prove our results we apply the method of moments that has been widely used in control theory of distributed parameter systems since the classical papers of h o fattorini and d l russell we use recent results about exponential bases in sobolev spaces with noninteger indices

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

on controllability of a non homogeneous elastic string with memory abstract we are motivated by the problem of control for a non homogeneous elastic string with memory we reduce the problem of controllability to a non standard moment problem the solution of the latter problem is based on an auxiliary riesz basis property result for a family of functions quadratically close to the nonharmonic exponentials this result requires the detailed analysis of an integro differential equation and is of interest in itself for function theory controllability of the string implies observability of a dual system

controllability of the linear one dimensional wave equation with inner moving forces we address the null inner controllability of the one dimensional wave equation with supports of the controls that may vary with respect to the time variable we consider theoretical and numerical aspects assuming a geometric optic condition on the support of the controls we first prove a generalized observability inequality for the homogeneous wave equation implying the controllability we then introduce and prove the well posedness of a mixed formulation that characterizes the controls of minimal square integrable norm a numerical approximation of this formulation based on a conformal space time finite element method is then presented several numerical experiments are finally reported

a spillover phenomenon in the optimal location of actuators in this paper we are interested in finding the optimal location and shape of the actuators in a stabilization problem namely we consider the one dimensional wave equation damped by an internal feedback supported on a subdomain omega of given length the criterion we want to optimize represents the rate of decay of the total energy of the system it theoretically involves all the eigenmodes of the operator from an engineering point of view it seems more realistic to consider only a finite number of modes say the n first ones in that context we are able to prove existence and uniqueness of an optimal domain omega n it is the better possible location for the actuators we characterize this optimal domain and we point out the following strange phenomenon at least for small lengths the optimal domain omega n which is the better one for the n first modes is actually the worse one for the n 1th mode this looks like the well known spillover phenomenon in control theory at last we will give some possible extension and open problems in higher dimension

averaged control we analyze the problem of controlling parameter dependent systems we introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter first we consider the problem of controllability for linear finite dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition in the spirit of the classical rank condition for linear control systems but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action we also describe some open problems and directions of possible research in particular on the average controllability of evolution partial differential equations in this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system

stabilization of the wave equation on 1 d networks in this paper we study the stabilization of the wave equation on general 1 d networks for that we transfer known observability results in the context of control problems of conservative systems see r dager and e zuazua wave propagation observation and control in 1 d flexible multi structures math appl 50 springer verlag berlin 2006 into a weighted observability estimate for dissipative systems then we use an interpolation inequality similar to the one proved in p begout and f soria j differential equations 240 2007 pp 324 356 to obtain the explicit decay estimates of the energy for smooth initial data the obtained decay rate depends on the geometric and topological properties of the network we also give some examples of particular networks in which our results apply yielding different decay rates

on the semigroup property for some structured iterations abstract nonlinear matrix equations play a crucial role in science and engineering problems however solutions of nonlinear matrix equations cannot in general be given analytically one standard way of solving nonlinear matrix equations is to apply the fixed point iteration with usually only the linear convergence rate to advance the existing methods we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order we realize our way by starting with examples of solving the scalar equations and also connect this method with some well known equations including but not limited to the stein matrix equation the generalized eigenvalue problem the generalized nonlinear matrix equation the discrete time algebraic riccati equations to express the capacity of this method

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

ensemble control of finite dimensional time varying linear systems in this article we investigate the problem of simultaneously steering an uncountable family of finite dimensional time varying linear systems with the same control signal this class of control problems motivates further research in the new subject of control theory called ensemble control a notion coming from the study of complex spin dynamics in nuclear magnetic resonance spectroscopy and imaging we derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for an ensemble of finite dimensional time varying linear systems we show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics in addition the problem of optimal ensemble control of harmonic oscillators is studied to demonstrate the controllability results we show that the optimal solutions are pertinent to the study of time frequency limited signals and prolate spheroidal wave functions a systematic study of ensemble control systems has immediate applications to dynamical systems with parameter uncertainty as well as to wide ranging areas such as neuroscience and quantum control the work in ensemble control will foster further developments in mathematical control and systems theory

simultaneous control problems for systems of elastic strings and beams abstract controllability problems for n strings or beams controlled from a common endpoint are studied we give necessary and sufficient conditions for simultaneous spectral and approximate controllability and describe the space of simultaneously reachable states as a function of the position of the endpoint for each type of controllability result the sharp controllability time is obtained the proofs are based on new results in nonharmonic fourier series which describe riesz bases of exponential divided differences

stable observation of additive superpositions of partial differential equations abstract we address the problem of observing the solutions of a given partial differential equation pde submitted to unknown additive perturbations fulfilling another pde possibly of different nature the problem under consideration is motivated by the issue of averaged control and observation in the latter one aims at determining the initial datum of a dynamics submitted to unknown perturbations by means of measurements done on the average of solutions with respect to the unknown parameters we do it in two different cases first we assume that a given dynamics is perturbed in an additive manner by a state governed by a pde but not necessarily of the same type in particular we consider the additive superposition of multi physics solutions of wave and heat type for instance assuming the pdes under consideration have constant coefficients we prove that the by now well known observability properties of pde mainly heat and wave equations from open subsets of the domain where the dynamics evolves are stable under these additive perturbations the proof is rather systematic and easy to apply it consists in composing all but one the observable one pde operators so to reduce the problem to the consideration of a single equation and applying its known observability properties the same proof leads to very interesting open problems for systems involving pdes with variable coefficients where commutators lead to non local lower order perturbations we then consider a particular case of a pde depending on a parameter in a continuous manner we show that the average is governed by the superposition of two distinct solutions corresponding to the extremal values of the parameter and experiencing an extra regularizing effect this allows showing that the averaged observability property holds for the continuous average but to the price of losing a finite number of derivatives this opens a wide range of interesting problems to be explored

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

dirichlet series and simultaneous observability two problems solved by the same approach abstract in a classical paper ingham gave a simple proof of an important theorem of polya on singular points of dirichlet series under a uniform gap assumption on the exponents bernstein generalized polya s theorem by weakening this gap condition we give a simpler proof of bernstein s theorem by applying a recent generalization of ingham s theorem furthermore we also solve a simultaneous observability problem by using this theory

wigner measures and observability for the schrodinger equation on the disk we analyse the structure of semiclassical and microlocal wigner measures for solutions to the linear schrodinger equation on the disk with dirichlet boundary conditions our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk we prove a structure theorem expressing the restriction of the wigner measures on each invariant torus in terms of second microlocal measures they are obtained by performing a finer localization in phase space around each of these tori at the limit of the uncertainty principle and are shown to propagate according to heisenberg equations on the circle our construction yields as corollaries a that the disintegration of the wigner measures is absolutely continuous in the angular variable which is an expression of the dispersive properties of the equation b an observability inequality saying that the l 2 norm of a solution on any open subset intersecting the boundary resp the l 2 norm of the neumann trace on any nonempty open set of the boundary controls its full l 2 norm resp h 1 norm these results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary

a note on stabilization of locally damped wave equations with time delay abstract in this paper we consider the wave equation with internal distributed time delay and local damping in a bounded and smooth domain r n when the local damping acts on a neighborhood of a suitable part of the boundary of we show that an exponential stability result holds if the coefficient of the delay term is sufficiently small

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

perturbation of frame sequences and its applications to shift invariant spaces abstract we generalize the main result in o christensen h o kim r y kim j k lim perturbation of frame sequences in shift invariant spaces j geom anal 15 2005 181 191 in order to make it comparable with existing results then we compare the special cases of the three results in the literature in the setting of the perturbation of the generating sets of finitely generated shift invariant spaces of l 2 r d

observability and control of schrodinger equations we propose an exact controllability result for schrodinger equations in bounded domains under the bardos lebeau rauch geometric control condition with an estimate of the control which is explicit with respect to the time of controllability also we prove an explicit in time logarithmic observability estimate for the schrodinger equation where no geometrical conditions are supposed on the domain

helices for mathematical modelling of proteins nucleic acids and polymers abstract helices occur in modelling a range of structures including ropes fibers polymers and biopolymers in a recent paper with thamwattana and hill the authors derived euler lagrange equations for modelling protein structure where the energy being minimised was assumed to depend only on the curvature and torsion of the protein backbone space curve and their first derivatives such a model is applicable to helices occurring in other scenarios and in this article the author considers more generally which energies will yield helices as solutions of their corresponding euler lagrange equations he finds in particular classes of energy for which all circular helices are solutions and an energy depending on curvature and its derivative which generates conical helices as solutions of the euler lagrange equations also included are some new results for energies depending only on curvature extending previous investigations by feoli et al a feoli v v nesterenko g scarpetta functionals linear in curvature and statistics of helical proteins nuclear phys b 705 2005 577 592

remarks on the asymptotic behavior of the solution to damped wave equations abstract we study the diffusion phenomena for damped wave equations we prove that the solution of an abstract damped wave equation becomes closer to the solution of a heat type equation as time tends to infinity under some assumption to the resolvent as an application of our approach we also study the asymptotic behavior of the damped wave equation in euclidean space under the geometric control condition

asymptotic stability of the wave equation on compact manifolds and locally distributed damping a sharp result let m g be a n dimensional n geqq 2 compact riemannian manifold with boundary where g denotes a riemannian metric of class c this paper is concerned with the study of the wave equation on m g with locally distributed damping described by left begin array l u tt delta bf g u a x g u t 0 quad hbox on thinspace m times left 0 infty right u 0 hbox on partial m times left 0 infty right end array right where m represents the boundary of m and a x g ut is the damping term the main goal of the present manuscript is to generalize our previous result in cavalcanti et al trans ams 361 9 4561 4580 2009 treating the conjecture in a more general setting and extending the result for n dimensional compact riemannian manifolds m g with boundary in two ways i by reducing arbitrarily the region m ast subset m where the dissipative effect lies this gives us a totally sharp result with respect to the boundary measure and interior measure where the damping is effective ii by controlling the existence of subsets on the manifold that can be left without any dissipative mechanism namely a precise part of radially symmetric subsets an analogous result holds for compact riemannian manifolds without boundary

explicit observability inequalities for the wave equation with lower order terms by means of carleman inequalities in this paper by means of carleman estimates and the usual energy estimate we obtain directly two observability inequalities for the linear wave equation with time variant nonsmooth lower order terms we do not need any unique continuation property of the linear equation a priori since this is actually one of the by products of our analysis furthermore the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation also we apply our observability estimates to exact controllability for wave equations

null controllability of a system of linear thermoelasticity we consider a linear system of thermoelasticity in a compact c infin n dimensional connected riemannian manifold this system consists of a wave equation coupled to a heat equation when the boundary of the manifold is non empty dirichlet boundary conditions are considered we study the controllability properties of this system when the control acts in the hyperbolic equation and not in the parabolic one and has its support restricted to an open subset of the manifold we show that if the control time and the support of the control satisfy the geometric control condition for the wave equation this system of thermoelasticity is null controllable more precisely any finite energy solution can be driven to zero at the control time an analogous result is proved when the control acts on the parabolic equation finally when the manifold has no boundary the null controllability of the linear system of three dimensional thermoelastic ity is proved

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

fast and strongly localized observation for the schrodinger equation we study the exact observability of systems governed by the schrodinger equation in a rectangle with homogeneous dirichlet respectively neumann boundary conditions and with neumann respectively dirichlet boundary observation generalizing previous results of ramdani takahashi tenenbaum and tucsnak we prove that these systems are exactly observable in arbitrarily small time moreover we show that the above results hold even if the observation regions have arbitrarily small measures more precisely we prove that in the case of homogeneous neumann boundary conditions with dirichlet boundary observation the exact observability property holds for every observation region whith non empty interior in the case of homogeneous dirichlet boundary conditions with neumann boundary observation we show that the exact observability property holds if and only if the observation region has an open intersection with an edge of each direction moreover we give explicit estimates for the blow up rate of the observability constants as the time and or the size of the observation region tend to zero the main ingredients of the proofs are an effective version of a theorem of beurling and kahane on non harmonic fourier series and an estimate for the number of lattice points in the neighbourhood of an ellipse

sharp observability estimates for heat equations the goal of this article is to derive new estimates for the cost of observability of heat equations we have developed a new method allowing one to show that when the corresponding wave equation is observable the heat equation is also observable this method allows one to describe the explicit dependence of the observability constant on the geometry of the problem the domain in which the heat process evolves and the observation subdomain we show that our estimate is sharp in some cases particularly in one space dimension and in the multi dimensional radially symmetric case our result extends those in fattorini and russell arch rational mech anal 43 272 292 1971 to the multi dimensional setting and improves those available in the literature namely those by miller j differ equ 204 1 202 226 2004 siam j control optim 45 2 762 772 2006 atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 17 4 351 366 2006 and tenenbaum and tucsnak j differ equ 243 1 70 100 2007 our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation in contrast to the standard application of transmutation methods which uses a reverse representation of the heat solution in terms of the wave one we shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side we will also comment on the applications of our techniques to controllability properties of heat type equations