instability results for the damped wave equation in unbounded domains abstract we extend some previous results for the damped wave equation in bounded domains in r d to the unbounded case in particular we show that if the damping term is of the form a with bounded a taking on negative values on a set of positive measure then there will always exist unbounded solutions for sufficiently large positive in order to prove these results we generalize some existing results on the asymptotic behaviour of eigencurves of one parameter families of schrodinger operators to the unbounded case which we believe to be of interest in their own right

exponential stability of variable coefficients rayleigh beams under boundary feedback controls a riesz basis approach in this paper we study the boundary stabilizing feedback control problem of rayleigh beams that have non homogeneous spatial parameters we show that no matter how non homogeneous the rayleigh beam is as long as it has positive mass density stiffness and mass moment of inertia it can always be exponentially stabilized when the control parameters are properly chosen the main steps are a detail asymptotic analysis of the spectrum of the system and the proving of that the generalized eigenfunctions of the feedback control system form a riesz basis in the state hilbert space as a by product a conjecture in guo j optim theory appl 112 3 2002 529 is answered

convergence analysis for gmsfem approximation of elliptic eigenvalue problems abstract in this paper we analyze the approximation of elliptic eigenvalue problems using generalized multiscale finite element method gmsfem and get error estimates for eigenfunctions and eigenvalues for the case of simple eigenvalues the approximation errors for eigenfunctions are considered in both energy error and l 2 error the derived error estimates clearly give the relation between the errors and the coarse mesh size local multiscale enrichment and the corresponding eigenvalues the convergence analysis shows that the approximation of eigenvalue problems using gmsfem does not depend on the contrastness of the coefficient when the diffusion coefficient is highly heterogeneous a few numerical examples are presented to illustrate the performance of the gmsfem approximation and the theoretic analysis for the eigenvalue problems

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

exponential stability of nondissipative linear system in banach space and application to unbounded bilinear systems abstract we consider linear control systems of the form d z t d t a z t b z t where is a real parameter a is the infinitesimal generator of a linear c 0 semigroup of contractions s 0 t on a banach space x and the linear perturbation b is a relatively bounded operator with respect to a we provide necessary and sufficient conditions for exponential stability of the above system our stability results are then applied to establish uniform exponential stabilization of linear and bilinear systems with unbounded control operators applications to hyperbolic like equations are considered

the damped string problem revisited abstract we revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it employing methods familiar from supersymmetric quantum mechanics we also derive completeness and riesz basis results with parentheses for the associated root functions under less smoothness assumptions on the coefficients than usual using operator theoretic methods rather than detailed eigenvalue and root function asymptotics only

a multigrid method for eigenvalue problem a multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem in this scheme solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space besides the multigrid scheme all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems the total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem therefore this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving some numerical experiments are presented to validate the efficiency of the method

weighted l2 estimates for dissipative wave equations with variable coefficients abstract we establish weighted l 2 estimates for the wave equation with variable damping u t t u a u t 0 in r n where a x a 0 1 x with a 0 0 and 0 1 in particular we show that the energy of solutions decays at a polynomial rate t n 2 1 if a x a 0 x for large x we derive these results by strengthening significantly the multiplier method this approach can be adapted to other hyperbolic equations with damping

energy decay rate of wave equations with indefinite damping abstract we consider the one dimensional wave equation with an indefinite sign damping and a zero order potential term using a shooting method we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients in addition if the damping coefficient is more positive than negative we prove that the energy of system decays uniformly exponentially to zero this sharp result generalizes a previous work of freitas and zuazua 1996

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

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

exponential stability of an abstract nondissipative linear system in this paper we consider an abstract linear system with perturbation of the form frac dy dt ay varepsilon by on a hilbert space cal h where a is skew adjoint b is bounded and varepsilon is a positive parameter motivated by a work of freitas and zuazua on the one dimensional wave equation with indefinite viscous damping p freitas and e zuazua j differential equations 132 1996 pp 338 352 we obtain a sufficient condition for exponential stability of the above system when b is not a dissipative operator we also obtain a hautus type criterion for exact controllability of system a g where g is a bounded linear operator from another hilbert space to cal h our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping as well as the exponential stabilization of the elastic systems with noncolocated observation and control

on linear and nonlinear integro differential equations abstract the decomposition method adomian nonlinear stochastic operator equations academic press new york in press stochastic systems academic press new york 1983 is shown to be applicable to integro differential operator equations

local and parallel finite element discretizations for eigenvalue problems based on the work of xu and zhou math comp 69 2000 pp 881 909 this paper combines the local defect correction technique and the shifted inverse power method to establish new local and parallel finite element three scale schemes for a class of eigenvalue problems it is proved that with these schemes the solution of an eigenvalue problem on a fine grid pi h is reduced to the solution of an eigenvalue problem on a much coarser grid pi h the solution of a linear algebraic system on a globally mesoscopic grid pi w and the solutions of linear systems on several locally fine grids in parallel the principle to determine the diameters of three different scale grids is given especially this paper devises a new local and parallel finite element multiscale discretization scheme theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out and can be used to solve singular eigenvalue problems efficiently

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

analytical solution of the droplet breakup equation by the adomian decomposition method abstract the droplet population balance equations pbes which despite its importance rarely has an analytical solution however few cases with assumed functional forms of breakup rate daughter droplet distribution exist where most of these solutions are for the batch stirred vessel a new framework for solving pbes for batch and continuous systems is proposed in this work which uses the adomian decomposition method this technique overcomes the crucial difficulties of numerical discretization and stability that often characterize previous solutions in this area the technique used in this work has been tested for the droplet breakup equation the solutions are presented for several cases for which analytical solutions are available for batch and continuous systems for droplet breakup in stirred vessels in all cases the predicted droplet size distributions converge exactly in a continuous form to that of the analytical solution

asymptotic stability for intermittently controlled second order evolution equations motivated by several works on ordinary differential equations we are interested in the asymptotic stability of intermittently controlled partial differential equations we give a condition of asymptotic stability for second order evolution equations uniformly damped by an on off feedback this result extends to the case of partial differential equations a previous result of r a smith concerning ordinary differential equations

two grid finite element discretization schemes based on shifted inverse power method for elliptic eigenvalue problems this paper discusses highly efficient discretization schemes for solving self adjoint elliptic differential operator eigenvalue problems several new two grid discretization schemes including the conforming and nonconforming finite element schemes are proposed by combining the finite element method with the shifted inverse power method for matrix eigenvalue problems with these schemes the solution of an eigenvalue problem on a fine grid pi h is reduced to the solution of an eigenvalue problem on a much coarser grid pi h and the solution of a linear algebraic system on the fine grid pi h theoretical analysis shows that the schemes have a high efficiency for instance the resulting solution can maintain an asymptotically optimal accuracy by using the conforming linear element or the nonconforming crouzeix raviart element by taking h o sqrt 4 h numerical experiments are presented to support the theoretical analysis in addition this paper establishes multigrid discretization schemes and proves their efficiency

stability results for the wave equation with indefinite damping abstract the one dimensional wave equation with damping of indefinite sign in a bounded interval with dirichlet boundary conditions is considered it is proved that solutions decay uniformly exponentially to zero provided the damping potential is in thebv class has positive average is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes this sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms the method of proof relies on the developments by s cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa

stabilization of the wave equations with potential and indefinite damping by means of global carleman type estimate we study the stabilization problem of the wave equations with potential and indefinite damping the energy decay rate of the system is given explicitly also we obtain an upper bound estimate on the negative damping to guarantee the energy of the system decays exponentially

exponential stability of the plate equations with potential of second order and indefinite damping we study the exponential stability of the plate equations with potential of second order and indefinite sign damping term by means of a global carleman type estimate and the usual perturbation method we show that the energy of the system decays exponentially provided that the negative damping is sufficiently small both the energy decay rate and the upper bound estimate on the negative damping are given explicitly

analytic solutions for nonlinear equations solutions obtained by the decomposition method for physical problems modeled by nonlinear differential and partial differential equations without linearization or pertubation are not only continuous but analytic as well thus the solution and all derivatives are obtained

stabilization of high eigenfrequencies of a beam equation with generalized viscous damping in this paper using the one dimensional vibrating beam equation with generalized viscous damping as a model of vibration of flexible robot arms it is shown that for such a system the high eigenmodes decay at a uniform rate the proof is obtained by perturbation theory of linear operators y h luo acta math sinica 32 1991 pp 556 563 and asymptotic estimates of eigenvectors based on an earlier work of g b birkhoff and m h stone feedback control for this class of system is investigated and a finite dimensional controller is presented for an exponentially stable closed loop system our method may be used to study nondissipative systems

exponential decay of energy of evolution equations with locally distributed damping consider an evolution equation with energy dissipation frac partial partial t y x t ay x t by x t on a bounded x domain omega where by signifies a dissipative perturbation to an otherwise energy conserving system this dissipation may be due to medium impurities viscous effects or artificially imposed dampers and stabilizers it is distributed over only part of the domain omega the question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined because of the locally distributed nature of energy dissipation the problem lacks coercivity and is not directly solvable by energy identities thus to get conditions sufficient for uniform exponential decay a different approach needs to be taken provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorbon the separated eigenmodes or clustered eigenmodes ofa the main theorems are general enough to treat the wave and bea

on exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback abstract the uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered the existence of both weak and strong solutions to the system is proven by the galerkin method the exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the riemannian manifold this equivalent energy function shows particularly that the system is essentially a dissipative system this result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in a guesmia a new approach of stabilization of nondissipative distributed systems siam j control optim 42 2003 24 52 where the system is claimed to be nondissipative

bresse system with indefinite damping abstract we consider the bresse system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign presented only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 1 l 0 l a x d x 0 and a a l 2 for small enough in the arguments we shall also give a new proof of exponential stability for the constant case a a

timoshenko systems with indefinite damping abstract we consider the timoshenko system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign present only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 0 l a x d x 0 and a a l 2 for small enough the decay rate will be described explicitly in the arguments we shall also give a new proof of exponential stability for the constant case a a moreover we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth sdg property i e the type of the induced semigroup coincides with the spectral bound for its generator

the multilevel mixed finite element discretizations based on local defect correction for the stokes eigenvalue problem abstract based on the work of xu and zhou 2000 this paper combines mixed finite element method and the local defect correction technique to establish new local and parallel multilevel discretization schemes for the stokes eigenvalue problem theoretical analysis and numerical experiments show that the computational approach proposed in this paper is simple and easy to carry out in parallel and can be used to solve singular stokes eigenvalue problem efficiently this paper also discusses local error estimates of mixed finite element approximations it is a new feature here that the estimates apply to the local domains containing corner points

on the spectrum of euler bernoulli beam equation with kelvin voigt damping abstract the spectral property of an euler bernoulli beam equation with clamped boundary conditions and internal kelvin voigt damping is considered the essential spectrum of the system operator is rigorously identified to be an interval on the left real axis under some assumptions on the coefficients it is shown that the essential spectrum contains continuous spectrum only and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity the asymptotic behavior of eigenvalues is presented

exponential stability of wave equations with potential and indefinite damping abstract first we consider the linear wave equation u t t u x x a x u t b x u 0 on a bounded interval 0 l r the damping function a is allowed to change its sign if a 1 l 0 l a x d x is positive and the spectrum of the operator x x b is negative exponential stability is proved for small a a l 2 explicit estimates of the decay rate are given in terms of a and the largest eigenvalue of x x b second we show the existence of a global small smooth solution of the corresponding nonlinear wave equation u t t u x x a x u t b x u 0 if additionally the negative part of a is small enough compared with

exponential stability for wave equations with non dissipative damping abstract we consider the nonlinear wave equation u t t u x x a x u t 0 in a bounded interval 0 l r 1 the function a is allowed to change sign but has to satisfy a 1 l 0 l a x d x 0 for this non dissipative situation we prove the exponential stability of the corresponding linearized system for i possibly large a l with small a a l 2 and ii a class of pairs a l with possibly negative moment 0 l a x sin 2 x l d x estimates for the decay rate are also given in terms of a moreover we show the global existence of smooth small solutions to the corresponding nonlinear system if additionally the negative part of a is small enough

shifted inverse iteration based multigrid methods for the quad curl eigenvalue problem abstract this paper is devoted to multgrid methods for solving the quad curl eigenvalue problem the multgrid methods based on the rayleigh quotient iteration and the inverse iteration with fixed shift are proposed we prove the error estimates for both methods numerical experiments confirm our theoretical analysis and validate the efficiency of both methods

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

an approximate solution of a nonlinear fractional differential equation by adomian decomposition method the aim of the present analysis is to apply adomian decomposition method for the solution of a nonlinear fractional differential equation finally the solution obtained by the decomposition method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method a good agreement of the results is observed

two grid methods for maxwell eigenvalue problems two new two grid algorithms are proposed for solving the maxwell eigenvalue problem the new methods are based on the two grid methodology recently proposed by xu and zhou math comp 70 2001 pp 17 25 and further developed by hu and cheng math comp 80 2011 pp 1287 1301 for elliptic eigenvalue problems the new two grid schemes reduce the solution of the maxwell eigenvalue problem on a fine grid to one linear indefinite maxwell equation on the same fine grid and an original eigenvalue problem on a much coarser grid the new schemes therefore save total computational cost the error estimates reveals that the two grid methods maintain asymptotically optimal accuracy and the numerical experiments presented confirm the theoretical results

two grid discretizations and a local finite element scheme for a non selfadjoint stekloff eigenvalue problem abstract in this paper for a new stekloff eigenvalue problem which is non selfadjoint and not h 1 elliptic we establish and analyze two kinds of two grid discretization scheme and a local finite element scheme we present the error estimates of approximations of two grid discretizations we also prove a local error estimate which is suitable for the case that the local refined region contains singular points lying on the boundary of domain numerical experiments are reported finally to show the efficiency of our schemes

nonconforming finite element analysis for poisson eigenvalue problem the main aim of this paper is to study a nonconforming quadrilateral finite element named modified quasi wilson element approximation to poisson eigenvalue problem firstly by employing a special property of this element when u h 3 the consistency error is of order o h 2 which is one order higher than its interpolation error o h and the interpolation postprocessing technique the superclose and superconvergence results of order o h 2 for the exact solution of eigenvector u in broken h 1 norm are deduced on generalized rectangular meshes and rectangular meshes respectively secondly it is proved that the consistency error even can reach o h 4 order for arbitrary quadrilateral meshes when u h 5 this is a new astonishing feature which has never been discovered subsequently based on the above characteristic and some asymptotic expansions of the conforming bilinear finite element the extrapolation solution of order o h 4 for eigenvalue is derived finally some numerical results are provided to verify the theoretical analysis

superconvergence two grid scheme based on shifted inverse power method for eigenvalue problems by function value recovery abstract in the paper an improved two grid scheme based on shifted inverse power method is proposed to solve the elliptic eigenvalue problems with this new scheme the solution of the elliptic eigenvalue problem on a fine grid t h is reduced to the solution of the elliptic eigenvalue problem and the recovered eigenfunction on a much coarser grid t h and the solution of an elliptic boundary value problem and the recovered solution on the fine grid t h theoretical analysis shows that the scheme allows a much coarser mesh to achieve the superconvergence rate finally some numerical experiments are carried out to confirm the theoretical analysis

boundary control of an anti stable wave equation with anti damping on the uncontrolled boundary abstract much of the boundary control of wave equations in one dimension is based on a single principle passivity under the assumption that control is applied through neumann actuation on one boundary and the other boundary satisfies a homogeneous dirichlet boundary condition we have recently expanded the scope of tractable problems by allowing destabilizing anti stiffness a robin type condition on the uncontrolled boundary where the uncontrolled system has a finite number of positive real eigenvalues in this paper we go further and develop a methodology for the case where the uncontrolled boundary condition has anti damping which makes the real parts of all the eigenvalues of the uncontrolled system positive and arbitrarily high i e the system is anti stable exponentially stable in negative time using a conceptually novel integral transformation we obtain extremely simple explicit formulae for the gain functions for the case with only boundary sensing available at the same end with actuation we design backstepping observers which are dual to the backstepping controllers and have explicit output injection gains we then combine the control and observer designs into an output feedback compensator and prove the exponential stability of the closed loop system

riesz basis property and exponential stability of controlled euler bernoulli beam equations with variable coefficients this paper studies the basis property and the stability of a distributed system described by a nonuniform euler bernoulli beam equation under linear boundary feedback control it is shown that there is a sequence of generalized eigenfunctions of the system which forms a riesz basis for the state hilbert space the asymptotic distribution of eigenvalues the spectrum determined growth condition and the exponential stability are concluded the results are applied to a nonuniform beam equation with viscous damping of variable coefficient as a generalization of existing results for the uniform beam

the adaptive finite element method based on multi scale discretizations for eigenvalue problems in this paper adaptive finite element methods for differential operator eigenvalue problems are discussed for multi scale discretization schemes based on rayleigh quotient iteration see scheme 3 in y yang h bi a two grid discretization scheme based on shifted inverse power method siam j numer anal 49 2011 1602 1624 a reliable and efficient a posteriori error indicator is given in addition a new adaptive algorithm based on the multi scale discretizations is proposed and we apply the algorithm to the schrodinger equation for hydrogen atoms the algorithm is performed under the package of chen and satisfactory numerical results are obtained

sign preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second order elliptic operators this paper is concerned with the p1 finite element approximation of the eigenvalue problem of second order elliptic operators subject to the dirichlet boundary condition the focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem it is shown that when the stiffness matrix is an irreducible m matrix the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain mesh conditions leading to such a stiffness matrix are also studied a sufficient condition is that the mesh is simplicial interiorly connected and acute when measured in the metric specified by the inverse of the diffusion matrix the acute requirement can be replaced by the delaunay condition in two dimensions numerical results show that when the stiffness matrix is not an m matrix a finite element approximation can be structurally different from the continuous eigenvalue problem the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue in modulus can even be complex for the case with nonsymmetric operators

a new nonconforming mixed finite element scheme for second order eigenvalue problem a new nonconforming mixed finite element method mfem for short is established for the laplace eigenvalue problem firstly the optimal order error estimates for both the eigenvalue and eigenpair the original variable u and the auxiliary variable p u are deduced the lower bound of eigenvalue is estimated simultaneously then by use of the special property of the nonconforming e q 1 r o t element the consistency error is of order o h2 in broken h1 norm which is one order higher than its interpolation error the techniques of integral identity and interpolation postprocessing we derive the superclose and superconvergence results of order o h2 for u in broken h1 norm and p in l2 norm furthermore with the help of asymptotic expansions the extrapolation solution of order o h3 for eigenvalue is obtained finally some numerical results are presented to validate our theoretical analysis

the damped wave equation with unbounded damping abstract we analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity we prove the generation of a contraction semigroup study the relation between the spectra of the semigroup generator and the associated quadratic operator function the convergence of non real eigenvalues in the asymptotic regime of diverging damping on a subdomain and we investigate the appearance of essential spectrum on the negative real axis we further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential these analytic results are illustrated by examples

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

asymptotic stability for bresse systems abstract in this paper consider a vibrating system of bresse type in a one dimensional bounded domain with a nonlinear damping acting in the equation of the shear angle displacement and nonlinear localized damping in other equations we show the asymptotic stability without imposing conditions about the equal speed wave propagation using a method developed by lasiecka and tataru 9

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

stability of a nonuniform rayleigh beam with indefinite damping abstract this is a continuation of our earlier work j m wang g q xu s p yung exponential stability for variable coefficients rayleigh beams under boundary feedback control a riesz basis approach systems control lett 51 1 2004 33 50 on the study of a nonhomogeneous rayleigh beam and this time the stabilization is achieved via an internal damping instead of the boundary feedbacks we continue to address a conjecture of guo basis property of a rayleigh beam with boundary stabilization j optim theory appl 112 3 2002 529 547 in this paper and demonstrate how the damping term can affect the decay rate asymptotically by a detailed spectral analysis we obtain a necessary condition for the stability and establish the riesz basis property as well as the spectrum determined growth condition for the system furthermore when the damping is indefinite we provide a condition on how negative the damping can be without destroying the exponential stability

exponential stabilization of laminated beams with structural damping and boundary feedback controls we study the boundary stabilization of laminated beams with structural damping which describes the slip occurring at the interface of two layered objects by using an invertible matrix function with an eigenvalue parameter and an asymptotic technique for the first order matrix differential equation we find out an explicit asymptotic formula for the matrix fundamental solutions and then carry out the asymptotic analyses for the eigenpairs furthermore we prove that there is a sequence of generalized eigenfunctions that forms a riesz basis in the state hilbert space and hence the spectrum determined growth condition holds furthermore exponential stability of the closed loop system can be deduced from the eigenvalue expressions in particular the semigroup generated by the system operator is a c 0 group due to the fact that the three asymptotes of the spectrum are parallel to the imaginary axis

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

optimal shape and position of the actuators for the stabilization of a string abstract the energy in a string subject to constant viscous damping k on a subset of length l 0 decays exponentially in time we consider the problem of optimizing the decay rate for the which are the unions of at most n intervals this rate is given by the spectral abscissa of the linear operator associated to the wave equation we are interested in small values of k therefore we consider the derivative of the spectral abscissa at k 0 we prove that except for the case l 1 2 when the number of intervals is not fixed a priori an optimal domain does not exist we study numerically the case of one or two intervals using a genetic algorithm these numerical results are not intuitive in particular the optimal position of one interval is never at the middle of the string