threshold effects in parameter estimation from compressed data in this paper we investigate threshold effects associated with the swapping of signal and noise subspaces in estimating signal parameters from compressed noisy data the term threshold effect refers to a sharp departure of mean squared error from the cramer rao bound when the signal to noise ratio falls below a threshold snr in many cases the threshold effect is caused by a subspace swap event when the measured data or its sample covariance is better approximated by a subset of components of an orthogonal subspace than by the components of a signal subspace we derive analytical lower bounds on the probability of a subspace swap in compressively measured noisy data in two canonical models a first order model and a second order model in the first order model the parameters to be estimated modulate the mean of a complex multivariate normal set of measurements in the second order model the parameters modulate the covariance of complex multivariate measurements in both cases the probability bounds are tail probabilities of f distributions and they apply to any linear compression scheme these lower bounds guide our understanding of threshold effects and performance breakdowns for parameter estimation using compression in particular they can be used to quantify the increase in threshold snr as a function of a compression ratio c we demonstrate numerically that this increase in threshold snr is roughly 10 log 10 c db which is consistent with the performance loss that one would expect when measurements in gaussian noise are compressed by a factor c

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

a mixed finite element formulation for the boundary controllability of the wave equation abstract this paper introduces the mixed finite element method as a viable numerical procedure for the boundary controllability of the linear wave equation another numerical implementation using galerkin finite elements has been investigated however due to approximation problems of the normal derivative on the boundary the method becomes unstable as the mesh is refined to correct for the ill posedness of the approximate problem a tychonoff regularization method was implemented the aforementioned paper also presents other possible remedies among them is the mixed finite element method the fixed finite element approximation is a plausible procedure to overcome these difficulties since the derivative at certain nodal values arises naturally from the formulation

a numerical method for exact boundary controllability problems for the wave equation abstract the computational approximation of exact boundary controllability problems for the wave equation in two dimensions is studied a numerical method is defined that is based on the direct solution of optimization problems that are introduced in order to determine unique solutions of the controllability problem the uniqueness of the discrete finite difference solutions obtained in this manner is demonstrated the convergence properties of the method are illustrated through computational experiments efficient implementation strategies for the method are also discussed it is shown that for smooth minimum l 2 norm dirichlet controls the method results in convergent approximations without the need to introduce regularization furthermore for the generic case of nonsmooth dirichlet controls convergence with respect to l 2 norms is also numerically demonstrated one of the strengths of the method is the flexibility it allows for treating other controls and other minimization criteria such generalizations are discussed in particular the minimum h 1 norm dirichlet controllability problem is approximated and solved as are minimum regularized l 2 norm dirichlet controllability problems with small penalty constants finally a discussion is provided about the differences between our method and existing methods these differences may explain why our methods provide convergent approximations for problems for which existing methods produce divergent approximations unless they are regularized in some manner

backstepping boundary control of a wave pde with spatially distributed time invariant unknown disturbances an observer based boundary control scheme is suggested for an infinite dimensional system modeled as a wave partial differential equation pde with in domain and boundary disturbances we consider a collocated case where the sensors and the actuators are located at the same boundary the aim of the controller is to regulate the rate of change of the opposite boundary around a given reference the unknown in domain disturbance is spatially distributed but it is constant in time same as the unknown boundary disturbance the problem is reformulated as a stabilization of an linear time invariant lti system with simultaneous delays in input and output channels the exponential stability of the equilibrium of the closed loop system is proven the performance of the controller is shown using numerical simulations

jacobi matrices and transversality the paper deals with smooth nonlinear ode systems in n f x such that the derivative f x has a matrix representation of jacobi type not necessarily symmetric with positive off diagonal entries a discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system f x t y two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system it is also proved that the nonwandering points are critical points a new class of morse smale systems in n is then explicitly constructed

impulse output rapid stabilization for heat equations abstract the main aim of this paper is to provide a new feedback law for the heat equations in a bounded domain with dirichlet boundary condition two constraints will be compulsory first the controls are active in a subdomain of and at discrete time points second the observations are made in another subdomain and at different discrete time points our strategy consists in linking an observation estimate at one time minimal norm impulse control approximate inverse source problem and rapid output stabilization

ensuring well posedness by analogy stokes problem and boundary control for the wave equation abstract in this article we give a comparative discussion of the finite element approximation of two partial differential equation problems these two problems which are apparantly quite unrelated are the stokes problem for incompressible viscous flow and an exact boundary controllability problem for the wave equation we show that straightforward discrete approximations to these problems yield approximate problems which are ill posed the analysis of the ill posedness of the above problems shows an identical cause namely the strong damping of the high frequency modes beyond a critical wave number from this analogy a well known cure for the discrete stokes problem i e using more accurate approximations for velocity than for pressure provides a simple way to eliminate the ill posedness of the discrete exact boundary controllability problem numerical examples concerning the control problem testify about the soundness of the new approach to conclude this paper one takes advantage of the previous analysis to give a brief discussion of the wavelet approximation of the stokes problem for dirichlet boundary conditions

structural stability for time periodic one dimensional parabolic equations denote q w x 0 l x r the nonlinear term f is assumed to satisfy the following conditions f l f q r is of class c f 2 f t l x u f t x u for all te w xe o l and uer define a self adjoint operator a in x l2 0 l by a t bxx for e d a h2 0 l a ha o l for each c1 e r let a be the fractional power of a and x d a be the fractional power space equipped with the graph norm ii 11 xn 1111 11x iia i ii x throughout this section we fix a c1 1 by the above assumptions f l f 2 for any u0 e x there exists a unique local in time classical solution u t x u t x of

rapid boundary stabilization of linear distributed systems we prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates our approach is direct and constructive and avoids many of the technical difficulties associated with the usual methods based on riccati equations we give several applications for the wave equation and for petrovsky systems

rapid exponential feedback stabilization with unbounded control operators we study the wellposedness and the main features of a class of feedback control systems the involved control system is composed of the generator of a strongly continuous group for the free part and of an unbounded control operator so that the results can be applied to boundary or point control problems for partial differential equations of hyperbolic or petrowski type the feedback operator is explicit and one can achieve an arbitrary large decay rate for the closed loop system these results are proved under a controllability assumption and the proofs rely on general results about the algebraic riccati equation associated with the linear quadratic regulator problem

computational methods for the fast boundary stabilization of flexible structures part 1 the case of beams an efficient active control strategy for flexible systems v komornik rapid boundary stabilization of linear distributed systems siam j control optim 35 5 1997 1591 1613 is thoroughly investigated from the numerical point of view a non standard computational framework is proved to be relevant both for simulation and control synthesis the proposed formulation proves necessary in so far as a standard numerical approach is shown to fail the observed properties of the state feedback law confirm the theory as far as beams are concerned in particular one can achieve an arbitrarily large decay rate of some weak norm of the system moreover the control law compares favourably with the integral force feedback in terms of efficiency finally smoothing procedures are introduced to decrease the control spill over associated with the possible lack of compatibility between boundary control and initial conditions this purely artificial spill over is proved to result from oversimplification of the control process modelling

asymptotic achievability of the cramer rao bound for noisy compressive sampling we consider a model of the form y ax n where x isin cm is sparse with at most l nonzero coefficients in unknown locations y isin cn is the observation vector a isin cn times m is the measurement matrix and n isin cn is the gaussian noise we develop a cramer rao bound on the mean squared estimation error of the nonzero elements of x corresponding to the genie aided estimator gae which is provided with the locations of the nonzero elements of x intuitively the mean squared estimation error of any estimator without the knowledge of the locations of the nonzero elements of x is no less than that of the gae assuming that l n is fixed we establish the existence of an estimator that asymptotically achieves the cramer rao bound without any knowledge of the locations of the nonzero elements of x as n rarr infin for a a random gaussian matrix whose elements are drawn i i d according to n 0 1

boundary stabilization for the von karman equations the boundary stabilization of a nonlinear plate model is studied the equations used take in consideration the in plane accelerations and the rotary inertia of the cross sections applying linear feedbacks the authors obtain the exponential decay of the energy

sensitivity to basis mismatch in compressed sensing the theory of compressed sensing suggests that successful inversion of an image of the physical world broadly defined to include speech signals radar sonar returns vibration records sensor array snapshot vectors 2 d images and so on for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis provided that the image is sparse in an apriori known basis for imaging problems in spectrum analysis and passive and active radar sonar this basis is usually taken to be a dft basis however in reality no physical field is sparse in the dft basis or in any apriori known basis no matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity in this paper we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases we start by analyzing the effect of basis mismatch on the best k term approximation error which is central to providing exact sparse recovery guarantees we establish achievable bounds for the l1 error of the best k term approximation and show that these bounds grow linearly with the image or grid dimension and the mismatch level between the assumed and actual bases for sparsity we then derive bounds with similar growth behavior for the basis pursuit l1 recovery error indicating that the sparse recovery may suffer large errors in the presence of basis mismatch although we present our results in the context of basis pursuit our analysis applies to any sparse recovery principle that relies on the accuracy of best k term approximations for its performance guarantees we particularly highlight the problematic nature of basis mismatch in fourier imaging where spillage from off grid dft components turns a sparse representation into an incompressible one we substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch for problem sizes common to radar and sonar

k dimensional invariant cones of random dynamical systems in rn with applications abstract we investigated a linear random dynamical system which strongly preserves a cone c of dimension k abbr k cone in r n under some general assumptions it is shown that such system admits a measurable family of k dimensional subspaces and a measurable family of n k dimensional subspaces which are complementary to each other and form into a tempered invariant splitting of r n we further apply the measurable bundles so obtained to study the linear random monotone cyclic feedback systems as well as the linear competitive cooperative tridiagonal systems this generalizes the floquet theory for these deterministic non autonomous or time periodic systems to the random systems

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

distributed lagrange multiplier methods for incompressible viscous flow around moving rigid bodies abstract in this article we discuss the application of a distributed lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the navier stokes equations around moving bodies we suppose the rigid bodies motion known a priori the solution method combines finite element approximations time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method numerical experiment results for two dimensional flow around a moving disk are presented

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

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

domain imbedding methods for the stokes equations we study direct and iterative domain imbedding methods for the stokes equations on certain non rectangular domains in two space dimensions we analyze a continuous analog of numerical domain imbedding for bounded smooth domains and give an example of a simple numerical algorithm suggested by the continuous analysis this algorithms is applicable for simply connected domains which can be covered by rectangular grids with uniformly spaced grid lines in at least one coordinate direction we also discuss a related fft based fast solver for stokes problems with physical boundary conditions on rectangles and present some numerical results

numerical dispersive schemes for the nonlinear schrodinger equation we consider semidiscrete approximation schemes for the linear schrodinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations for the conservative finite difference semidiscretization scheme we show that as the mesh size tends to zero the semidiscrete approximate solutions lose the dispersion property this fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete laplacian vanish therefore this phenomenon is due to the presence of numerical spurious high frequencies to recover the dispersive properties of the solutions at the discrete level we introduce two numerical remedies fourier filtering and a two grid preconditioner for each of them we prove strichartz like estimates and a local space smoothing effect uniform in the mesh size the methods we employ are based on classical estimates for oscillatory integrals these estimates allow us to treat nonlinear problems with l 2 initial data without additional regularity hypotheses we prove the convergence of the two grid method for nonlinearities that cannot be handled by energy arguments and which even in the continuous case require strichartz estimates

exact controllability of a nonlinear korteweg de vries equation on a critical spatial domain we consider the boundary controllability problem for a nonlinear korteweg de vries equation with the dirichlet boundary condition we study this problem for a spatial domain with a critical length for which the linearized control system is not controllable in order to deal with the nonlinearity we use a power series expansion of second order we prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough

modal analysis using co prime arrays we address the problem of estimating mode parameters from noisy observations of a linear combination of the corresponding modes this problem arises in line spectrum estimation vibration analysis speech processing system identification and direction of arrival estimation our results differ from standard results of modal analysis to the extent that we consider co prime samplings in space or equivalently co prime samplings in time our main result is a characterization of the orthogonal subspace for this problem this is the subspace that is orthogonal to the signal subspace spanned by the columns of the generalized vandermonde matrix of modes in co prime samplings this characterization is derived in a form that allows us to adapt modern methods of subspace signal processing to co prime sampled signals several numerical examples are presented to demonstrate the application of the proposed modal estimation method we state and prove theorems on identifiability of the modes and calculate a cramer rao bound that allows us to analyze the performance of co prime arrays that are subsamplings of uniform linear arrays of the same apertures

comparison of preconditioners for collocation chebyshev approximation of 2d and 3d generalized stokes problem the aim of the paper is concerned with the iterative resolution of the generalized stokes problem in the framework of collocation chebyshev approximation more precisely we analyze the performance of several preconditioners improving the classical uzawa method we first recall that the general stokes problem gsp is an elementary substep of general interest for computing not only incompressible flows but also low mach number flows we then remark that performances of the classical uzawa algorithm for solving the gsp are in fact closely related to the reynolds number in order to overcome this dependence preconditioners are needed the preconditioners we analyze here are recommended by fourier analysis of the pressure operator we additionally give interpretation of the preconditioners in terms of influence or capacitance techniques we give a detailed analysis of the conditioning of the discrete collocation chebyshev versions of the operators numerical comparison is conducted in 2d as well as in 3d rectangular geometry our comparative study shows that the fictitious wall permeability fwp method is the most efficient preconditioner if complemented with a suitable pressure filtering its efficiency still increases

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

arbitrary decay rate for euler bernoulli beam by backstepping boundary feedback we consider a problem of stabilization of the euler bernoulli beam the beam is controlled at one end using position and moment actuators and has the ldquoslidingrdquo boundary condition at the opposite end we design the controllers that achieve any prescribed decay rate of the closed loop system removing a long standing limitation of classical ldquoboundary damperrdquo controllers the idea of the control design is to use the well known representation of the euler bernoulli beam model through the schrodinger equation and then adapt recently developed backstepping designs for the latter in order to stabilize the beam we derive the explicit integral transformation and its inverse of the closed loop system into an exponentially stable target system the transformation is of a novel volterra fredholm type the design is illustrated with simulations

boundary observability controllability and stabilization of linear elastodynamic systems in 1988 lions obtained observability and exact controllability results for linear homogeneous isotropic elastodynamic systems siam rev 30 1988 pp 1 68 applying some new identities we extend his theorems to nonisotropic systems in 1991 lagnese obtained uniform stabilizability results for two dimensional linear homogeneous isotropic systems by applying a somewhat artificial feedback nonlinear anal 16 1991 pp 35 54 then he asked whether analogous results hold for a natural and physically implementable boundary feedback using some new identities and applying a method introduced in 1987 by zuazua and the second author j math pures appl 69 1990 pp 33 54 we give an affirmative answer to this question in all dimensions and also for nonisotropic systems moreover we obtain good decay estimates finally applying a recent general method of uniform stabilization we construct boundary feedbacks leading to arbitrarily large energy decay rates

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems ii finite dimensional systems abstract this is the second part in a series of papers concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part focuses on applications of general theory developed in the authors paper j mierczy ski w shen principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory trans amer math soc in press http dx doi org 10 1090 s0002 9947 2013 05814 x preprint available at http arxiv org abs 1209 3475 to positive random dynamical systems on finite dimensional ordered banach spaces it is shown under some quite general assumptions that measurable linear skew product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type k strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations

uniform boundary controllability of a semi discrete 1 d wave equation a numerical scheme for the controlled semi discrete 1 d wave equation is considered we analyze the convergence of the boundary controls of the semi discrete equations to a control of the continuous wave equation when the mesh size tends to zero we prove that if the high modes of the discrete initial data have been filtered out there exists a sequence of uniformly bounded controls and any weak limit of this sequence is a control for the continuous problem the number of the eliminated frequencies depends on the mesh size and the regularity of the continuous initial data the case of the hum controls is also discussed

a two level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems this work is concerned with the boundary observability of an abstract system of two coupled second order evolution equations the coupling operator being a compact perturbation of the uncoupled system we assume that only one of the two components of the unknown is observed this is indirect observability we prove that by observing only one component one can get back a full weakened energy of both components under a compatibility condition linking the operators of each equation and for small coupling using the hilbert uniqueness method we then establish an indirect exact controllability result we apply this abstract result to several coupled systems of partial differential equations wave wave coupled elastodynamic systems petrowsky petrowsky and wave petrowsky systems

output feedback stabilization for heat equations with sampled data controls abstract in this paper we build up an output feedback law to stabilize a sampled data controlled heat equation with a potential in a bounded domain the feedback law abides the following rules first we divide equally the time interval 0 into infinitely many disjoint time periods and divide each time period into three disjoint subintervals second for each time period we observe a solution over an open subset of in the first subinterval take sample from outputs at one time point of the first subinterval add a time invariant output feedback control over another open subset of in the second subinterval let the equation evolve free in the last subinterval thus the corresponding feedback control is of sampled data our feedback law has the following advantages the sampling period which is the length of the above time period can be arbitrarily taken the feedback law has an explicit expression in terms of the sampling period the behaviors of the norm of the feedback law when the sampling period goes to zero or infinity are clear the construction of the feedback law is based on two kinds of approximate null controllability for heat equations one has time invariant controls while another has impulse controls the studies of the aforementioned controllability with time invariant controls need a new observability inequality for heat equations built up in the current work

uniform exponential long time decay for the space semi discretization of a locally damped wave equation via an artificial numerical viscosity we consider the finite difference space semi discretization of a locally damped wave equation the damping being supported in a suitable subset of the domain under consideration so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity the decay rate of the semi discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero we prove that adding a suitable vanishing numerical viscosity term leads to a uniform with respect to the mesh size exponential decay of the energy of solutions this numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept we discuss this problem in 1d and 2d in the interval and the square respectively our method of proof relies on discrete multiplier techniques

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

exact boundary controllability of the second order maxwell system theory and numerical simulation the exact controllability of the second order time dependent maxwell equations for the electric field is addressed through the hilbert uniqueness method a two grid preconditioned conjugate gradient algorithm is employed to inverse the h u m operator and to construct the numerical control the underlying initial value problems are discretized by lagrange finite elements and an implicit newmark scheme two dimensional numerical experiments illustrate the performance of the method

rapid boundary stabilization of the wave equation the boundary stabilization of the wave equation in bounded domains is studied it is shown that a particular choice of the feedback leads to fast energy decay explicit decay rate estimates are obt

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

on the null asymptotic stabilization of the two dimensional incompressible euler equations in a simply connected domain we study the asymptotic stabilization of the origin for the two dimensional 2 d euler equation of incompressible inviscid fluid in a bounded domain we assume that the controls act on an arbitrarily small nonempty open subset of the boundary we prove the null global asymptotic stabilizability by means of explicit feedback laws if the domain is connected and simply connected

on the quadratic finite element approximation of one dimensional waves propagation observation and control we study the propagation observation and control properties of the quadratic p 2 classical finite element semidiscretization of the one dimensional wave equation on a bounded interval a careful fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum the acoustic one of physical nature and the optic one related to the perturbations that this second order finite element approximation introduces with respect to the p 1 one on both modes there are high frequencies with vanishing group velocity as the mesh size tends to zero this shows that the classical property of continuous waves of being observable from the boundary fails to be uniform for this discretization scheme as a consequence of this the controls of the discrete waves may blow up as the mesh size tends to zero to remedy these high frequency pathologies we design filtering mechanisms based on the fourier truncation method or on a bi grid algorithm for which one can recover the uniformity of the

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory this series of papers is concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part of the series focuses on the development of general theory first the notions of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations for general positive random dynamical systems in ordered banach spaces are introduced which extend the classical notions of principal floquet subspaces principal lyapunov exponents and exponential separations for strongly positive deterministic systems in strongly ordered banach to general positive random dynamical systems in ordered banach spaces under some quite general assumptions it is then shown that a positive random dynamical system in an ordered banach space admits a family of generalized principal floquet subspaces a generalized principal lyapunov exponent and a generalized exponential separation we will consider in the forthcoming part s the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations including random leslie matrix models random cooperative systems of ordinary differential equations and random parabolic equations institute of mathematics and computer science wroc law university of technology wybrzeze wyspianskiego 27 pl 50 370 wroc law poland url http www im pwr wroc pl mierczyn index html e mail address mierczyn pwr wroc pl department of mathematics and statistics auburn university auburn university al 36849 usa e mail address wenxish auburn edu 2010 mathematics subject classification primary 37h15 37l55 37a30 secondary 15b52 34f05 35r60

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

exponential separation and principal lyapunov exponent spectrum for random nonautonomous parabolic equations abstract the purpose of the paper is to extend the principal eigenvalue and principal eigenfunction theory for time independent and periodic parabolic equations to random and general nonautonomous ones in the random case a notion of principal lyapunov exponent serving as an analog of principal eigenvalue is introduced it is shown that the principal lyapunov exponent is deterministic and of simple multiplicity it is also shown that there is a one dimensional invariant random subbundle corresponding to the solutions that are globally defined and of the same sign which serves as an analog of principal eigenfunction in addition monotonicity of the principal lyapunov exponent with respect to the zero order terms both in the equation and in the boundary condition is proved when the second and first order terms are deterministic it is proved that the principal lyapunov exponent is greater than or equal to the principal eigenvalue of the associated time averaged equation in the general nonautonomous case the concepts of principal spectrum which serves as an analog of principal eigenvalue and principal lyapunov exponents are introduced as is known the principal spectrum is a compact interval it is proved in the paper that the principal spectrum contains all the principal lyapunov exponents when the second and first order terms are time independent a lower estimate of the infimum of the principal spectrum is given in terms of an associated time averaged equation

dirichlet feedback control for the stabilization of the wave equation a numerical approach abstract in j differential equations 66 1987 340 a uniform stabilization method of the wave equation by boundary control a la dirichlet has been discussed in this article we investigate the numerical implementation of the above stabilization process by a numerical scheme which mimics the energy decay properties of its continuous counterpart the practical implementation of that scheme leads to a biharmonic problem of a new type which is solved by a method directly inspired by some related work of glowinski and pironneau on the solution of the dirichlet problem for the biharmonic operator siam rev 21 2 1979 167 numerical experiments show that the decay properties of the energy are well preserved by our numerical methodology

analysis of fisher information and the cramer rao bound for nonlinear parameter estimation after random compression in this paper we analyze the impact of compression with complex random matrices on fisher information and the cramer rao bound crb for estimating unknown parameters in the mean value function of a complex multivariate normal distribution we consider the class of random compression matrices whose distribution is right unitarily invariant the compression matrix whose elements are i i d standard complex normal random variables is one such matrix we show that for all such compression matrices the fisher information matrix has a complex matrix beta distribution we also derive the distribution of crb these distributions can be used to quantify the loss in crb as a function of the fisher information of the noncompressed data in our numerical examples we consider a direction of arrival estimation problem and discuss the use of these distributions as guidelines for choosing compression ratios based on the resulting loss in crb

partial exact controllability for spherical membranes in this paper the partial exact controllability of an elastic spherical membrane is proved the reachability problem for the integrodifferential equation introduced for the vibrations of the meridional displacement is solved the main result is a generalization of a ingham s theorem on nonharmonic fourier series

numerical analysis for the wave equation with locally nonlinear distributed damping in this paper we present spectral methods in order to solve wave equation subject to a locally distributed nonlinear damping thanks to the efficiency and the accuracy of spectral method we can check that discrete energy decreases to zero as time goes to infinity uniformly with respect to the mesh size when the damping is supported in a suitable subset of the domain of consideration we prove the convergence of the full fourier galerkin discretization thus we apply our schemes to illustrate the uniform discrete energy decay rates of the solution for a wide range of damping functions

approximation of the controls for the wave equation with a potential this article deals with the approximation of the boundary controls of a 1 d linear wave equation with a variable potential by using a finite difference space semi discrete scheme due to the high frequency numerical spurious oscillations the semi discrete model is not uniformly controllable with respect to the mesh size and the convergence of the approximate controls cannot be guaranteed in this paper we analyze how do the initial data to be controlled and their discretization affect the approximation of the controls under certain conditions on the potential we prove that the convergence of the scheme is ensured if the highest frequencies of the discrete initial data have been previously filtered out several filtration procedures are proposed and analyzed moreover we identify a class of regular continuous initial data which can be controlled uniformly without any special treatment

an efficient feedback control algorithm for beams experimental investigations an experimental implementation of a rapid active control strategy for flexible structures combined with a modal filter for state reconstruction is presented the control law which builds upon komornik s original idea is designed so as to achieve an arbitrarily large decay rate of the energy of the system at the price of solving once and for all a well conditioned small size linear system instead of a riccati equation it is applied to a slender beam the fundamental ideas and properties of this simple but efficient strategy are explained the emphasis is put on the practical implementation and on the experimental results which show an important increase of the apparent modal damping even for values of the resulting gain that do not generate spill over the proposed feedback outperforms the standard lq control the actual behaviour of the closed loop system is accurately predicted by a complete electro mechanical model that accounts for the identified dynamics of the shaker

generalized inf sup conditions for chebyshev spectral approximation of the stokes problem a mixed problem and its approximation in an abstract framework are considered they are proved to be well posed if and only if several inf sup conditions are satisfied these results are applied to the stokes equations in a square formulated in chebyshev weighted sobolev spaces and their approximations two kinds of spectral discretizations are analyzed a galerkin method and a collocation method at the chebyshev nodes

numerical solution of a flow control problem vorticity reduction by dynamic boundary action in order to laminarize an unsteady internal flow the vorticity field is minimized in a least squares sense using an optimal control approach the flow model is the navier stokes equation for a viscous incompressible fluid and the flow is controlled by suction and blowing on a part of the boundary a quasi newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term the navier stokes equations are approximated using a finite difference scheme in time and finite element approximations in space accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm therefore first order necessary conditions for a minimizer of the objective function are derived in the fully discrete case a memory saving device is discussed without which problems of any realistic size especially in three space dimensions would remain computationally intractable the feasibility of the optimal control approach for flow control problems is demonstrated by numerical experiments for a two dimensional flow in a rectangular cavity at a reynolds number high enough for nonlinear effects to be important

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

boundary controllability of a linear semi discrete 1 d wave equation derived from a mixed finite element method in this article one discusses the controllability of a semi discrete system obtained by discretizing in space the linear 1 d wave equation with a boundary control at one extremity it is known that the semi discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero see 8 here we introduce a new semi discrete model based on a mixed finite element method with two different basis functions for the position and velocity we show that the controls obtained with these semi discrete systems can be chosen uniformly bounded in l 2 0 t and in such a way that they converge to the hum control of the continuous wave equation i e the minimal l 2 norm control we illustrate the mathematical results with several numerical experiments