exact controllability of stochastic differential equations with memory abstract in this work we mainly study the exact controllability of stochastic differential equations with memory for time invariant systems by applying forward backward stochastic differential equations some rank criteria ensuring systems exact controllability are provided for time variant systems equivalent conditions guaranteeing systems exact controllability are given for deterministic systems as a special case of stochastic ones more easily checkable rank criteria for time invariant systems and alternative conditions for time variant systems are proposed

infinite time exact observability of volterra systems in hilbert spaces abstract we study infinite time exact observability of volterra systems in hilbert spaces sufficient conditions are established under which infinite time exact observability of a volterra system follows from that of the corresponding cauchy system without convolution term this is done by combining the composition operator method with a result about bounded below composition operators on the hardy space

product formula methods for time dependent schrodinger problems this paper introduces a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time dependent hermitian operator rigorous upper bounds are derived for two different time ordered approximation schemes and for errors resulting from approximating a time ordered exponential by an ordinary exponential operator the properties and the usefulness of the product formula algorithms are examined by applying them to the problem of zener tunnelling the most efficient algorithm is employed to solve the time dependent schrodinger equation for a wavepacket incident on a time modulated rectangular barrier

approximate controllability for linearized compressible barotropic navier stokes system in one and two dimensions abstract in this paper we consider compressible barotropic navier stokes equations in one and two dimensions linearized around a constant steady state with dirichlet boundary conditions we explore the controllability of this linearized system using a control only for the velocity equation we prove that the system with homogeneous dirichlet boundary conditions is approximately controllable by a localized interior control when time is sufficiently large

maximum entropy and bayesian data analysis entropic prior distributions the problem of assigning probability distributions which reflect the prior information available about experiments is one of the major stumbling blocks in the use of bayesian methods of data analysis in this paper the method of maximum relative entropy me is used to translate the information contained in the known form of the likelihood into a prior distribution for bayesian inference the argument is inspired and guided by intuition gained from the successful use of me methods in statistical mechanics for experiments that cannot be repeated the resulting entropic prior is formally identical with the einstein fluctuation formula for repeatable experiments however the expected value of the entropy of the likelihood turns out to be relevant information that must be included in the analysis the important case of a gaussian likelihood is treated in detail

approximate controllability of the fitzhugh nagumo equation in one dimension abstract the fitzhugh nagumo fhn equation is a simplified model of a nerve axon we explore the controllability of this model using a localized interior control only for the first equation the linearized system is not null controllable using a localized interior control since the spectrum of the linearized system has an accumulation point though it is approximate controllable we show that the solution of the fhn equation fails to be globally approximate controllable in a given time but it is possible to move from any steady state to any other steady state arbitrarily close after some appropriate time by a localized interior control provided that both steady states are in the same connected component of the set of steady states finally we make some additional remarks and comments and we mention some open questions for our system for the sake of completeness we give the details of the existence uniqueness and uniform bound of the solution in appendix

heat equation with memory lack of controllability to rest abstract we prove that the one dimensional heat equation with memory cannot be controlled to rest for large classes of memory kernels and controls the approach is based on the application of the theory of interpolation in paley wiener spaces

gradient descent learning in and out of equilibrium relations between the off thermal equilibrium dynamical process of on line learning and the thermally equilibrated off line learning are studied for potential gradient descent learning the approach of opper to study on line bayesian algorithms is used for potential based or maximum likelihood learning we look at the on line learning algorithm that best approximates the off line algorithm in the sense of least kullback leibler information loss the closest on line algorithm works by updating the weights along the gradient of an effective potential which is different from the parent off line potential a few examples are analyzed and the origin of the potential annealing is discussed

reachability problems for a class of integro differential equations abstract we study control problems for some integro differential equations using the hilbert uniqueness method to do that we follow a harmonic analysis approach our results can be applied to concrete examples in viscoelasticity theory

null controllability of a class of functional differential systems we consider a class of functional differential equations with both discrete and distributed delays distributed delays are represented by integrals whose bounds are commensurate with the discrete delays and by weight functions which are impulse responses of finite dimensional time invariant linear systems the properties of such systems can be studied by using simple algebraic techniques polynomials in one variable under the natural assumption of spectral controllability we give an explicit algebraic algorithm to compute a control function which steers any given initial datum to the equilibrium position in finite time this algorithm also constitutes a direct constructive and simple proof that spectral controllability implies null controllability for the class of systems considered the algorithm requires after the computation of the roots of a polynomial in one variable a finite number of steps

null controllability properties of the wave equation with a second order memory term abstract we study the internal controllability of a wave equation with memory in the principal part defined on the one dimensional torus t r 2 z we assume that the control is acting on an open subset t t which is moving with a constant velocity c r 1 0 1 the main result of the paper shows that the equation is null controllable in a sufficiently large time t and for initial data belonging to suitable sobolev spaces its proof follows from a careful analysis of the spectrum associated with our problem and from the application of the classical moment method

product formulas for exponentials of commutators we provide a recursive method for systematically constructing product formula approximations to exponentials of commutators giving approximations that are accurate to arbitrarily high order using these formulas we show how to approximate unitary exponentials of possibly nested commutators using exponentials of the elementary operators and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance by presenting an algorithm for quantum search using evolution according to a commutator we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal finally we discuss applications of our product formulas to quantum control and to implementing anticommutators providing new methods for simulating many body interaction hamiltonians

the stability and exponential stabilization of the heat equation with memory abstract in this paper we investigate the stability of heat equation with a constant memory kernel we prove that the system does not decay for the positive kernel and is polynomially stable but not exponentially for the negative kernel in particular a more detailed description of the decay property for the negative kernel is obtained by dividing initial values into two categories moreover we show that the system can be exponentially stabilized by boundary control for any constant memory kernel

high order quantum algorithm for solving linear differential equations linear differential equations are ubiquitous in science and engineering quantum computers can simulate quantum systems which are described by a restricted type of linear differential equations here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations which describe many classical physical systems we examine the use of high order methods where the error over a time step is a high power of the size of the time step to improve the efficiency these provide scaling close to t2 in the evolution time t as with other algorithms of this type the solution is encoded in amplitudes of the quantum state and it is possible to extract global features of the solution

performance of the bayesian online algorithm for the perceptron in this letter we derive continuum equations for the generalization error of the bayesian online algorithm bona for the one layer perceptron with a spherical covariance matrix using the rosenblatt potential and show by numerical calculations that the asymptotic performance of the algorithm is the same as the one for the optimal algorithm found by means of variational methods with the added advantage that the bona does not use any inaccessible information during learning

banach lie poisson spaces and reduction the category of banach lie poisson spaces is introduced and studied it is shown that the category of w algebras can be considered as one of its subcategories examples and applications of banach lie poisson spaces to quantization and integration of hamiltonian systems are given the relationship between classical and quantum reduction is discussed

approximate controllability and lack of controllability to zero of the heat equation with memory abstract in this paper we consider the heat equation with memory in a bounded region r d d 1 in the case that the propagation speed of the signal is infinite i e the coleman gurtin model the memory kernel is of class c 1 we examine its controllability properties both under the action of boundary controls and when the controls are distributed in a subregion of we prove approximate controllability of the system and in contrast to this we prove the existence of initial conditions which cannot be steered to hit the target 0 in a certain time t of course when the memory kernel is not identically zero in both the cases we derive our results from well known properties of the memoryless heat equation

integral type approximate controllability of linear parabolic integro differential equations abstract this paper addresses the study of the integral type approximate controllability of linear parabolic integro differential equations this new controllability is defined by imposing some additional integral type constraints on the usual approximate controllability and therefore can be used to keep the state close to the target the paper is concerned with a special choice of integral kernels which are multiples of the same exponential function we reduce the problem of new controllability to the obtention of a unique continuation property for the suitable adjoint system

exact controllability for multidimensional semilinear hyperbolic equations in this paper we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients for this purpose we establish an observability estimate for the linear hyperbolic equation with an unbounded potential in which the crucial observability constant is estimated explicitly by a function of the norm of the potential such an estimate is obtained by a combination of a pointwise estimate and a global carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem

boundary controllability and source reconstruction in a viscoelastic string under external traction abstract treatises on vibrations devote large space to study the dynamical behavior of an elastic system subject to known external tractions since usually a system is part of a chain of mechanisms which disturb the system for example due to the periodic rotation of shafts these disturbances affect the horizontal component of traction in a time dependent way in this paper we shall study control and source identification problems for a viscoelastic string subject to external traction using moment theory

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

fourth order algorithms for solving the multivariable langevin equation and the kramers equation we develop a fourth order simulation algorithm for solving the stochastic langevin equation the method consists of identifying solvable operators in the fokker planck equation factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically a key contribution of this paper is to show how certain double commutators in the factorization process can be simulated in practice the method is general applicable to the multivariable case and systematic with known procedures for doing fourth order factorizations the fourth order convergence of the resulting algorithm allowed very large time steps to be used in simulating the brownian dynamics of 121 yukawa particles in two dimensions the converged result of a first order algorithm can be obtained by using time steps 50 times as large to further demonstrate the versatility of our method we derive two new classes of fourth order algorithms for solving the simpler kramers equation without requiring the derivative of the force the convergence of many fourth order algorithms for solving this equation are compared

on the control of viscoelastic jeffreys fluids abstract this paper is devoted to analyzing the control of vicoelastic fluids of the jeffreys kind also known as oldroyd models we will present the interesting problems with special emphasis in the difficulties that they involve then we will consider appropriate linear approximations and we will establish some partial approximate finite dimensional controllability results in an arbitrarily small time with distributed or boundary controls supported by arbitrarily small sets the proofs rely on some specific unique continuation properties which are implied by the structure of the solutions

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

control problems for weakly coupled systems with memory abstract we investigate control problems for wave petrovsky coupled systems in the presence of memory terms by writing the solutions as fourier series we are able to prove ingham type estimates and hence reachability results our findings have applications in viscoelasticity theory and linear acoustic theory

convergence of general decompositions of exponential operators a general theorem is proved concerning the convergence of decompositions of exponential operators in a banach space or normed space as a corollary the convergence of fractal decompositions is proved the convergence of generalized trotter like formulas is also shown to result from the general theorem

spectral gap amplification many problems can be solved by preparing a specific eigenstate of some hamiltonian h the generic cost of quantum algorithms for these problems is determined by the inverse spectral gap of h for that eigenstate and the cost of evolving with h for some fixed time the goal of spectral gap amplification is to construct a hamiltonian h with the same eigenstate as h but a bigger spectral gap requiring that constant time evolutions with h and h are implemented with nearly the same cost we show that a quadratic spectral gap amplification is possible when h satisfies a frustration free property and give h for these cases this results in quantum speedups for optimization problems it also yields improved constructions for adiabatic simulations of quantum circuits and for the preparation of projected entangled pair states which play an important role in quantum many body physics defining a suitable black box model we establish that the quadratic amplification is optimal for frustration f

wave equations fractional derivatives and a new instance of the lack of robustness of velocity feedbacks due to the attenuation of disturbances under integration it has been proposed that the time derivative of order in the stabilizing feedback of a hyperbolic system be replaced with caputo fractional derivatives of lower order it has already been noted that in this way exponential stability is not preserved we first complement this result we then prove that even dissipativity is not preserved if the caputo fractional derivative has to be numerically computed

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

control of a second order integro differential equation reachability for a second order integro differential equation is proved the method is based upon a new kind of unique continuation property the main significance of the result is that there is no restriction on the size of the memory kernel

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

lack of controllability of the heat equation with memory abstract we consider a model for the heat equation with memory which has infinite propagation speed like the standard heat equation we prove that in spite of this for every t 0 there exist square integrable initial data which cannot be steered to hit zero at time t using square integrable controls we show that the counterexample we present complies with the restrictions imposed by the second principle of thermodynamics

efficient numerical integrators for stochastic models abstract the efficient simulation of models defined in terms of stochastic differential equations sdes depends critically on an efficient integration scheme in this article we investigate under which conditions the integration schemes for general sdes can be derived using the trotter expansion it follows that in the stochastic case some care is required in splitting the stochastic generator we test the trotter integrators on an energy conserving brownian model and derive a new numerical scheme for dissipative particle dynamics we find that the stochastic trotter scheme provides a mathematically correct and easy to use method which should find wide applicability

bracket formulation of diffusion convection equations abstract an abstract diffusion convection equation is introduced it is shown that many well known and experimentally tested time evolution equations of physical systems e g the boltzmann and the enskog kinetic equations and the navier stokes fourier fluid mechanics equations are identical to or well approximated by realizations of the abstract diffusion convection equation the physical content of the abstract diffusion convection equation is compatibility with equilibrium thermodynamics it means that properties of solutions of the abstract diffusion convection equation are mathematical formulations of the experience on which equilibrium thermodynamics is based the requirement that all admissible time evolution equations are realizations of the abstract diffusion convection equation is used to find admissible closures in the bbgky and the maxwell grad hierarchies and admissible constitutive relations in generalized hydrodynamics

nearly optimal lattice simulation by product formulas we consider simulating an n qubit hamiltonian with nearest neighbor interactions evolving for time t on a quantum computer we show that this simulation has gate complexity nt 1 o 1 using product formulas a straightforward approach that has been demonstrated by several experimental groups while it is reasonable to expect this complexity in particular this was claimed without rigorous justification by jordan lee and preskill we are not aware of a straightforward proof our approach is based on an analysis of the local error structure of product formulas as introduced by descombes and thalhammer and significantly simplified here we prove error bounds for canonical product formulas which include well known constructions such as the lie trotter suzuki formulas we also develop a local error representation for time dependent hamiltonian simulation and we discuss generalizations to periodic boundary conditions constant range interactions and higher dimensions combined with a previous lower bound our result implies that product formulas can simulate lattice hamiltonians with nearly optimal gate complexity

general theory of fractal path integrals with applications to many body theories and statistical physics a general scheme of fractal decomposition of exponential operators is presented in any order m namely exp x a b sm x o xm 1 for any positive integer m where sm x et1a et2b et3a et4b etma with finite m depending on m a general recursive scheme of construction of tj is given explicitly it is proven that some of tj should be negative for m 3 and for any finite m nonexistence theorem of positive decomposition general systematic decomposition criterions based on a new type of time ordering are also formulated the decomposition exp x a b sm x n n o xm 1 nm yields a new efficient approach to quantum monte carlo simulations

interior approximate and null controllability of the heat equation with memory in this paper we study the approximate and null controllability of the heat equation with memory we mainly consider the distributed control case with an arbitrarily small support of the control function for each positive final time t we deduce that the approximate controllability property holds and the null controllability property fails

controllability and observability of a heat equation with hyperbolic memory kernel abstract the exact controllability and observability for a heat equation with hyperbolic memory kernel in anisotropic and nonhomogeneous media are considered due to the appearance of such a kind of memory the speed of propagation for solutions to the heat equation is finite and the corresponding controllability property has a certain nature similar to hyperbolic equations and is significantly different from that of the usual parabolic equations by means of carleman estimate we establish a positive controllability and observability result under some geometric condition on the other hand by a careful construction of highly concentrated approximate solutions to hyperbolic equations with memory we present a negative controllability and observability result when the geometric condition is not satisfied

essential equivalence of the general equation for the nonequilibrium reversible irreversible coupling generic and steepest entropy ascent models of dissipation for nonequilibrium thermodynamics by reformulating the steepest entropy ascent sea dynamical model for nonequilibrium thermodynamics in the mathematical language of differential geometry we compare it with the primitive formulation of the general equation for the nonequilibrium reversible irreversible coupling generic model and discuss the main technical differences of the two approaches in both dynamical models the description of dissipation is of the entropy gradient type sea focuses only on the dissipative i e entropy generating component of the time evolution chooses a sub riemannian metric tensor as dissipative structure and uses the local entropy density field as potential generic emphasizes the coupling between the dissipative and nondissipative components of the time evolution chooses two compatible degenerate structures poisson and degenerate co riemannian and uses the global energy and entropy functionals as potentials as an illustration we rewrite the known generic formulation of the boltzmann equation in terms of the square root of the distribution function adopted by the sea formulation we then provide a formal proof that in more general frameworks whenever all degeneracies in the generic framework are related to conservation laws the sea and generic models of the dissipative component of the dynamics are essentially interchangeable provided of course they assume the same kinematics as part of the discussion we note that equipping the dissipative structure of generic with the leibniz identity makes it automatically sea on metric leaves

higher order decompositions of ordered operator exponentials we present a decomposition scheme based on lie trotter suzuki product formulae to approximate an ordered operator exponential with a product of ordinary operator exponentials we show using a counterexample that lie trotter suzuki approximations may be of a lower order than expected when applied to problems that have singularities or discontinuous derivatives of appropriate order to address this problem we present a set of criteria that is sufficient for the validity of these approximations prove convergence and provide upper bounds on the approximation error this work may shed light on why related product formulae fail to be as accurate as expected when applied to coulomb potentials

self averaging and on line learning conditions are given under which one may prove that the stochastic dynamics of on line learning can be described by the deterministic evolution of a finite set of order parameters in the thermodynamic limit a global constraint on the average magnitude of the increments in the stochastic process is necessary to ensure self averaging in the absence of such a constraint convergence may only be in probability

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

the rosenblatt bayesian algorithm learning in a nonstationary environment in this letter we study online learning in neural networks nns obtained by approximating bayesian learning the approach is applied to gibbs learning with the rosenblatt potential in a nonstationary environment the online scheme is obtained by the minimization maximization of the kullback leibler divergence cross entropy between the true posterior distribution and the parameterized one the complexity of the learning algorithm is further decreased by projecting the posterior onto a gaussian distribution and imposing a spherical covariance matrix we study in detail the particular case of learning linearly separable rules in the case of a fixed rule we observe an asymptotic generalization error egpropalpha 1 for both the spherical and the full covariance matrix approximations however in the case of drifting rule only the full covariance matrix algorithm shows a good performance this good performance is indeed a surprise since the algorithm is obtained by projecting without the benefit of the extra information on drifting

optimal generalization in perceptions a new learning algorithm for the one layer perceptron is presented it aims to maximize the generalization gain per example analytical results are obtained for the case of single presentation of each example the weight attached to a hebbian term is a function of the expected stability of the example in the teacher perceptron this leads to the obtention of upper bounds for the generalization ability this scheme can be iterated and the results of numerical simulations show that it converges within errors to the theoretical optimal generalization ability of the bayes algorithm analytical and numerical results for an algorithm with maximized generalization in the learning strategy with selection of examples are obtained and it is proved that as expected orthogonal selection is optimal exponential decay of the generalization error is obtained for the single presentation of selected examples

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

efficient quantum algorithms for simulating sparse hamiltonians we present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse hamiltonian h over a given time t in terms of a procedure for computing the matrix entries of h in particular when h acts on n qubits has at most a constant number of nonzero entries in each row column and h is bounded by a constant we may select any positive integer k such that the simulation requires o log n t1 1 2k accesses to matrix entries of h we also show that the temporal scaling cannot be significantly improved beyond this because sublinear time scaling is not possible

on the null controllability of heat equation with memory abstract the aim of this article is to analyze the null controllability properties of the heat equations with memory by a new carleman inequality with weighted functions that do not vanish at t 0 we prove the controllability for the system with a located internal controller under a condition on the kernel

generalization of the dynamical lack of fit reduction from generic to generic the lack of fit statistical reduction developed and formulated first by bruce turkington is a general method taking liouville equation for probability density detailed level and transforming it to reduced dynamics of projected quantities less detailed level in this paper the method is generalized the hamiltonian liouville equation is replaced by an arbitrary hamiltonian evolution combined with gradient dynamics generic the boltzmann entropy is replaced by an arbitrary entropy and the kinetic energy by an arbitrary energy the gradient part is a generalized gradient dynamics generated by a dissipation potential the reduced evolution of the projected state variables is shown to preserve the generic structure of the original detailed level evolution the dissipation potential is obtained by solving a hamilton jacobi equation in summary the lack of fit reduction can start with generic and obtain generic for the reduced state variables

a trotter suzuki approximation for lie groups with applications to hamiltonian simulation we present a product formula to approximate the exponential of a skew hermitian operator that is a sum of generators of a lie algebra the number of terms in the product depends on the structure factors when the generators have large norm with respect to the dimension of the lie algebra or when the norm of the effective operator resulting from nested commutators is less than the product of the norms the number of terms in the product is significantly less than that obtained from well known results we apply our results to construct product formulas useful for the quantum simulation of some continuous variable and bosonic physical systems including systems whose potential is not quadratic for many of these systems we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local hilbert spaces where such a dimension is usually determined by the energy scale of the problem our results emphasize the power of quantum computers for the simulat

admissibility of the observation operators for volterra systems with exponential kernels we study admissibility of the observation operators for volterra systems in hilbert spaces with exponentially decaying kernels sufficient conditions for infinite time admissibility of the observation operators are established and some illustrating examples are given

stabilization of semilinear heat equations with fading memory by boundary feedbacks abstract in this paper we design stabilizing dirichlet boundary feedback laws for heat equations with fading memory the linear finite dimensional controllers are easily manageable from the computational point of view because of their simple feedback form involving only the first n n eigenfunctions of the laplace operator two examples are provided at the end of the paper in order to illustrate the acquired results

boundary controllability and observability of a viscoelastic string in this paper we consider an integrodifferential system which governs the vibration of a viscoelastic one dimensional object we assume that we can act on the system at the boundary and we prove that it is possible to control both the position and the velocity at every point of the body and at a certain time t large enough we shall prove this result using moment theory and we shall prove that the solution of this problem leads to identification of a riesz sequence which solves controllability and observability the results presented here are constructive and can lead to simple numerical algorithms