characteristic of left invertible semigroups and admissibility of observation operators in this paper we discuss the characteristic property of the left invertible semigroups on general banach spaces and admissibility of the observation operators for such semigroups we obtain a sufficient and necessary condition about their generators further for the left invertible and exponentially stable semigroup in hilbert space we show that there is an equivalent norm under which it is contractive based on these results we prove that for any observation operator satisfying the resolvent condition is admissible for the left invertible semigroup if its range is finite dimensional in addition we prove that any observation operator satisfying the resolvent condition can be approximated by the admissible observation operators finally we give a sufficient condition of exact observability of the left invertible semigroup

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

mean square power series solution of random linear differential equations this paper deals with the construction of random power series solutions of linear differential equations containing uncertainty through the diffusion coefficient the source term as well as the initial condition under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent in a certain interval whose length depends on the mean square norm of the random variable coefficient also the main statistical functions of the approximating stochastic process solution generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples

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

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

sufficient conditions for admissibility sufficient conditions for the finite and infinite time admissibility of an observation operator are given it is shown that the estimates of weiss are close to being sufficient if the semigroup is surjective then the estimate is sufficient

l infty exact observability of the heat equationwith scanning pointwise sensor the problem of exact observability of the heat equation in an arbitrary space dimension with scanning pointwise sensor is considered in the case when the space for outputs is l infty varepsilon theta varepsilon 0 a new method for the construction of observation curves for sensors that are able to ensure l infty varepsilon theta exact observability at final time is given based on the maximum principle for the heat equation an application of the method to the observability problem with discrete time scanning observations and related approximate controllability results are also discussed

the approximate solutions of some stochastic differential equations using transformations in this paper the transformation technique together with a numerical technique is used to obtain an approximate probability density function pdf for the solution process of some stochastic differential equations the approach is applied on some first order and second order differential equations with making comparisons with the exact solution if possible

admissibility of control operators for solution families to volterra integral equations we reduce the admissibility of finite dimensional control operators for an evolution system satisfying certain mild regularity conditions to that for a semigroup control system for which there are plenty of results available in the literature one such necessary and sufficient condition is the carleson measure criterion we prove this condition explicitly for evolution systems without the regularity assumption mentioned above we also look at some examples which show that our respective hypotheses are necessary

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

error estimates for inconsistent load lumping approach in finite element solution of differential equations the finite element method fem is very widely used for solving differential equations for large problem sizes computational cost becomes a very important factor in fem implementation stiffness matrix and load vector calculations requires numerical integration to be carried out and is generally done using gaussian quadrature gq however the exact load vector cannot always be obtained using gq especially when the forcing is complicated or periodic in nature hence an approximate load vector is generally employed which can be obtained using the inconsistent load lumping ill approach however the ill approach increases the error in the solution in the present work it is shown via both numerical experiments and a pseudo analytical proof in the case of constant coefficient differential equations that the order of the additional error due to ill is greater than or equal to the discretization error in the finite element solution

a remark on null exact controllability of the heat equation it is well known that the heat equation u t delta u f chi omega in 0 t times omega with homogeneous dirichlet boundary conditions is null exactly controllable for any t 0 and any open nonempty subset omega of omega in this note we show that this property may be obtained as a singular limit of the exact controllability properties of singularly perturbed damped wave equations with a changing controller

second order linear differential equations with analytic uncertainties stochastic analysis via the computation of the probability density function abstract this paper concerns the analysis of random second order linear differential equations usually solving these equations consists of computing the first statistics of the response process and that task has been an essential goal in the literature a more ambitious objective is the computation of the solution probability density function we present advances on these two aspects in the case of general random non autonomous second order linear differential equations with analytic data processes the frobenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series we demonstrate that the convergence requires the boundedness of the random input coefficients further the mean square error of the frobenius method is proved to decrease exponentially with the number of terms in the series although not uniformly in time regarding the probability density function of the solution at a given time which is the focus of the paper we rely on the law of total probability to express it in closed form as an expectation for the computation of this expectation a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set we prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation the pointwise convergence turns out to be exponential under a lipschitz hypothesis as the density functions are expressed in terms of expectations we propose a crude monte carlo sampling algorithm for their estimation this algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper after that the efficiency of the algorithm is improved by employing the control variates method numerical examples corroborate the variance reduction of the monte carlo approach

solution of the stochastic radiative transfer equation with rayleigh scattering using rvt technique abstract this paper considers the solution of the radiative transfer equation in a semi infinite continuous stochastic medium with rayleigh scattering the available solutions in literature for this stochastic integro differential equation side are represented only by the ensemble averaged solutions here the deterministic solution of the problem is obtained at first then the random variable transformation rvt technique together with an integral transformation to the input stochastic process is implemented to get the complete stochastic solution of the problem this solution is represented by the probability density function of the solution process moreover the nth statistical moment of the solution is derived numerical results are given for exponential and gaussian distributions of the input stochastic process also these results are given for different boundary conditions

skew product semi flows and non autonomous control systems in this paper we give a necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite dimensional non autonomous control systems this is done by employing skew product semi flow technique finally we apply these results to prove the controllability of a broad class of non autonomous reaction diffusion equations in hilbert spaces

on the generalization of probabilistic transformation method abstract the probabilistic transformation method with the finite element analysis is a new technique to solve random differential equation the advantage of this technique is finding the exact expression of the probability density function of the solution when the probability density function of the input is known however the disadvantage is due to the characteristics geometrics and materials of the analyzed structure included in the random differential equation in this paper a developed formula is used to generalize this technique by obtaining the exact joint probability density function of the solution in any situations as well as the proposed technique for the non linear case

riesz bases and exact controllability of c0 groups with one dimensional input operators abstract this paper considers linear infinite dimensional systems with c 0 group generators and one dimensional admissible input operators the exact controllability and riesz basis generation property are discussed the corresponding results of jacob and zwart advances in mathematical systems theory birkhauser boston ma 2000 under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity

solution of the stochastic transport equation of neutral particles with anisotropic scattering using rvt technique much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation sde the random variable transformation technique rvt is a powerful technique to get the complete solution for the sde represented by the probability density function of the solution process in this paper the rvt technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one speed transport equation for neutral particles in a semi infinite stochastic medium with linear anisotropic scattering the extinction function of the medium input stochastic process is assumed to be a continuous random function of position the probability density function and hence the higher order statistical moments of the solution process are presented numerical results are given for different distributions of the input stochastic process

admissible input elements for systems in hilbert space and a carleson measure criterion we study the control system dot x ax bu quad x in x quad u text scalar where a generates a semigroup on the hilbert space x but in general the control input element b notin x many boundary value control systems point control force situations etc can be studied in this context we define and analyze admissible input elements b and develop sufficient conditions for b to be admissible in terms of the carleson measure theorem of h p theory

probabilistic solution of the homogeneous riccati differential equation this paper deals with the determination of the first probability density function of the solution stochastic process to the homogeneous riccati differential equation taking advantage of both linearization and random variable transformation techniques the study is split in all possible casuistries regarding the deterministic random character of the involved input parameters an illustrative example is provided for each one of the considered cases

controllability of laguerre and jacobi equations in this paper we study the controllability of the controlled laguerre equation and the controlled jacobi equation for each case we find conditions which guarantee when such systems are approximately controllable on the interval 0 t 1 moreover we show that these systems can never be exactly controllable

computing probabilistic solutions of the bernoulli random differential equation the random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of the solution stochastic process in this paper that technique is applied to construct a closed form expression of the solution for the bernoulli random differential equation in order to account for the general scenario all the input parameters coefficients and initial condition are assumed to be absolutely continuous random variables with an arbitrary joint probability density function the analysis is split into two cases for which an illustrative example is provided finally a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data

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

a 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

random hermite differential equations mean square power series solutions and statistical properties abstract this paper deals with the construction of random power series solution of second order linear differential equations of hermite containing uncertainty through its coefficients and initial conditions under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent we provide conditions in order to obtain random polynomial solutions and as a consequence random hermite polynomial are introduced also the main statistical functions of the approximate stochastic process solution generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples comparing the numerical results with respect to those provided by other available approaches including monte carlo simulation

random differential operational calculus theory and applications in this article we obtain a product rule and a chain rule for mean square derivatives an application of the chain rule to the mean square solution of random differential equations is shown however to achieve such mean square differentiation rules fourth order properties were needed and therefore we first studied a mean fourth order differential and integral calculus results are applied to solve random linear variable coefficient differential problems

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

a proposed technique of sfem on solving ordinary random differential equation the stochastic finite element sfem is a new method to solve random differential equation using the known deterministic finite element technique fem adapted to stochastic techniques for solving stochastic differential equations in this paper a proposed technique of fem and random variable transformation rvt is used to solve a differential equation with random excitation the technique shows high accuracy when solving a case study compared with the exact solution

admissibility and controllability of diagonal volterra equations with scalar inputs this article studies volterra evolution equations from the point of view of control theory in the case that the generator of the underlying semigroup has a riesz basis of eigenvectors conditions for admissibility of the system s control operator are given in terms of the carleson embedding properties of certain discrete measures moreover exact and null controllability are expressed in terms of a new interpolation question for analytic functions providing a generalization of results known to hold for the standard cauchy problem the results are illustrated by examples involving heat conduction with memory

the probability density function to the random linear transport equation we present a formula to calculate the probability density function of the solution of the random linear transport equation in terms of the density functions of the velocity and the initial condition we also present an expression for the joint probability density function of the solution in two different points our results have shown good agreement with monte carlo simulations

a developed solution of the stochastic milne problem using probabilistic transformations abstract this paper considers the solution of the milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium properties of the medium are assumed to be continuous random functions of the spatial dimensions the available solutions in literature for this stochastic integro differential equation side are represented only by the ensemble average of the radiant energy density in this paper a developed algorithm based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties is introduced a complete stochastic solution represented by the probability density function p d f of the radiant energy density is obtained using the closed form of the p d f the nth moment of the stochastic solution is evaluated in realization of this work exponential and gaussian statistics for the medium properties are assumed results are physically acceptable and found to be compatible with those in the literature

approximate controllability of three dimensional navier stokes equations the paper is devoted to studying the problem of controllability for 3d navier stokes equations in a bounded domain we develop the method introduced by agrachev and sarychev in the 2d case and establish a sufficient condition under which the problem in question is approximately controllable by a finite dimensional force in the particular case of a torus it is shown that our sufficient condition is fulfilled for a control of low dimension not depending on the viscosity

a general necessary condition for exact observability suppose a generates an exponentially stable strongly continuous semigroup on the hilbert space x y is another hilbert space and c d a rightarrow y is an admissible observation operator for this semigroup this means that to any initial state in x we can associate an output function in l 2 0 infty y this paper gives a necessary condition for the exact observability of the system defined by a and c this condition called bf e is related to the hautus lemma from finite dimensional systems theory it is an estimate in terms of the operators a and c alone in particular it makes no reference to the semigroup this paper shows that bf e implies approximate observability and if a is bounded it implies exact observability this paper conjectures that bf e is in fact equivalent to exact observability the paper also shows that for diagonal semigroups bf e takes on a very simple form and applies the results to sequences of complex exponential functions

an approximate probabilistic solution of a random sir type epidemiological model using rvt technique abstract this paper adapts a multi dimensional random variable transformation technique to derive a comprehensive stochastic description of the susceptible infected recovered epidemiological model an approximate solution of a system of nonlinear differential equations that characterizes this model is deterministically obtained and from this approximation we derive the first probability density functions for the solution processes of susceptible infected and recovered percentages these probability density functions are used to find the approximate mean and variance functions as well as the confidence intervals taking a general situation the infection contact rate the recovery rate and the initial conditions are taken to be random variables with arbitrary distributions to test the validity of the theoretical findings associated to the proposed random epidemiological model some numerical results are tabulated and graphically presented through an illustrative example

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

polynomial chaos for random fractional order differential equations fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models where such effects are difficult to incorporate and often neglected also many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations thus to be able to deal with many real life problems it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty the aim of this paper is to study the application of the generalized polynomial chaos gpc to random fractional ordinary differential equations the method of polynomial chaos has played an increasingly important role when dealing with uncertainties the main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses however to apply polynomial chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations in order to avoid more complex numerical computations and obtain accurate solutions we rely on richardson extrapolation it is shown that the application of generalized polynomial chaos method in conjunction with richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations

solving second order linear differential equations with random analytic coefficients about ordinary points a full probabilistic solution by the first probability density function abstract this paper deals with the approximate computation of the first probability density function of the solution stochastic process to second order linear differential equations with random analytic coefficients about ordinary points under very general hypotheses our approach is based on considering approximations of the solution stochastic process via truncated power series solution obtained from the random regular power series method together with the so called random variable transformation technique the validity of the proposed method is shown through several illustrative examples

topological crossovers in the forced folding of self avoiding matter abstract we study the scaling properties of forced folding of thin materials of different geometry the scaling relations implying the topological crossovers from the folding of three dimensional plates to the folding of two dimensional sheets and further to the packing of one dimensional strings are derived for elastic and plastic manifolds these topological crossovers in the folding of plastic manifolds were observed in experiments with predominantly plastic aluminum strips of different geometry elasto plastic materials such as paper sheets during the fast folding under increasing confinement force are expected to obey the scaling force diameter relation derived for elastic manifolds however in experiments with paper strips of different geometry we observed the crossover from packing of one dimensional strings to folding two dimensional sheets only because the fractal dimension of the set of folded elasto plastic sheets is the thickness dependent due to the strain relaxation after a confinement force is withdrawn

a resolvent test for admissibility of volterra observation operators necessary and sufficient conditions are given for finite time admissibility of a linear system defined by a volterra integral equation when the underlying semigroup is equivalent to a contraction semigroup in terms of a pointwise bound on the resolvent of the infinitesimal generator this generalizes an analogous result known to hold for the standard cauchy problem for infinite time admissibility however it is shown by means of an example that the natural generalization of the weiss resolvent test is no longer valid

using fem rvt technique for solving a randomly excited ordinary differential equation with a random operator the technique of stochastic finite element sfem which is the finite element technique fem adapted to stochastic problems can be re described to use random variable transformation technique rvt a new fem rvt technique was successfully used in solving stochastic problems with random excitation m el tawil w el tahhan a hussein a proposed technique of sfem on solving ordinary random differential equation j appl math comput 161 2005 35 47 in this paper the technique is adapted to solve a randomly excited differential equation with a random operator the technique shows high accuracy when solving a case study compared with the exact solution finally a problem with unknown exact solution is solved using this technique

stochastic beam equations under random dynamic loads the fourth order partial differential equation representing beams under random loading is considered a general solution for this equation is obtained using the eigenfunction and variation of parameters techniques also the average and the variance of the beam deflection shear and bending moment are obtained the load is divided into a deterministic function and a randomly perturbed function representing the expected error in the deterministic load the general closed form solution is obtained in stochastic integral terms some important statistical moments of the solution process are computed and illustrated

weak admissibility does not imply admissibility for analytic semigroups two conjectures on admissible control operators by george weiss are disproved in this paper one conjecture says that an operator b defined on an infinite dimensional hilbert space u is an admissible control operator if for every element u in u the vector bu defines an admissible control operator the other conjecture says that b is an admissible control operator if a certain resolvent estimate is satisfied the examples given in this paper show that even for analytic semigroups the conjectures do not hold in the last section we construct a semigroup example showing that the first estimate in the hille yosida theorem is not sufficient to conclude boundedness of the semigroup

random surfaces on hard spheres the relation between mass m and size r for self avoiding random surfaces enclosing a hard sphere of radius r is examined it is suggested that m approximately r2f r r r with f proportional to the surface roughness these functions are investigated using experimental data and theoretical arguments

necessary conditions for exact controllability with a finite dimensional input space we derive necessary conditions for exact controllability of infinite dimensional systems described by x ax bu where the generator a has a riesz basis of eigenvectors and the input space is finite dimensional these conditions are in terms of the eigenvalues of a and the degree of unboundedness of b but no further information about b is needed our results easily imply lack of exact controllability for a wide range of distributed parameter systems it is known from partial differential equation pde examples that the addition of damping into an exactly controllable second order system can destroy exact controllability we discuss the effect of damping on such systems using our general conditions we also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension

the operator carleson measure criterion for admissibility of control operators for diagonal semigroups on 1 2 abstract suppose a control system is described by a diagonal semigroup on the state space l 2 and by an unbounded control operator b defined on the input space l 2 we formulate a condition called the operator carleson measure criterion and show that this condition is necessary for the admissibility of b if the semigroup is analytic or invertible then the condition is sufficient as well our results extend the ones of ho russell and weiss concerning the case where the input space is one dimensional

random matrix difference models arising in long term medical drug strategies abstract this paper deals with the construction of random discrete solutions of coupled linear difference equations incorporating uncertainty into both the initial condition and the source term first sufficient conditions in order to guarantee mean square stability of the solution are provided then the main statistical functions such as mean and covariance of the discrete solution stochastic process are given finally illustrative examples of potential interest in long time medical drug strategies are shown

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

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

solving initial and two point boundary value linear random differential equations a mean square approach abstract this paper deals with the construction of mean square real valued solutions to both initial and boundary value problems of linear differential equations whose coefficients are assumed to be stochastic processes and initial and boundary conditions are random variables a key result to conduct our study is the extension of the leibniz integral rule to the random framework taking advantage of the so called random fourth calculus exact expressions for the main statistical functions average and variance associated to the solutions to both problems are also provided illustrative examples computing the average and standard deviation are included

the crumpled state of some non equilibrium fractal surfaces the authors study how the three dimensional air or pythagorean distance r q q between two points q and q on a non equilibrium crumpled fractal surface cs with the topology of the plane transforms in the internal or geodesic distance x q q with probability p x r after the unfolding of the cs on a plane the probability distribution p x r governing this process is examined for the first time among other results they find that 1 the width of p x r diverges for r near the ensemble average radius r of the cs and 2 x approximately r13

exact observability of diagonal systems with a finite dimensional output operator in this paper equivalent conditions for exact observability of diagonal systems with a finite dimensional output operator are given one of these equivalent conditions is the conjecture of russell and weiss siam j control opt 32 1 1994 1 23 the other conditions are in terms of the eigenvalues and the lyapunov solutions of finite dimensional subsystems

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

random analytic solution of coupled differential models with uncertain initial condition and source term this paper deals with the construction of random power series solution of vector initial value problems containing uncertainty in both initial condition and source term under appropriate hypothesis on the data we prove that the random series solution constructed by a random frobenius method is convergent in the mean square sense also the main statistical functions of the approximating stochastic process solution generated by truncation of the exact series solution are given finally we apply the proposed technique to several illustrative examples

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

analytic numerical approximating processes of diffusion equation with data uncertainty this paper deals with the construction of approximating processes for solving mixed problems related to the diffusion equation where the initial condition is assumed to be a stochastic process and the diffusion coefficient is a random variable expectation and standard deviation of the approximation processes are given

controllability of 2d euler and navier stokes equations by degenerate forcing we study controllability issues for the 2d euler and navier stokes ns systems under periodic boundary conditions these systems describe the motion of the homogeneous ideal or viscous incompressible fluid on a two dimensional torus mathbb t 2 we assume the system to be controlled by a degenerate forcing applied to a fixed number of modes

construction and computation of variable coefficient sylvester differential problems abstract in this paper initial value problems for sylvester differential equations x t a t x t x t b t f t with analytic matrix coefficients are considered first an exact series solution of the problem is obtained given a bounded domain and an admissible error a finite analytic numerical series solution is constructed so that the error with respect to the exact series solution is uniformly upper bounded by in an iterative procedure for the construction of the approximate solutions is included

counterexamples concerning observation operators for c0 semigroups this paper concerns systems of the form dot x t ax t y t cx t where a generates a c 0 semigroup two conjectures which were posed in 1991 and 1994 are shown not to hold the first conjecture by g weiss states that if the range of c is one dimensional then c is admissible if and only if a certain resolvent estimate holds the second conjecture by d russell and g weiss states that a system is exactly observable if and only if a test similar to the hautus test for finite dimensional systems holds the c 0 semigroup in both counterexamples is analytic and possesses a basis of eigenfunctions using the a c pair from the second counterexample we construct a generator a e on a hilbert space such that si a e is uniformly left invertible but its semigroup does not have this property

necessary condition of linear distributed parameter systems with exact controllability abstract in this paper we studied the necessary condition of distributed parameter system with exact controllability let 0 t be the control mapping we introduce new a class of control operators that is called the i class control which satisfy r 0 t r t t is closed set for t 0 if the system is exactly controllable in finite time then the semigroup t t must have closed range in particular if the generator of t t has compact resolvent its spectra distribute in a strip parallel to imaginary axis as an application we assert that the systems associated with the immediately norm continuous semigroups are never exact controllable for zero class or i class controls

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

a zabczyk type method for the study of the exponential trichotomy of discrete dynamical systems the aim of this paper is to present new discrete time characterizations for uniform exponential trichotomy of dynamical systems we obtain necessary and sufficient conditions for the existence of uniform exponential trichotomy of discrete dynamical systems using a set of computational conditions of zabczyk type the main results are applied to deduce new criteria for the detection of the uniform exponential trichotomy of dynamical systems modeled by skew product flows

random variable transformation for generalized stochastic radiative transfer in finite participating slab media abstract the stochastic radiative transfer problem is studied in a participating planar finite continuously fluctuating medium the problem is considered for specular and diffusly reflecting boundaries with linear anisotropic scattering random variable transformation rvt technique is used to get the complete average for the solution functions that are represented by the probability density function pdf of the solution process in the rvt algorithm a simple integral transformation to the input stochastic process the extinction function of the medium is applied this linear transformation enables us to rewrite the stochastic transport equations in terms of the optical random variable x and the optical random thickness l then the transport equation is solved deterministically to get a closed form for the solution as a function of x and l so the solution is used to obtain the pdf of the solution functions applying the rvt technique among the input random variable l and the output process the solution functions the obtained averages of the solution functions are used to get the complete analytical averages for some interesting physical quantities namely reflectivity and transmissivity at the medium boundaries in terms of the average reflectivity and transmissivity the average of the partial heat fluxes for the generalized problem with internal source of radiation are obtained and represented graphically

state reconstruction of a distributed parameter system with exact observability abstract we study the problem of state reconstruction for exactly observable systems here we present two approaches for state reconstruction the riesz basis approach and the dual hilbert uniqueness method hum based on an iteration algorithm the riesz basis approach gives an explicit representation of the initial state of the system whereas the hum dual method gives an efficient approximation of the initial state

geometrical and electrical properties of crumpled wires geometrical and electrical properties of crumpled wires are investigated power laws connecting the geometrical and electrical properties of these random systems are studied and critical exponents are calculated in particular the authors have obtained the resistance exponent zeta which seems to be compatible with a previous conjecture

some recommendations for applying gpc generalized polynomial chaos to modeling an analysis through the airy random differential equation in this paper we study the use of the generalized polynomial chaos method to differential equations describing a model that depends on more than one random input this random input can be in the form of parameters or of initial or boundary conditions we investigate the effect of the choice of the probability density functions for the inputs on the output stochastic processes the study is performed on the airy s differential equation this equation is a good test case since its solutions are highly oscillatory and errors can develop both in the amplitude and the phase several different situations are considered and finally conclusions are presented

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

random airy type differential equations mean square exact and numerical solutions this paper deals with the construction of power series solutions of random airy type differential equations containing uncertainty through the coefficients as well as the initial conditions under appropriate hypotheses on the data we establish that the constructed random power series solution is mean square convergent over the whole real line in addition the main statistical functions such as the mean and the variance of the approximate solution stochastic process generated by truncation of the exact power series solution are given finally we apply the proposed technique to several illustrative examples which show substantial speed up and improvement in accuracy compared to other approaches such as monte carlo simulations