nodal sets and morse indices of solutions of super linear elliptic pdes in this paper we develop a sturm liouville type theory for the nodal sets and morse indices of solutions of super linear elliptic pdes with dirichlet boundary condition it shows that there are some relationships between analytic properties e g lp norm vanishing order of the nodal point andn 1 dimensional hausdorff measure of the nodal set of the solutions as the functions on the domain and morse indices of the solutions as the critical points of the functionalj u 12 u 2 f x u dxonh10

carleman inequalities for the dirac and laplace operators and unique continuation on demontre les meilleures inegalites de type carleman possibles dans le cas de l operateur de dirac on obtient la propriete de prolongement unique pour d v ou v l loc r n 3n 2 2 n 3

an approximate numerical method for the complex eigenproblem in systems characterised by a structural damping matrix this paper presents an efficient numerical method that approximates the complex eigenvalues and eigenvectors in structural systems with viscoelastic damping materials characterised by the complex structural damping matrix this new method begins from the solution of the undamped system and approximates the complex eigenpair by finite increments using the eigenvector derivatives and the rayleigh quotient it is implemented in three different approaches single step incremental and iterative schemes the single step technique is presented for systems with low and medium damping from numerical examples it can be verified that the errors committed by the approximate single step technique with respect to the exact solutions obtained by the iram method are less than 0 2 when the loss factor of the material damping is lower than 1 this is a considerable improvement on other approximate methods for higher damped systems an improvement to the previous approach is proposed by an incremental technique that keeps the accuracy without significantly increasing the computational time the complex eigenproblem for materials whose mechanical properties are dependent on frequency is solved by a fast iterative approach whose validity is proved using a four parameter fractional derivative model

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

efficient modal analysis of systems with local stiffness uncertainties the characterization of the uncertainty in modal quantities of an uncertain linear structural system is essential to the rapid determination of its response to arbitrary loadings although the size of many computational structural models used is extremely large i e thousands of equations the uncertainty to be analyzed is oftentimes localized to very small regions of the model this paper addresses the development of an efficient computational methodology for the modal analysis of linear structural systems with local stiffness uncertainties the newly developed methodology utilizes an enriched basis that consists of the sub spectrum of a nominal structural system augmented with additional basis vectors generated from a knowledge of the structure of the stiffness uncertainty in addition methods for determining bounds on the approximate modal frequencies and mode shapes are discussed numerical results demonstrate that the algorithm produces highly accurate results with greatly reduced computational effort copyright 2009 john wiley sons ltd

on predicting the response of non conservative linear vibrating systems by using dynamical matrix solutions abstract formulae are obtained for determining the response of non conservative linear systems without any particular assumption about the nature of the matrix coefficients multiplicity of eigenvalues or external forcing the results apply equally to passive or active systems this is accomplished in terms of the dynamical matrix solution of the given system this solution corresponds to a zero initial matrix displacement and a unit matrix impulse when in standard form moreover it is directly related to the receptance or transfer matrix of the system

comparison and validation of multi phase closure models the purpose of this paper is to propose a simple one or two parameter multiphase flow model suitable for the description of rayleigh taylor and richtmyer meshkov mixing layers we justify model closure assumptions in this model by comparison to rayleigh taylor and richtmyer meshkov simulation data we show that the relative errors related to model closure terms are about 10 and are about two to four times smaller than related closure models of abgrall and saurel

a quantitative boundary unique continuation for stochastic parabolic equations abstract this paper is addressed to the boundary unique continuation property for forward stochastic parabolic equations that is to determine the value of the solution by virtue of the observation on an arbitrary open subset of the boundary by means of a global carleman estimate we establish a quantitative version of this property

sensitivities of frequency response functions of linear dynamic systems to variations in design parameter values abstract expressions for the sensitivities of the frequency response functions of linear constant coefficient systems with distinct eigenvalues to perturbations in the values of their design parameters are derived the expressions are of each of two forms the one involving complex number phasor representation of the responses and the sensitivities the other involving separate amplitude and phase representations the relations derived contain eigenvalue eigenvector and adjoint eigenvector sensitivities engineering applications of the theory are suggested

carleman inequalities and the heat operator we prove a strong unique continuation theorem for functions u vanishing to infinite order in the space time variables at 0 0 and verifying the inequality 4u tu v x t u on rn 0 t when t1 n 2r 1 s v x t lt locl r x n 2r 1 s 1 n 2 r

on the modal analysis of non conservative linear systems this paper applies the results of classical eigenvalue theory to the analysis of a non conservative vibratory system general results are obtained for the response of a multiple degree of freedom system which is represented by a set of linear second order differential equations with constant coefficients for the case when the eigenvalues of the system are distinct these results confirm those obtained by 1 fawzy and bishop by a different method the latter results are restricted to distinct eigenvalues but the more general analysis presented here allows the case of multiple eigenvalues to be included the physical interpretation of complex normal co ordinate vectors is discussed and some simple numerical examples illustrate the results the approach described here lends itself to numerical computation using computer library programs for eigenvalue determination

continuous dependence of optimal control to controlled domain of actuator for heat equation abstract in this paper we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi dimensional heat equation both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval and visa versa it is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the lebesgue measure

impulse output feedback stabilization of fisher s equation abstract an explicit impulse output feedback controller has been designed in phung 2016 for stabilization of the linear heat equation the aim of this work is to show that the feedback controller of the same form as in phung 2016 can locally exponentially stabilize the equilibrium solutions to fisher s equation in l 2 0 1

minimum time and minimum energy for linear control systems abstract in this paper we give estimates concerning regularity properties of the minimum time function and the minimum norm controls involving the controllability function given by the observability inequalities we further study the continuity of the minimum time function in two variables the initial point and the radius of the ball that limits the control we prove a maximum principle for both time and norm optimal control problems

discrete equations for physical and numerical compressible multiphase mixtures we have recently proposed in 21 a compressible two phase unconditionally hyperbolic model able to deal with a wide range of applications interfaces between compressible materials shock waves in condensed multiphase mixtures homogeneous two phase flows bubbly and droplet flows and cavitation in liquids one of the difficulties of the model as always in this type of physical problems was the occurrence of non conservative products in 21 we have proposed a discretisation technique that was without any ambiguity only in the case of material interfaces not in the case of shock waves this model was extended to several space dimensions in 24 in this paper thanks to a deeper analysis of the model we propose a class of schemes that are able to converge to the correct solution even when shock waves interact with volume fraction discontinuities this analysis provides a more accurate estimate of closure terms but also an accurate resolution method for the conservative fluxes as well as non conservative terms even for situations involving discontinuous solutions the accuracy of the model and method is clearly demonstrated on a sequence of difficult test problems

quantitative estimates of unique continuation for parabolic equations and inverse initial boundary value problems with unknown boundaries in this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations we apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain in r n from the knowledge of overdetermined boundary data for parabolic boundary value problems

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

dynamical evolution of rayleigh taylor and richtmyer meshkov mixing fronts dynamic behavior of mixing fronts plays a crucial role in multifluid turbulent mixing in this paper we derive an analytic solution for the dynamic evolution of mixing fronts driven by constant acceleration rayleigh taylor rt and impulsive acceleration richtmyer meshkov instabilities from a simple physics model expressed as a pair of ordinary differential equations an approximate closed form asymptotic evaluation of the rt solution is obtained through terms of order o 1 as t infinity this three term expansion including lower order terms is used to interpret experimental and simulation data our solutions improve on previous analyses in their agreement with experimental data in that we can fit both the slope and the intercept of the z b vs agt 2 experimental plots by adjusting parameters in our model since the experimental data are close to self similar the improvement due to the lower order contributions in the asymptotic expansion is modest we also apply this analysis to simulation data for which preasymptotic data exist we reexamine previous simulation data and determine an improved growth rate alpha b 0 0625 the present paper provides concepts and tools to explore the preasymptotic aspects of these data

uniqueness of self similar shrinkers with asymptotically conical ends in this paper we show the uniqueness of smooth embedded selfshrinkers asymptotic to generalized cylinders of infinite order also we construct non rotationally symmetric self shrinking ends asymptotic to generalized cylinders with rate as fast as any given polynomial

solution of eigenvalue problems for non classically damped systems with multiple frequencies abstract an efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of non classically damped structural systems with multiple eigenvalues the proposed method is obtained by applying the modified newton raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem through matrix augmentation of the quadratic eigenvalue problem in the iteration methods such as the inverse iteration method and the subspace iteration method singularity may occur during the factorizing process when the shift value is close to an eigenvalue of the system however even though the shift value is an eigenvalue of the system the proposed method provides non singularity and that is analytically proved since the modified newton raphson technique is adapted to the proposed method initial values are needed because the lanczos method effectively produces better initial values than other methods the results of the lanczos method are taken as the initial values of the proposed method two numerical examples are presented to demonstrate the effectiveness of the proposed method and the results are compared with those of the well known subspace iteration method and the lanczos method

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

verification and validation for turbulent mixing abstract verification and validation constitute a relatively new requirement for numerical analysis it is one which results from the growing practical use of computation as a basis for engineering decisions while a number of procedures have been proposed for these tasks their application to the more difficult cases remains a research issue here we explain methods developed by ourselves and collaborators for verification and validation of simulations for chaotic multiscale flows specifically for turbulent mixing simulations

perturbations of time optimal control problems for a class of abstract parabolic systems in this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge in an appropriate sense to a given strictly negative operator our main application to pdes systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients as we encounter in homogenization theory our main results assert that provided that the target is a closed ball centered at the origin and of positive radius the solutions of the time optimal control problems for the systems with oscillating coefficients converge in the usual norms to the solution of the corresponding problem for the homogenized system in order to prove our main theorem we provide several new results which could be of a broader interest on time and norm optimal control problems

controllability of impulse controlled systems of heat equations coupled by constant matrices abstract this paper studies the approximate and null controllability for impulse controlled systems of heat equations coupled by a pair a b of constant matrices we present a necessary and sufficient condition for the approximate controllability which is exactly kalman s controllability rank condition of a b we prove that when such a system is approximately controllable the approximate controllability over an interval 0 t can be realized by adding controls at arbitrary q a b different control instants 0 1 2 q a b t provided that q a b 1 d a where d a min im a and q a b n we also show that in general such systems are not null controllable

dynamic response of some dissipative systems by means of functions of matrices abstract a certain class of linear viscously damped dynamical systems the matrix coefficients of which satisfy a certain commutativity condition are known to exhibit the same normal modes as the ones that exist in the absence of damping these systems are said to have classical normal modes in the present study such systems are shown to be particular cases of dynamical systems that possess a zero commutator which form a subset of separable systems the response of the latter is investigated by means of functions of matrices and its connection with the classical eigenfunction expansion approach is established the paper continues the author s previous work where functions of matrices were used to a develop an algorithm for the numerical integration of the equations of motion for such systems and b to establish their conservation laws

eigenvalue counting methods for non proportionally damped systems abstract this paper presents numerical methods of counting the number of eigenvalues for non proportionally damped system in some interested regions on the complex plane most of the eigenvalue analysis methods for proportionally damped systems use the well known sturm sequence property to check the missed eigenvalues when only a set of the lowest modes is used however in the case of the non proportionally damped systems such as the soil structure interaction system the structural control system and composite structures no counterpart of the sturm sequence property for undamped systems has been established yet in this study a numerical method based on argument principle is explained with emphasis on the discretization of the contour and a new method based on gleyse s theorem is proposed to verify the applicability of the methods two numerical examples are considered

on eigenproblem solution of damped vibrations associated with gyroscopic moments a new efficient approach is presented for solving the quadratic eigenvalue problem of weakly nonproportionally damped vibration systems in the analysis of these systems gyroscopic moments and external damping are both considered traditional restriction of symmetry of inertia damping and stiffness matrices is slightly relaxed a second order perturbation theory is developed such that the perturbed solution is based on the eigensolution of an unperturbed subproblem this subproblem considers the unperturbed system in two different forms i a conservative gyroscopic part of an original problem or ii a nonconservative gyroscopic part of an original problem that is proportionally damped to cope with asymmetry of the system matrices a duncan s like state formulation is used to bring these matrices into a suitable form for perturbations two numerical examples are introduced for explaining the detailed implementation of the presented approach additionally a practical problem of rotor supported by two tilting pad bearings is investigated the eigensolutions obtained by the current approach match to a great extent other solutions obtained by time consuming exact methods the investigation procedure given here gives a framework to handle vibration problems of weakly nonproportional damping and or weakly asymmetric inertia damping and stiffness matrices

taussky s theorem symmetrizability and modal analysis revisited this paper is concerned with symmetrization and diagonalization of real matrices and their implications for the dynamics of linear second order systems governed by equations of motion having asymmetric coefficient matrices results in the light of taussky s theorem are presented the connection of the symmetrizers with the eigenvalue problem is brought out an alternative proof of taussky s theorem for real matrices is presented diagonalization of two real symmetric but not necessarily positive definite matrices is discussed in the context of undamped non gyroscopic systems a commutator of two matrices with respect to a given third matrix is defined this commutator is found to play an interesting role in deciding simultaneous diagonalizability of two or three matrices errors in a few previously known results are brought out pseudo conservative systems are studied and their connection with the so called symmetrizable systems is critically examined results for modal analysis of general non conservative systems are presented illustrative examples are given

equivalent conditions on periodic feedback stabilization for linear periodic evolution equations abstract this paper studies the periodic feedback stabilization for a class of linear t periodic evolution equations several equivalent conditions on the linear periodic feedback stabilization are obtained these conditions are related to the following subjects the attainable subspace of the controlled evolution equation under consideration the unstable subspace of the evolution equation with the null control provided by the kato projection the poincare map associated with the evolution equation with the null control and two unique continuation properties for the dual equations on different time horizons 0 t and 0 n 0 t where n 0 is the sum of algebraic multiplicities of distinct unstable eigenvalues of the poincare map it is also proved that a t periodic controlled evolution equation is linear t periodic feedback stabilizable if and only if it is linear t periodic feedback stabilizable with respect to a finite dimensional subspace some applications to heat equations with time periodic potentials are presented

equivalence of three different kinds of optimal control problems for heat equations and its applications this paper presents an equivalence theorem for three different kinds of optimal control problems which are optimal target control problems optimal time control problems and optimal norm control problems controlled systems in this study are internally controlled heat equations with the aid of this theorem we establish an optimal norm feedback law and build up some explicit algorithms for solutions of optimal norm and optimal time control problems

an improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem a technique is developed and presented which improves the accuracy of the predictions for the eigenvalues of the usual first order perturbation scheme for the generalized symmetric linear algebraic eigenvalue problem the improved method very simply replaces the usual b norms appearing in the denominator of the expressions for the first order changes in the eigenvalues by bp norms where bp is the perturbed b matrix with increasing sizes for the perturbations the improved technique produces results whose accuracy approaches those of the usual second order predictions numerical studies are presented to support the analysis

quantitative uniqueness for elliptic equations with singular lower order terms we use a carleman type inequality of koch and tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms first we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure

on backward uniqueness for the heat operator in cones abstract consider the system t u u m u u u x t m e m x 2 in c 0 t and u x 0 0 in c where c is a cone with opening angle l escauriaza constructed an example to show that such system has a nonzero solution when 90 and it s conjectured that the system has only zero solution when 90 lu li and v sverak proved that the conjecture is true when 109 5 here we improve their result and prove that the conjecture is true when 99 by exploring a new type of carleman inequality which is of independent interest

the eigenproblem for damped linear vibrating systems with multiple eigenvalues abstract vibration systems with damping are analyzed with the equations written in second order form as well as first order state space form the existence of eigenpairs is studied for vibrating systems with null eigenvalues a detailed study is carried out of the effect of velocity constraints on rigid body motion and on the number of associated linearly independent eigenvectors the results are useful for complex modal synthesis in which subsystems include rigid body motion the analysis is extended to multiple non null eigenvalues by using an eigenvalue shifting technique for the double non null eigenvalue problem a necessary condition is derived for the associated eigenvector in terms of a convenient orthogonality relation

a multiphase godunov method for compressible multifluid and multiphase flows we propose a new model and a solution method for two phase compressible flows the model involves six equations obtained from conservation principles applied to each phase completed by a seventh equation for the evolution of the volume fraction this equation is necessary to close the overall system the model is valid for fluid mixtures as well as for pure fluids the system of partial differential equations is hyperbolic hyperbolicity is obtained because each phase is considered to be compressible two difficulties arise for the solution one of the equations is written in non conservative form non conservative terms exist in the momentum and energy equations we propose robust and accurate discretisation of these terms the method solves the same system at each mesh point with the same algorithm it allows the simulation of interface problems between pure fluids as well as multiphase mixtures several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible the method provides reliable results is able to compute strong shock waves and deals with complex equations of state

approximation of time optimal controls for heat equations with perturbations in the system potential in this paper we study a certain approximation property for a time optimal control problem of the heat equation with l infty potential we prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation where the potential has a small perturbation are close to those for the original problem we also verify that for the heat equation with a small perturbation in the potential one can construct a new time optimal control problem which has the same target as that of the original problem but has a different control constraint bound from that of the original problem such that the new and the original problems share the same optimal time and meanwhile the optimal control of the new problem is close to that of the original one the main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration this theorem was first established by wa

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

quantitative uniqueness of solutions to parabolic equations abstract we investigate the quantitative uniqueness of solutions to parabolic equations with lower order terms on compact smooth riemannian manifolds quantitative uniqueness is a quantitative form of strong unique continuation property we characterize quantitative uniqueness by the rate of vanishing we can obtain the sharp vanishing order for solutions in term of the c 1 1 norm of the potential functions as well as the l norm of the coefficient functions some new quantitative carleman estimates and three cylinder inequalities are established

symmetric state space method for a class of nonviscously damped systems multiple degree of freedom linear nonviscously damped systems are considered it is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions the traditional state space approach well known for viscously damped systems is extended to such nonviscously damped systems using a set of internal variables suitable numerical examples are provided to illustrate the proposed approach i introduction v iscousdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems this model e rst introduced by rayleigh 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping viscous damping models are used widely for their simplicity and mathematical convenience even though the behavior of real structural materials is at best poorly mimicked by simple viscous models forthisreason itiswellrecognizedthat ingeneral aphysically realistic model of damping will not be viscous damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models mathematically any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model clearly a wide range of choice is possible either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments here we will use a particular type of damping model that is not viscous and throughout the paper the terminology nonviscous damping will refer to this specie c model only possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions 2 the equations of motion of an n degree of freedom linear system with such damping can be expressed by

an improved subspace iteration method with shift for structures with multiple natural frequencies an efficient and numerically stable eigensolution method for structures with multiple natural frequencies is presented the proposed method is developed by improving the well known subspace iteration method with shift a major difficulty of the subspace iteration method with shift is that because of singularity problem a shift close to an eigenvalue cannot be used resulting in slower convergence in this paper the above singularity problem has been solved by introducing side conditions without sacrifice of convergence the proposed method is always non singular even if a shift is on a distinct eigenvalue or multiple ones this is one of the significant characteristics of the proposed method the non singularity is proved analytically the convergence of the proposed method is at least equal to that of the subspace iteration method with shift and the operation counts of the above two methods are almost the same when a large number of eigenpairs are required to show the effectiveness of the proposed method two numerical examples are considered

the maximum principle for control systems described by linear parabolic equations we prove a version of the maximum principle for linear parabolic distributed parameter systems that avoids some of the usual smoothness assumptions on the target

uncertainty quantification for chaotic computational fluid dynamics we seek error models for simulations that model chaotic flow stable statistics for the solution and for the error are obtained after suitable averaging procedures