stability of linear time varying delay systems and applications to control problems this paper deals with the stability of a class of linear time varying systems with multiple delays using the lyapunov function method we give sufficient delay dependent conditions for the exponential stability with a given convergence rate which are described in terms of linear matrix inequalities lmi and the solution of riccati differential equations rde the results are applied to the problem of stabilization of linear time varying control systems with multiple delays numerical examples are given to illustrate the results

on the u invariant of a real function field we obtain a new upper bound on the dimensions of anisotropic quadratic torsion forms over a field that is an extension of finite transcendence degree of a real closed field

lq regulator design method for system with delay based on spectral decomposition of the hamiltonian a design method to construct an lq regulator for a system with time delay is proposed it is based on a new technique for composing the solution of an infinite dimensional riccati equation using the spectral decomposition of the hamiltonian this design method has the feature that the closed loop poles can be calculated a priori from the open loop poles and one scalar design parameter so that we can choose the parameter to obtain an optimal control law which results in the desired degree of exponential stability of the closed loop system

sharp l p bounds on spectral clusters for lipschitz metrics we establish l p bounds on l 2 normalized spectral clusters for self adjoint elliptic dirichlet forms with lipschitz coefficients in two dimensions we obtain best possible bounds for all 2 le p le infty up to logarithmic losses for 6 p le 8 in higher dimensions we obtain best possible bounds for a limited range of p

global existence for energy critical waves in 3 d domains we prove that the defocusing quintic wave equation with dirichlet boundary conditions is globally well posed on for any smooth compact domain the main ingredient in the proof is an spectral projector estimate obtained recently by smith and sogge combined with a precise study of the boundary value problem

a multistage reduction technique for feedback stabilizing distributed time lag systems abstract a direct pole relocation theory is advanced for linear time invariant systems with distributed delays in both state and control variables the principal tools of the theory include i the finite cardinality of the unstable spectrum ii a set of matrices each of which is a left zero of the system characteristic quasi polynomial matrix and iii a linear transformation which reduces the delay system to a sufficiently high order delay free system whose spectrum contains the delay system unstable spectrum it is shown that if the delay system is spectrally stabilizable then it shares a common feedback stabilizing control law with its delay free counterpart this point of contact with a delay free system permits the determination of the control law using well established ordinary system methods the workability of the approach hinges on the ability to partition the unstable spectrum augmented with additional poles from the stable spectrum if necessary into n symmetric sets when this partition is impossible a spectral controllability invariance theorem facilitates resolution of the problem

on the lp norm of spectral clusters for compact manifolds with boundary we use microlocal and paradifferential techniques to obtain l8 norm bounds for spectral clusters associated with elliptic second order operators on two dimensional manifolds with boundary the result leads to optimal lq bounds in the range 2 q for l2 normalized spectral clusters on bounded domains in the plane and more generally for two dimensional compact manifolds with boundary we also establish new sharp lq estimates in higher dimensions for a range of exponents qn q

stabilization of linear systems with non commensurate delays a stabilization method for linear systems with non commensurate delays in both state and control variables is presented this method is obtained by combining the algebraic stabilization method with the finite spectrum assignment method the present method can stabilize systems that algebraic methods cannot

output feedback stabilization for a class of stochastic time delay nonlinear systems this note focuses on a class of stochastic time delay nonlinear systems the problem we address is to design a controller such that the closed loop system is exponentially stable it is shown that the stabilization via output feedback can be solved by a lyapunov based recursive design method

approximation schemes for the linear quadratic optimal control problem associated with delay equations the linear regulator problem for delay equations is discussed we propose a theoretical solution involving riccati integral equations and then axiomatically discuss a general approximation scheme the details are given for spline and averaging approximations

pitfalls in parameter estimation for delay differential equations given a set of data u gamma i approx u gamma i sfp star corresponding to the em delay differential equation vspace 3pt begin array rcll u t sfp f t u t sfp u alpha t sfp sfp sfp hbox for t ge t 0 sfp u t sfp psi t sfp hbox for t le t 0 sfp end array the basic problem addressed here is that of calculating the vector sfp star in blackboardd r n we also consider neutral differential equations most approaches to parameter estimation calculate sfp star by minimizing a suitable objective function that is assumed by the minimization algorithm to be sufficiently smooth in this paper we use the derivative discontinuity tracking theory for delay differential equations to analyze how jumps can arise in the derivatives of a natural objective function these jumps can occur when estimating parameters in lag functions and estimating the position of the initial point and as such are not expected to occur in parameter estimation problems for ordinary differential equations

on the design of a stable adaptive filter for state estimation in high dimensional systems design of specific parametrized gain structures provides a new tool for ensuring stability of reduced order filter for high dimensional systems which is optimized adaptively in the minimum prediction error framework

global existence for energy critical waves in 3 d domains neumann boundary conditions we prove that the defocusing quintic wave equation with neumann boundary conditions is globally well posed on h 1 l 2 for any smooth compact domain r 3 the proof relies on one hand on l p estimates for the spectral projector and on the other hand on a precise analysis of the boundary value problem which turns out to be much more delicate than in the case of dirichlet boundary conditions

estimation of variable coefficients in parabolic distributed systems we present techniques based on cubic spline approximations for estimating coefficients e g diffusion convector velocity etc depending on time and the spatial variable in parabolic distributed systems such as those that arise in transport models convergence results and a summary of numerical performance of the resulting algorithms are given

convergence of controllers designed using state space techniques in this paper coprime factorization theory is used to examine convergence of finite dimensional controllers for infinite dimensional systems designed using approximations under assumptions of uniform stabilizability detectability the controllers stabilize the original system for large enough model order and the corresponding sequence of closed loops converge

on the approximation of eigenvalues associated with functional differential equations abstract in this paper an approximation method based upon spline functions is developed for the eigenvalue problem associated with functional differential equations convergence results are established and the rate of convergence is investigated numerical results for cubic and quintic spline based methods are given the paper concludes with a brief discussion of other possible approximation methods

approximation schemes for nonlinear neutral optimal control systems abstract we discuss methods of approximating stable neutral functional differential equations and associated optimal control problems by sequences of optimal control problems for ordinary differential equations by introducing a class of mollified neutral functional differential equations convergence of the linear interpolating spline and the averaging approximation scheme is proved a number of numerical examples are included

spline approximation for retarded systems and the riccati equation the purpose of this paper is to introduce a new spline approximation scheme for retarded functional differential equations the special feature of this approximation scheme is that it preserves the product space structure of retarded systems and approximates the adjoint semigroup in a strong sense these facts guarantee the convergence of the solution operators for the differential riccati equation in a strong sense numerical findings indicate a significant improvement in the convergence behaviour over both the averaging and the previous spline approximation scheme

essential dimension of quadrics let x be an anisotropic projective quadric over a field f of characteristic not 2 the essential dimension dimes x of x a s def ined by oleg izhboldin is dimes x dim x i x 1 where i x is the first witt index of x i e the witt index of x over its function field let y be a complete possibly singular algebraic variety over f with all closed points of even degree and such that y has a closed point of odd degree over f x our main theorem states that dimes x dim y and that in the case dimes x dim y the quadric x is isotropic over f y applying the main theorem to a projective quadric y we get a proof of izhboldin s conjecture stated as follows if an anisotropic quadric y becomes isotropic over f x t hen dim es x dimes y and the equality holds if and only if x is isotropic over f y we also solve knebusch s problem by proving that the smallest transcendence degree of a generic splitting field of a quadric x is equal to dimes x let v be a non degenerate quadratic form of dimension at least 2 over a field f of characteristic not 2 and let x q be the quadric hypersurface given by the equation x 0 in the projective space p v we say that the quadric x is anisotropic if is an anisotropic quadratic form by springer s theorem every closed point of an anisotropic quadric x has

newton polygons as lattice points let m be the reduction modulo p of a shimura variety and let n m be the partially ordered set predicted to be the set of all types occurring in the stratification of m by newton polygons in the context of f isocrystals with additional structures we prove that n m is ranked or catenary in the sense that any two maximal chains have the same length we also give a lie theoretic formula for the distance between two comparable elements in n m this formula can be used to give the expected dimension of the newton strata of m

parameter identification in classes of hereditary systems of neutral type in this paper we prove theoretical convergence for a variety of parameter identification schemes based on approximations by equations with piecewise constant arguments for classes of neutral differential equations

a factorization on the semi infinite interval ii applications to stable regulator problems abstract a factorization approach to the synthesis of feedback control laws for stable regulator problems is developed new feedback representations are obtained for hereditary problems with and without control delay also new derivations of representations of the optimal gain for regulator problems in hilbert space are presented

special factorization and riccati integral equations abstract a class of operator riccati integral equations is associated with a factorization problem in a certain banach algebra recent results concerning factorization in this algebra are used to obtain existence uniqueness and continuous dependence results for the riccati equations

strichartz and localized energy estimates for the wave equation in strictly concave domains we prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous dirichlet or neumann conditions are imposed by restricting the solution to small frequency dependent space time collars of the boundary it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates mixed norm estimates of strichartz and square function type follow as a result using the energy estimates to control error terms which arise in a wave packet parametrix construction while the latter estimates are not new for dirichlet conditions the present approach provides an avenue for treating these estimates when neumann conditions are imposed the method also treats schr o dinger equations with time independent coefficients

mesh independence of kleinman newton iterations for riccati equations in hilbert space in this paper we consider the convergence of the infinite dimensional version of the kleinman newton algorithm for solving the algebraic riccati operator equation associated with the linear quadratic regulator problem in a hilbert space we establish mesh independence for this algorithm and apply the result to systems governed by delay equations numerical examples are presented to illustrate the results

dispersive estimates for principally normal pseudodifferential operators the aim of these notes is to describe some recent re sults concerning dispersive estimates for principally normal pseu dodifferential operators the main motivation for this comes from unique continuation problems such estimates can be used to prove l q carleman inequalities which in turn yield unique continuation results for various partial differential operators with rough poten tials dispersive estimates are l q estimates for nonelliptic partial differ ential operators which are a consequence of the decay properties of their fundamental solutions these decay properties follow from spa tial spreading of the singularities of the solutions since solutions prop agate in directions conormal to the characteristic set of the operator this spreading can be related to nonzero curvatures of the characteristic set dispersive estimates for constant coefficient operators are closely related to the restriction theorem in harmonic analysis various types of dispersive estimates are known to be true for op erators such as the wave operator the schrodinger operator and the linear kdv see ginibre velo 4 keel tao 11 they have proved to be useful in the study of nonlinear problems as well as of problems with unbounded potentials more recently similar estimates have been obtained for wave op erators with variable coefficients beginning with the smooth case in kapitanskii 10 mockenhaupt seeger and sogge 14 up to operators with c 2 coefficients in smith 15 and tataru 21 23 similar results were obtained for the schrodinger equation in staffilani tataru 19 c 2 coefficients and in burq gerard tzvetkov 1 smooth coeffic ients in the variable coefficient elliptic case one should also mention sogge s l q

invariance results for delay and volterra equations in fractional order sobolev spaces on deduit l invariance des trajectoires d equations de type volterra et a retard infini dans des espaces de sobolev d ordre fractionnaire pour des hypotheses minimales sur les donnees du probleme

quantum ergodicity and l p norms of restrictions of eigenfunctions we prove an analogue of sogge s local l p estimates for l p norms of restrictions of eigenfunctions to submanifolds and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of burq gerard tzvetkov hu and chen sogge the improvements are logarithmic on negatively curved manifolds without boundary and by o 1 for manifolds with or without boundary with ergodic geodesic flows in the case of ergodic billiards with piecewise smooth boundary we get o 1 improvements on l infty estimates of cauchy data away from a shrinking neighborhood of the corners and as a result using the methods of ghosh et al jung and zelditch jung and zelditch we get that the number of nodal domains of 2 dimensional ergodic billiards tends to infinity as lambda to infty these results work only for a full density subsequence of any given orthonormal basis of eigenfunctions we also present an extension of the l p estimates of burq gerard tzvetkov hu chen sogge for the restrictions of dirichlet and neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary this part does not assume ergodicity on the manifolds

the length and other invariants of a real field the length of a field is the smallest integer m such that any totally positive quadratic form of dimension m represents all sums of squares we investigate this field invariant and compare it to others such as the u invariant the pythagoras number the hasse number and the mordell function related to sums of squares of linear forms

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

approximating pointwise products of laplacian eigenfunctions abstract we consider laplacian eigenfunctions on a d dimensional bounded domain m or a d dimensional compact manifold m with dirichlet conditions these operators give rise to a sequence of eigenfunctions e l l n we study the subspace of all pointwise products a n span e i x e j x 1 i j n l 2 m clearly that vector space has dimension dim a n n n 1 2 we prove that products e i e j of eigenfunctions are simple in a certain sense for any e 0 there exists a low dimensional vector space b n that almost contains all products more precisely denoting the orthogonal projection b n l 2 m b n we have 1 i j n e i e j b n e i e j l 2 e and the size of the space dim b n is relatively small for every 0 dim b n m e n 1 we obtain the same sort of bounds for products of arbitrary length as well for approximation in h 1 norm pointwise products of eigenfunctions are low rank this has implications among other things for the validity of fast algorithms in electronic structure computations

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

optimal control of linear retarded systems in banach spaces this paper deals with standard optimal control problems namely the fixed time integral convex cost problem and the time optimal control problem for linear retarded systems in banach spaces for the basis of optimal control theory the fundamental solution is constructed and a variation of constant formula of mild solutions is established after the controlled system description and the formulation of optimal control problems are given the retarded adjoint system is introduced for the integral convex cost problem two existence theorems of optimal controls and necessary conditions of optimality are given these conditions are precisely characterized by the solution of retarded adjoint system the pointwise maximum principle for time varying control domain is derived from the optimality conditions the bang bang principle is also established for the terminal value cost problem under some regularity condition of the adjoint system for the time optimal control problem to a target set an existence theorem is shown in the case where the target set has interior the maximum principle and the bang bang principle are established for the time optimal control finally a convergence theorem of time optimal controls to a point target set is given this paper also contains illustrative examples which give technologically important control problems

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

strichartz estimates for operators with nonsmooth coefficients and the nonlinear wave equation the aim of this article is threefold first we use the fbi transform to set up a calculus for partial differential operators with nonsmooth coefficients next this calculus allows us to prove strichartz type estimates for the wave equation with nonsmooth coefficients finally we use these strichartz estimates to improve the local theory for second order nonlinear hyperbolic equations 1 introduction the fbi transform is in a way similar to the complex fourier transform in that for each function in rn it provides a representation as a holomorphic function in r2n however in the case of the fbi transform we can identify naturally r2n with the phase space t rn for a pseudodifferential operator with smooth symbol acting on functions in rn one can produce by conjugation a corresponding formal series acting on func tions in r2n9 for which the first term is exactly the multiplication by the symbol this series converges and has a nice representation in the weyl calculus provided that the symbol of the operator is analytic this is how the fbi transform has been used in the study of partial differential operators with analytic coefficients see 12 13 where this machinery is developed here in a way we do the opposite we look at operators with nonsmooth coefficients approximate the conjugated operator by a partial sum of the formal series and prove error estimates in the simplest case the approximate conjugate operator is exactly the multiplication by the symbol this is also related to the cordoba fefferman wave packet transform in 3 in the third section we use the error estimates to reduce the strichartz estimates for second order hyperbolic operators with nonsmooth coefficients to weighted lp lq l2 estimates for the fbi transform these in turn can be proved in the usual fashion using appropriate oscillatory integral esti mates in the last part of the article we explain how one can use the new strichartz type estimates to improve the local theory for nonlinear hyperbolic equations beyond the classical setup these results are not sharp and will be improved in subsequent articles

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

counterexamples to strichartz estimates for the wave equation in domains we consider the wave equation inside a strictly convex domain of dimension 2 and provide counterexamples to optimal strichartz estimates such estimates inside convex domains lose regularity when compared to the flat case at least for a subset of the usual range of indices mainly due to microlocal phenomena such as caustics which are generated in arbitrarily small time near the boundary

lq optimal control of infinite dimensional systems by spectral factorization abstract we consider the standard lq optimal control of an exponentially stabilizable and detectable infinite dimensional semigroup state space system with bounded sensing and control it is reported that the reachable restriction of the lq optimal state feedback operator can be identified 1 by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper stable transfer function algebra and 2 by obtaining any constant solution of an operator diophantine equation over the properties of the restricted solution feedback operator are discussed it is shown in particular that the lq optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable these theoretical results are applied to a simple model of heat diffusion with collocated sensor and actuator leading to an approximation procedure converging exponentially fast to the lq optimal state feedback operator this procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus

estimation of discontinuous coefficients in parabolic systems applications to reservoir simulation we present spline based techniques for estimating spatially varying parameters that appear in parabolic distributed systems typical of those found in reservoir simulation problems in particular we discuss the problem of determining discontinuous coefficients estimating both the functional shape and points of discontinuity for such parameters in addition our ideas may also be applied to problems with unknown initial conditions and unknown parameters appearing in terms representing external forces convergence results and a summary of numerical performance of the resulting algorithms are given

linear quadratic optimal actuator location in control of vibrations diffusion and many other problems governed by partial differential equations there is freedom in the choice of actuator location the actuator location should be chosen to optimize performance objectives in this paper we consider linear quadratic performance two types of cost are considered the choice depends on whether the response to the worst initial condition is to be minimized or whether the initial condition is regarded as random in practice approximations are used in controller design and thus in selection of the actuator locations the optimal cost and location of the approximating sequence should converge to the exact optimal cost and location in this work conditions for this convergence are given in the case of linear quadratic control examples are provided to illustrate that convergence may fail when these conditions are not satisfied

estimation of delays and other parameters in nonlinear functional differential equations we discuss a spline based approximation scheme for nonlinear nonautonomous delay differential equations convergence results using dissipative type estimates on the underlying nonlinear operators are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient type parameters a brief summary of some of our related numerical findings is also given

strichartz type estimates and the well posedness of an energy critical 2d wave equation in a bounded domain abstract we study the well posedness of the cauchy problem with dirichlet or neumann boundary conditions associated to an h 1 critical semilinear wave equation on a smooth bounded domain r 2 first we prove an appropriate strichartz type estimate using the l q spectral projector estimates of the laplace operator our proof follows burq lebeau and planchon 2008 4 then we show the global well posedness when the energy is below or at the threshold given by the sharp moser trudinger inequality finally in the supercritical case we prove an instability result using the finite speed of propagation and a quantitative study of the associated ode with oscillatory data

powers of the fundamental ideal in the witt ring abstract let k be a field of characteristic different from 2 in the algebraic theory of quadratic forms one studies the witt ring w k of equivalence classes of non degenerate quadratic forms the witt ring has a filtration given by the powers in k of the fundamental ideal i k of even dimensional forms the ideal in k is generated by the set pn k of n fold pfister forms a1 an ni 1 1 ai where ai k k 0 many questions about this filtration and its quotients have arisen in the study of w k the spectacular work of voevodsky together with his collaborative work with orlov and vishik allows one to answer many old questions the purpose of this paper is to indicate some of these solutions

feedback stabilization of linear autonomous time lag systems a stabilization theory which employs well established finite dimensional control system tools is developed for the stabilization of linear autonomous time lag systems the main ideas include 1 a set whose elements are matrices each of which is a left zero of the system characteristic quasi polynomial matrix and 2 a linear transformation which reduces the delay system to a delay free system whose state matrix is a direct sum of n elements of the matrix set where n is some positive integer from the definition of this matrix set it is shown that each of its elements inherits its spectrum from that of the delay system so that by design the system unstable poles may be embedded in the spectrum of the delay free system under the assumption of spectral stabilizability it is then shown how to obtain a stabilizing feedback control law on the basis of the delay free system numerical examples are presented to confirm the theory

lower bounds for nodal sets of dirichlet and neumann eigenfunctions let phi be a dirichlet or neumann eigenfunction of the laplace beltrami operator on a compact riemannian manifold with boundary we prove lower bounds for the size of the nodal set phi 0

the hilbert schmidt property of feedback operators in this article we prove new sufficient conditions under which the feedback operator associated with the linear quadratic regulator control design for distributed parameter systems is nuclear or hilbert schmidt examples illustrating the main results are also given

bounds for eigenfunctions of the laplacian on noncompact riemannian manifolds we deal with eigenvalue problems for the laplacian on noncompact riemannian manifolds m of finite volume sharp conditions ensuring l q m and l infinity m bounds for eigenfunctions are exhibited

decomposability for division algebras of exponent two and associated forms this article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated witt ring is torsion free

spectral cluster estimates for c 1 1 metrics in this paper we establish l p norm bounds for spectral clusters on compact manifolds under the assumption that the metric is c 1 1 precisely we show that the l p estimates proven by sogge in the case of smooth metrics hold under this limited regularity assumption it is known by examples of smith sogge that such estimates fail for c 1 metrics if 1

random data cauchy theory for supercritical wave equations i local theory we study the local existence of strong solutions for the cubic nonlinear wave equation with data in hs m s 1 2 where m is a three dimensional compact riemannian manifold this problem is supercritical and can be shown to be strongly ill posed in the hadamard sense however after a suitable randomization we are able to construct local strong solution for a large set of initial data in hs m where s 1 4 in the case of a boundary less manifold and s 8 21 in the case of a manifold with boundary

spline approximation for systems of linear neutral delay differential equations abstract we derive a new finite dimensional semidiscrete approximation scheme for systems of linear neutral delay differential equations and prove convergence results our construction extends to neutral delay equations results which were previously only available for retarded delay equations

on the solutions of a system of linear retarded and advanced differential equations by the bessel collocation approximation in this study we present a numerical approximation for the solutions of the system of high order linear retarded and advanced differential equations with variable coefficients under the mixed conditions this method is based on taking the truncated bessel expansion of the functions in the retarded and advanced differential equation system by the aid of the matrix operations and the collocation points the problem is transformed into a matrix equation with the unknown bessel coefficients by solving this matrix equation the unknown coefficients of the approximate solutions are computed in addition examples that illustrate the pertinent features of the method are presented and the results of this investigation are discussed

a twofold spline approximation for finite horizon lqg control of hereditary systems in this paper an approximation scheme is developed for the solution of the linear quadratic gaussian lqg control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay the solution here proposed is achieved by means of two approximating subspaces the first one to approximate the riccati equation for control and the second one to approximate the filtering equations since the approximating subspaces have finite dimension the resulting equations can be implemented the convergence of the approximated control law to the optimal one is proved simulation results are reported on a wind tunnel model showing the high performance of the method

approximation of solutions of riccati equations this paper deals with two interrelated issues one is an invariant subspace approach to finding solutions for the algebraic riccati equation for a class of infinite dimensional systems the second is approximation of the solution of the algebraic riccati equation by finite dimensional approximants the theory of exponentially dichotomous operators and bisemigroups is instrumental in our approach

dispersion for the wave equation inside strictly convex domains i the friedlander model case we consider a model case for a strictly convex domain r d of dimension d 2 with smooth boundary 6 and we describe dispersion for the wave equation with dirichlet boundary conditions more specically we obtain the optimal xed time decay rate for the smoothed out green function a t 1 4 loss occurs with respect to the boundary less case due to repeated occurrences of swallowtail type singularities in the wave front set

strichartz estimates and the nonlinear schrodinger equation on manifolds with boundary we establish strichartz estimates for the schrodinger equation on riemannian manifolds g with boundary for both the compact case and the case that is the exterior of a smooth non trapping obstacle in euclidean space the estimates for exterior domains are scale invariant the range of lebesgue exponents p q for which we obtain these estimates is smaller than the range known for euclidean space but includes the key l 4 t l infty x estimate which we use to give a simple proof of well posedness results for the energy critical schrodinger equation in 3 dimensions our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index we use these to establish well posedness for finite energy data of certain semilinear schrodinger equations on general compact manifolds with boundary

global stabilization for linear continuous time varying systems in this paper stabilization problem via static output feedback controls for linear time varying systems is investigated based on the lyapunov function techniques we show that for linear time varying systems the global null controllability guarantees the output feedback stabilization the verifiable stabilizability conditions and output feedback control design are stated the result can be applicable to the output feedback stabilizability of a class of nonlinear time varying systems numerical examples illustrated the conditions are given

computational approaches to parameter estimation and model selection in immunology one of the significant challenges in biomathematics and other areas of science is to formulate meaningful mathematical models our problem is to decide on a parametrized model which is in some sense most likely to represent the information in a set of observed data in this paper we illustrate the computational implementation of an information theoretic approach associated with a maximum likelihood treatment to modelling in immunology the approach is illustrated by modelling lcmv infection using a family of models based on systems of ordinary differential and delay differential equations the models which use parameters that have a scientific interpretation are chosen to fit data arising from experimental studies of virus cytotoxic t lymphocyte kinetics the parametrized models that result are arranged in a hierarchy by the computation of akaike indices the practical illustration is used to convey more general insight because the mathematical equations that comprise the models are solved numerically the accuracy in the computation has a bearing on the outcome and we address this and other practical details in our discussion

spline approximations for linear nonautonomous delay systems we develop a semi discrete approximation framework for linear nonautonomous nonhomogeneous functional differential equations of retarded type the approximation schemes are constructed and convergence results are obtained through the approximation of an associated abstract evolution operator computer implementation of the schemes is outlined and a spline based method included in the framework is constructed extensions of the semi discrete methods to schemes incorporating full discretization and difference equation approximation are also discussed numerical results for several examples demonstrating the feasibility of the schemes are presented

estimation of time and state dependent delays and other parameters in functional differential equations a parameter estimation algorithm is developed that can be used to estimate unknown time or state dependent delays and other parameters e g initial condition appearing within a nonlinear nonautonomous functional differential equation the original infinite dimensional differential equation is approximated using linear splines which are allowed to move with the variable delay the variable delays are approximated using linear splines as well the approximation scheme produces a system of ordinary differential equations with nice computational properties the unknown parameters are estimated within the approximating systems by minimizing a least squares fit to data criterion convergence theorems are proved for time dependent delays and state dependent delays within two classes which say essentially that fitting the data by using approximations will in the limit provide a fit to the data using the original system numerical test examples are presented that illustrate the method for all types of delay

on asymptotic properties of biharmonic steklov eigenvalues abstract in this paper by explicitly calculating the principal symbols of pseudodifferential operators we establish two weyl type asymptotic formulas with sharp remainder estimates for the counting functions of the two classes of biharmonic steklov eigenvalue problems of smooth bounded domains in a riemannian manifold

solving the linear quadratic optimal control problem for infinite dimensional systems abstract calculation of the solutions to linear quadratic optimal control problems for infinite dimensional systems is considered the sequence of solutions to a sequence of approximating finite dimensional problems converges to the optimal control for the infinite dimensional system if certain assumptions such as uniform stabilizability are satisfied we use this result to calculate controllers for the infinite dimensional system that are arbitrarily close to optimal a one dimensional heat equation and a problem of acoustic noise control are used to illustrate the algorithm the numerical results are discussed

upper and lower bounds for normal derivatives of spectral clusters of dirichlet laplacian abstract in this paper we prove the upper and lower bounds for normal derivatives of spectral clusters u s f of dirichlet laplacian m c s u l 2 m u l 2 m c s u l 2 m where the upper bound is true for any riemannian manifold and the lower bound is true for some small 0 s s m where s m depends on the manifold only provided that m has no trapped geodesics see theorem 1 3 for a precise statement which generalizes the early results for single eigenfunctions by hassell and tao in 2002

on nonconvergence of adjoint semigroups for control systems with delays it is shown that the adjoints of a spline based approximation scheme for delay equations do not converge strongly

pythagoras numbers of fields the study of sums of squares in a ring or a field is a classic topic in algebra and number theory in this context several questions arise naturally for example which elements can be represented as sums of squares and if an element can be written as a sum of squares how many squares are actually needed for instance for an integer n to be a sum of squares of integers we must obviously have that n 0 and it is well known that each such n can in fact be written as a sum of four integer squares the first published proof of this result being due to lagrange furthermore in general one needs four squares as for example 7 cannot be written as a sum of three integer squares this result readily carries over to the rational numbers q x c q is a sum of squares of elements in q if and only if x 0 and in this case x can be written as a sum of 0 for all a al a c ir where f is defined f is then called positive semi definite is f a sum of squares in f it can readily be shown that in order to be a sum of squares f must necessarily be positive semi definite this problem has been given a positive answer by artin in 1927 a one can furthermore show that in this situation p f 3 the precise value is not known for this and more on generalized versions of hilbert s 17th problem we refer to the exposition given in chapters 6 and 7 of pfister s beautiful book pf where one also finds an account of the current state of knowledge regarding problems related to the pythagoras number also for rings one question asked there was which

spline approximations for neutral functional differential equations based on an abstract approximation theorem for text c 0 semigroups trotter kato theorem we present an algorithm where linear autonomous functional differential equations of neutral type are approximated by sequences of ordinary differential equations of increasing dimensions numerical examples using cubic splines and cubic hermite splines illustrate the theoretical results

numerical approximation for the infinite dimensional discrete time optimal linear quadratic regulator problem an abstract approximation framework is developed for the finite and infinite horizon discrete time linear quadratic regulator problems for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional hilbert space the schemes included in the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law convergence arguments are given examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered spline based finite element schemes for these classes of problems together with numerical results are presented and discussed