random products of contractions in banach spaces we show that the random product of a finite number of w contractions converges weakly in all smooth reflexive banach spaces if one of the contractions is compact then the convergence is uniform let x 1 be a real banach space and t x x a linear operator recall that t is called a contraction if i tvi i v i for all vectors v in x we say that a contraction t satisfies condition w if whenever v is bounded and ivni itvni 0 it follows that the weak limn o vn tvn 0 the algebraic semigroup s generated by a possibly infinite set of contractions will be said to satisfy condition w if for any bounded sequence of vectors vn c x and a sequence of words wn from s such that i v n wnvni 0 the weak limn oo vn wnvn 0 now let t1 t2 be a sequence of contractions that satisfy condition w and let r be a self mapping of the set of natural numbers a random product of these contractions is the sequence sn n 1 2 defined by sn tr n tr n 1 tro 1 our purpose in this paper is to study the convergence properties of such products such a study is not only of intrinsic interest but is also motivated by applications to the numerical solution of partial differential equations 2 14 and linear inequalities 19 approximation theory 16 9 and computer tomography 18 11 we begin by noting several properties of contractions which satisfy condition w we shall call such contractions w contractions for short we then consider the algebraic semigroup generated by a finite number of w contractions and establish a weak convergence theorem for random products of these contractions in all smooth reflexive banach spaces theorem 1 this seems to be the first such result outside hilbert space it includes the hilbert space theorem due to amemiya and ando 1 we continue with a weak convergence result for semigroups generated by an infinite sequence of contractions theorem 2 finally we obtain a uniform convergence theorem for random received by the editors october 24 1988 and in revised form april 4 1989 1980 mathematics subject classification 1985 revision primary 47a05 secondary 47b05 65j10

weak compactness and fixed point property for affine mappings abstract it is shown that a closed convex bounded subset of a banach space is weakly compact if and only if it has the generic fixed point property for continuous affine mappings the class of continuous affine mappings can be replaced by the class of affine mappings which are uniformly lipschitzian with some constant m 1 in the case of c0 the class of affine mappings which are uniformly lipschitzian with some constant m 6 in the case of quasi reflexive james space j and the class of nonexpansive affine mappings in the case of l embedded spaces

energy decay rate of wave equations with indefinite damping abstract we consider the one dimensional wave equation with an indefinite sign damping and a zero order potential term using a shooting method we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients in addition if the damping coefficient is more positive than negative we prove that the energy of system decays uniformly exponentially to zero this sharp result generalizes a previous work of freitas and zuazua 1996

new convexity and fixed point properties in hardy and lebesgue bochner spaces we show that for the hardy class of functions h1 with domain the ball or polydisc in cn a certain type of uniform convexity property the uniform kadec klee huff property holds with respect to the topology of pointwise convergence on the interior which coincides with both the topology of uniform convergence on compacta and the weak topology on bounded subsets of h1 also we show that if a banach space x has a uniform kadec klee huff property then the lebesgue bochner space lp x 1 p must have a related uniform kadec klee huff property consequently by known results the above spaces have normal structure properties and fixed point properties for non expansive mappings

exponential stability of an abstract nondissipative linear system in this paper we consider an abstract linear system with perturbation of the form frac dy dt ay varepsilon by on a hilbert space cal h where a is skew adjoint b is bounded and varepsilon is a positive parameter motivated by a work of freitas and zuazua on the one dimensional wave equation with indefinite viscous damping p freitas and e zuazua j differential equations 132 1996 pp 338 352 we obtain a sufficient condition for exponential stability of the above system when b is not a dissipative operator we also obtain a hautus type criterion for exact controllability of system a g where g is a bounded linear operator from another hilbert space to cal h our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping as well as the exponential stabilization of the elastic systems with noncolocated observation and control

energy methods for quasilinear hyperbolic initial boundary value problems applications to elastodynamics the aim of this paper is to show how energy methods can be used to establish local in time existence of smooth solutions to certain initial boundary value problems for systems of quasilinear hyperbolic partial differential equations our particular objective is to treat the equations of nonlinear elastodynamics

on uniform opial condition and uniform kadec klee property in banach and metric spaces in this paper we define a property that implies uniform opial condition or uniform kadec klee property in banach spaces this will unify many examples in the literature finally we will extend these properties into hyperbolic metric spaces

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

new doubling algorithm for the discrete periodic riccati equation the periodic riccati equation that results from periodic state space models plays an important role in many fields of mathematics science and engineering in most applications it is essential that the solution to the riccati equation be obtained in the shortest possible time such a computationally effective doubling algorithm that solves the discrete periodic riccati equation is proposed in this paper moreover the memory requirements and the calculation burden needed for the sequential implementation of the proposed algorithm are established and compared to the memory requirements and the calculation burden needed for the sequential implementation of classical algorithms the basic conclusion of the above comparison is that the calculation time required to solve the periodic riccati equation using the classical algorithms is in general much greater than the calculation time required to solve the periodic riccati equation by using the proposed algorithm finally the numerical behavior of the proposed algorithm is tested through simulation examples it is established that the proposed algorithm is fast computationally efficient and numerically stable and possesses very good parallelism efficiency

stability results for the wave equation with indefinite damping abstract the one dimensional wave equation with damping of indefinite sign in a bounded interval with dirichlet boundary conditions is considered it is proved that solutions decay uniformly exponentially to zero provided the damping potential is in thebv class has positive average is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes this sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms the method of proof relies on the developments by s cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa

stabilization of the wave equations with potential and indefinite damping by means of global carleman type estimate we study the stabilization problem of the wave equations with potential and indefinite damping the energy decay rate of the system is given explicitly also we obtain an upper bound estimate on the negative damping to guarantee the energy of the system decays exponentially

a survey of recursive algorithms for the solution of the discrete time riccati equation abstract the riccati equation plays a fundamental role in many fields of mathematics science and engineering its solution constitutes an integral prerequisite to the solution of important problems in the above fields due to the importance of the riccati equation there exists considerable literature on its algebraic as well as algorithmic solution a very large number of those studies are devoted to the continuous time riccati equation in this paper we present a survey of classical as well as more recent recursive algorithms that solve the discrete time riccati equatbn emanating from the kaiman filter as well as from the lainiotis filter equations either using per step calculations or the doubling principle it is established that these algorithms converge fast and are numerically stable

perturbing the critically damped wave equation we consider the wave equation with viscous damping the equation is said to be critically damped when the damping is that value for which the spectral abscissa of the associated wave operator is minimized within the class of constant dampings the critically damped wave operator possesses a nonsemisimple eigenvalue we present a detailed study of the splitting of this eigenvalue under bounded perturbations of the damping and subsequently show that the critical choice is a local minimizes of the spectral abscissa over lines in the class of all bounded dampings

exponential stability of the plate equations with potential of second order and indefinite damping we study the exponential stability of the plate equations with potential of second order and indefinite sign damping term by means of a global carleman type estimate and the usual perturbation method we show that the energy of the system decays exponentially provided that the negative damping is sufficiently small both the energy decay rate and the upper bound estimate on the negative damping are given explicitly

exponential decay of energy of evolution equations with locally distributed damping consider an evolution equation with energy dissipation frac partial partial t y x t ay x t by x t on a bounded x domain omega where by signifies a dissipative perturbation to an otherwise energy conserving system this dissipation may be due to medium impurities viscous effects or artificially imposed dampers and stabilizers it is distributed over only part of the domain omega the question of when the dissipation is effective enough to cause uniform exponential decay of energy is examined because of the locally distributed nature of energy dissipation the problem lacks coercivity and is not directly solvable by energy identities thus to get conditions sufficient for uniform exponential decay a different approach needs to be taken provided here is a set of tight sufficient conditions in terms of the influence of the dissipative operatorbon the separated eigenmodes or clustered eigenmodes ofa the main theorems are general enough to treat the wave and bea

eigenfactor solution of the matrix riccati equation a continuous square root algorithm this paper introduces a new algorithm for solving the matrix riccati equation differential equations for the eigenvalues and eigenvectors of the solution matrix are developed in which their derivatives are expressed in terms of the eigenvalues and eigenvectors themselves and not as functions of the solution matrix the solution of these equations yields then the time behavior of the eigenvalues and eigenvectors of the solution matrix a reconstruction of the matrix itself at any desired time is immediately obtained through a trivial similarity transformation this algorithm serves two purposes first being a square root solution it entails all the advantages of square root algorithms such as nonnegative definiteness and accuracy secondly it furnishes the eigenvalues and eigenvectors of the solution matrix continuously without resorting to the complicated route of solving the equation directly and then decomposing the solution matrix into its eigenvalues and eigenvectors the algorithm which handles cases of distinct as well as multiple eigenvalues is tested on several examples through these examples it is seen that the algorithm is indeed more accurate than the ordinary one moreover it is seen that the algorithm works in cases where the ordinary algorithm fails and even in cases where the closed form solution cannot be computed as a result of numerical difficulties

local existence of solution for the initial boundary value problem of fully nonlinear wave equation on considere l existence locale par rapport au temps des solutions pour le probleme aux valeurs limites et initiales d une equation d onde totalement non lineaire

bresse system with indefinite damping abstract we consider the bresse system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign presented only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 1 l 0 l a x d x 0 and a a l 2 for small enough in the arguments we shall also give a new proof of exponential stability for the constant case a a

timoshenko systems with indefinite damping abstract we consider the timoshenko system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign present only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 0 l a x d x 0 and a a l 2 for small enough the decay rate will be described explicitly in the arguments we shall also give a new proof of exponential stability for the constant case a a moreover we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth sdg property i e the type of the induced semigroup coincides with the spectral bound for its generator

exponential stability of wave equations with potential and indefinite damping abstract first we consider the linear wave equation u t t u x x a x u t b x u 0 on a bounded interval 0 l r the damping function a is allowed to change its sign if a 1 l 0 l a x d x is positive and the spectrum of the operator x x b is negative exponential stability is proved for small a a l 2 explicit estimates of the decay rate are given in terms of a and the largest eigenvalue of x x b second we show the existence of a global small smooth solution of the corresponding nonlinear wave equation u t t u x x a x u t b x u 0 if additionally the negative part of a is small enough compared with

exponential stability for wave equations with non dissipative damping abstract we consider the nonlinear wave equation u t t u x x a x u t 0 in a bounded interval 0 l r 1 the function a is allowed to change sign but has to satisfy a 1 l 0 l a x d x 0 for this non dissipative situation we prove the exponential stability of the corresponding linearized system for i possibly large a l with small a a l 2 and ii a class of pairs a l with possibly negative moment 0 l a x sin 2 x l d x estimates for the decay rate are also given in terms of a moreover we show the global existence of smooth small solutions to the corresponding nonlinear system if additionally the negative part of a is small enough

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

asymptotic stability for bresse systems abstract in this paper consider a vibrating system of bresse type in a one dimensional bounded domain with a nonlinear damping acting in the equation of the shear angle displacement and nonlinear localized damping in other equations we show the asymptotic stability without imposing conditions about the equal speed wave propagation using a method developed by lasiecka and tataru 9

asymptotic behavior for almost orbits of a reversible semigroup of non lipschitzian mappings in a metric space let m be a metric space and a hausdorff topology on m such that m is compact let s be a right reversible semitopological semigroup and t s s s a representation of s as asymptotically nonexpansive type self mappings of m and u a bounded almost orbit of we study the convergence of the net u s s s in m when the triplet m satisfies various types of opial conditions our results extend and unify many previously known results 2002 elsevier science usa all rights reserved