the cauchy problem in one dimensional nonlinear viscoelasticity abstract the initial value problem for a nonlinear hyperbolic volterra equation which models the motion of an unbounded viscoelastic bar is studied under physically motivated assumptions the existence of a unique globally defined classical solution is established provided the initial data are sufficiently smooth and small boundedness and asymptotic behavior are also discussed this analysis is based on energy estimates in conjunction with properties of strongly positive definite kernels

weak solutions of a class of quasilinear hyperbolic integro differential equations describing viscoelastic materials global weak solutions of scalar second order quasilinear hyperbolic integro differential equations with singular kernels are constructed perturbations of rest states are shown to propagate with finite speed smoothing effects of the solution operator are exhibited and conditions for the asymptotic stability of rest states are given the equations arise in viscoelasticity

a decay result to a viscoelastic problem in rn with an oscillating kernel in this paper we consider a linear cauchy viscoelastic problem we show that for compactly supported initial data and for not necessarily decreasing oscillating relaxation function the decay of the first energy of solutions is polynomial the proof relies on some appropriately chosen functionals the combination of which satisfies a certain integral inequality

nonlinear stability of rarefaction waves for a viscoelastic material with memory in this paper we will discuss the stability of rarefaction waves for a viscoelastic material with memory the rarefaction waves for which the sta bility is tested are not themselves solutions to the integrodifferential equations 1 1 governing the viscoelastic material they are solutions to a related equi librium system of conservation laws given by 1 11 we shall show that if the forcing term and the past history are small and if the initial data are close to the rarefaction waves the solutions to 1 1 will approach the rarefaction waves in sup norm as the time goes to infinity j oo with the initial data u x 0 v x 0 u0 x v0 x xer l2 lim uo x v0 x u v x oo

solutions of fractional logistic equations by euler s numbers abstract in this paper we solve in the convergence set the fractional logistic equation making use of euler s numbers to our knowledge the answer is still an open question the key point is that the coefficients can be connected with euler s numbers and then they can be explicitly given the constrained of our approach is that the formula is not valid outside the convergence set the idea of the proof consists to explore some analogies with logistic function and euler s numbers and then to generalize them in the fractional case

uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings abstract this paper is concerned with the mixed initial boundary value problem for semilinear wave equations with complementary frictional dampings and memory effects we successfully establish uniform exponential and polynomial decay rates for the solutions to this initial boundary value problem under much weak conditions concerning memory effects more specifically we obtain the exponential and polynomial decay rates after removing the fundamental condition that the memory effect region includes a part of the system boundary while the condition is a necessity in the previous literature moreover for the polynomial decay rates we only assume minimal conditions on the memory kernel function g without the usual assumption of g controlled by g

global solutions to the equation of viscoelasticity with fading memory abstract we consider the initial history value problem for the one dimensional equation of thermoelasticity with fading memory it is proved that if the data are smooth and small then a unique smooth solution exists globally in time and converges to the constant equilibrium state as time goes to infinity our proof is based on a technical energy method which makes use of the strict convexity of the entropy function and the properties of strongly positive definite kernels

control of underactuated spacecraft with bounded inputs bounded feedback control laws are designed for stabilization and tracking of underactuated spacecraft the flat outputs of the system are computed and are used to generate reference trajectories for the tracking problem

on a class of quasilinear partial integrodifferential equations with singular kernels abstract we prove local and global existence theorems for a model equation in nonlinear viscoelasticity in contrast to previous studies we allow the memory function to have a singularity we approximate the equation by equations with regular kernels and use energy estimates to prove convergence of the approximate solutions

group actions on c algebras 3 cocycles and quantum field theory we study group extensions where acts on a c algebraa given a twisted covariant representation v of the paira we construct 3 cocycles on with values in the centre of the group generated byv these 3 cocycles are obstructions to the existence of an extension of byv which acts ona compatibly with the main theorems of the paper introduce a subsidiary invariant which classifies actions of onv and in terms of which a necessary and sufficient condition for the the cohomology class of the 3 cocycle to be non trivial may be formulated examples are provided which show how non trivial 3 cocycles may be realised the framework we choose to exhibit these essentially mathematical results is influenced by anomalous gauge field theories we show how to interpret our results in that setting in two ways one motivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge groups in order to make comparisons with the usual approach to cohomology in gauge theory we conclude with a lie algebra version of the invariant and the 3 cocycle

nonlinear stability of travelling wave solutions for viscoelastic materials with fading memory abstract in this paper we shall discuss the stability of smooth monotone travelling wave solutions for viscoelastic materials with memory it is known that a smooth monotone travelling wave solution exists for 1 1 if the end states are close and satisfy the rankine hugoniot condition for such a travelling wave we shall show that if the initial data are close to a travelling wave solution the solutions to 1 1 will approach the travelling wave solution in sup norm as the time goes to infinity for the constitutive relations we shall discuss two cases convex and nonconvex

integro differential equations of hyperbolic type with positive definite kernels abstract the main purpose of this work is to study the damping effect of memory terms associated with singular convolution kernels on the asymptotic behavior of the solutions of second order evolution equations in hilbert spaces for kernels that decay exponentially at infinity and possess strongly positive definite primitives the exponential stability of weak solutions is obtained in the energy norm it is also shown that this theory applies to several examples of kernels with possibly variable sign and to a problem in nonlinear viscoelasticity

nonlinear waves for viscoelasticity with fading memory abstract nonlinear waves for a model of viscoelastic materials are constructed and analyzed there are traveling waves expansion waves and diffusion waves the stress in the model depends on the present strain as well as its past history traveling waves are studied based on the admissibility criterion for the models with equilibrium or instantaneous elastic response expansion waves and diffusion waves are constructed in the time asymptotic sense asymptotic states with given end states are constructed and consist of these elementary waves

error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws we consider a general class of finite volume schemes on unstructured but quasi uniform meshes for first order systems of hyperbolic balance laws on unstructured meshes provided the system is equipped with at least one entropy entropy flux tuple and the associated cauchy problem allows for a classical solution u we give conditions such that the finite volume approximation u h converges to u if the mesh parameter h tends to zero in fact we prove an error estimate of the form u u h l 2 le c sqrt h where c is independent of h the proof relies on a stability result for classical solutions in the class of entropy solutions due to dafermos arch rational mech anal 94 1979 pp 373 389 and diperna indiana univ math j 28 1979 pp 137 188 finally we present examples such that the conditions to apply the general convergence estimate can be satisfied at least in part the examples cover general scalar equations weakly coupled systems and the system of elastodynamics in one dimension moreover we generalize the concept of entropy conservative methods due to tadmor math comp 49 1987 pp 91 103 and show how this can be used to establish the convergence of finite volume methods for the system s case

determination of optimal feedback terminal controllers for general boundary conditions using generating functions given a nonlinear system and a performance index to be minimized we present a general approach to expressing the finite time optimal feedback control law applicable to different types of boundary conditions starting from the necessary conditions for optimality represented by a hamiltonian system we solve the hamilton jacobi equation for a generating function for a specific canonical transformation this enables us to obtain the optimal feedback control for fundamentally different sets of boundary conditions only using a series of algebraic manipulations and partial differentiations furthermore the proposed approach reveals an insight that the optimal cost functions for a given dynamical system can be decomposed into a single generating function that is only a function of the dynamics plus a term representing the boundary conditions this result is formalized as a theorem the whole procedure provides an advantage over methods rooted in dynamic programming which require one to solve the hamilton jacobi bellman equation repetitively for each type of boundary condition the cost of this favorable versatility is doubling the dimension of the partial differential equation to be solved

optimal control of partitioned hybrid systems via discrete time hamilton jacobi theory abstract computational framework for optimal control of hybrid systems with a partitioned state space is presented it is shown that necessary conditions for optimality for a discrete time dynamic system can be solved concurrently for various boundary conditions according to the recent development of discrete time hamilton jacobi theory this unique property is utilized to construct computationally efficient numerical optimization of hybrid systems where discrete switching dynamics occurs at the boundary between partitions of the configuration space a benchmark example shows that the proposed approach has substantial computational advantages compared with the existing ones

existence and decay of solutions of a viscoelastic equation with a nonlinear source abstract in a bounded domain we consider u tt u 0 t g t u d u u where 0 and g is a nonnegative and decaying function we prove a local existence theorem and show for certain initial data and suitable conditions on g and that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function g

pointwise time asymptotic behavior of solutions to a general class of hyperbolic balance laws abstract in this paper we are interested in a general class of hyperbolic balance laws within the frame work of existence of a convex entropy function that symmetrizes the system in certain sense and the kawashima shizuta condition we study the large time behavior of the solution to the cauchy problem in one space dimension for a solution around a constant equilibrium state we predetermine the asymptotic solution as a superposition of the constant state and diffusion waves along the equilibrium characteristic directions we estimate the remainder in the pointwise sense both in space and in time the decay rates in various directions are optimal which further give the optimal l p rates with 1 p applications to several examples including the kerr debye model are given

decay rates of solutions of an anisotropic inhomogeneous n dimensional viscoelastic equation with polynomially decaying kernels we consider the anisotropic and inhomogeneous viscoelastic equation and we prove that the first and second order energy decay polynomially as time goes to infinity when the relaxation function also decays polynomially to zero that is if the kernelg ijkl satisfies dot g ijkl leqq c 0 g ijkl 1 frac 1 p and g ijkl g ijkl 1 frac 1 p in l 1 mathbb r for p 2such that 2 m 1 p then the first and second order energy decay as frac 1 1 t q withq 2 m 1

three cocycle in mathematics and physics it is shown that the three cocycle arises when a representation of a transformation group is nonassociative so that the jacobi identity fails a physical setting is given when the translation group in the presence of a magnetic monopole is represented by gauge invariant operators a trivial three cocycle occurs insisting that finite translations be associative leads to dirac s monopole quantization condition attention is called to the possible relevance of three cocycles in the quark model s u 6 ensuremath bigotimes u 6 algebra

h norm computation of linear continuous time periodic systems by a structure preserving algorithm a new reliable structure preserving algorithm for computing h norm of linear continuous time periodic systems is proposed in this paper in the computation of the h norm no riccati differential equations are needed to solve and only eigenvalues of a monodromy matrix of the associated periodic hamiltonian system will be evaluated first the monodromy matrix is expressed as the product of state transition matrices of the hamiltonian system second these state transition matrices which have been proved to be symplectic matrices are evaluated by a structure preserving magnus series method then in order to preserve the standard symplectic form of the monodromy matrix the structure preserving matrices obtained by state transition matrices are employed to compute the monodromy matrix at last the effectiveness and the high accuracy of the proposed structure preserving algorithm are demonstrated by numerical examples

convergence to diffusion waves of solutions to nonlinear viscoelastic model with fading memory we study the large time behavior inl2 of solutions to a model for the motion of an unbounded homogeneous viscoelastic bar with fading memory decay rates for the solutions are obtained under the assumption that the initial data and histories are smooth and small moreover convergence of the solutions to diffusion waves which are solutions of burgers equations is proved and rates are obtained our method is based on the study of properties of the solutions to the linearized system in the fourier space

global solution for the optimal feedback control of the underactuated heisenberg system we present a global solution for an optimal feedback controller of the underactuated heisenberg system or nonholonomic integrator employing a recently developed technique based on generating functions appearing in the hamilton jacobi theory we circumvent a singularity caused by under actuation to develop a nonlinear optimal feedback control in an implicitly analytical form the systematic procedure to deal with under actuation indicates that generating functions should be effective tools for solving general underactuated optimal control problems

on the mickelsson faddeev extension and unitary representations the mickelsson faddeev extension is a 3 space analogue of a kac moody group where the central charge is replaced by a space of functions of the gauge potential this extension is a pullback of a universal extension where the gauge potentials are replaced by operators in a schatten ideal as in non commutative differential geometry our main result is that the universal extension cannot be faithfully represented by unitary operators on a separable hilbert space we also examine potential consequences of the existence of unitary representations for the mickelsson faddeev extension

local analyticity in weighted spaces and applications to stability problems for volterra equations we study the qualitative properties of the solutions of linear convolution equations such as x x t f x a x f and x t f we are especially concerned with finding conditions which ensure that these equations have resolvents which belong to or are determined up to a term belonging to certain weighted l1 spaces our results are obtained as consequences of more general banach algebra results on functions that are locally analytic with respect to the elements of a weighted l1 space in particular we derive a proposition of wiener levy type for weighted l1 spaces which underlies all subsequent results our method applies equally well to equations more general than those mentioned above we unify and sharpen the results of several recent papers on the asymptotic behavior of volterra convolution equations of the types mentioned above and indicate how many of them can be extended to the fredholm case in addition we give necessary and sufficient conditions on the perturbation term f for the existence of bounded or integrable solutions x in some critical cases when the corresponding limit equations have nontrivial solutions

existence and decay to contact problems for thermoviscoelastic plates we consider the dynamic contact problem for thermoviscoelastic plates we show the existence of weak solutions as well as exponential decay of the solution

realizing 3 cocycles as obstructions the occurrence of a 3 cocycle in quantum mechanics or quantum field theory has been interpreted somewhat paradoxically as a breakdown of the jacobi identity the main result of this paper is that the 3 cocycle in chiral qcd arises as an obstruction which prevents the existence of a certain extension of one lie algebra by another this obstruction may be avoided by constructing a modified lie algebra extension consisting of derivations on the algebra generated by the fields however the 3 cocycle then appears when an attempt is made to implement these derivations by commutation with unbounded operators in the canonical equal time formalism assuming the existence of these unbounded operators is what leads to the violation of the jacobi identity