stabilization of the variable coefficient structural acoustic model with curved middle surface and delay effects in the structural component abstract this paper is concerned with the uniform energy decay rates of a structural acoustic model which describes the interactions between the acoustic medium and the elastic structural in an acoustic chamber where one wall is flexible and curved the coupled system considered consists of a variable coefficient wave equation and a variable coefficient plate equation defined on the riemannian manifold with the coupling on the interface between the acoustic medium and the flexible wall both the components of the dynamics are subject to nonlinear boundary damping furthermore a nonlinear delay term acting in the boundary feedbacks of the structure component is considered the riemannian geometry method is applied to derive the energy estimates of the nonlinear coupled system with variable coefficients the uniform energy decay rates of the nonlinear variable coefficient structural acoustic system with delay effects on the riemannian manifold are quantified by a solution to a constructed nonlinear ode

on a structural acoustic model which incorporates shear and thermal effects in the structural component abstract in this paper we consider a structural acoustic model which takes account of thermal effects over and above displacement rotational inertia and shear effects in the flat flexible structural component of the model thus the structural medium is a reissner mindlin plate into which an additional degree of freedom viz temperature variation in the plate has been introduced and the constitutive equations for the structural acoustic model couple parabolic dynamics with hyperbolic dynamics we show unique solvability of the mathematical model and investigate the effect of the presence of thermal effects on the mechanical dissipation devices needed to attain uniform stabilization of the two dimensional model in which the structural component is a timoshenko beam it turns out that as in linear structural acoustic models which use the euler bernoulli equation or the kirchoff equation to describe the deflections of the thermo elastic structural medium uniform stabilization of the energy associated with the model can be attained without introducing mechanical dissipation at the free edge of the beam open problems with regard to the stabilization of the three dimensional model are outlined

uniform stability in structural acoustic models with flexible curved walls abstract the aim of this paper is twofold first we develop an explicit extension of the kirchhoff model for thin shells based on the model developed by michel delfour and jean paul zolesio this model relies heavily on the oriented distance function which describes the geometry once this model is established we investigate the uniform stability of a structural acoustic model with structural damping the result no longer requires that the active wall be a plate it can be virtually any shell provided that the shell is thin enough to accommodate the curvatures

existence and energy decay rates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback abstract this paper is concerned with the existence and energy decay estimates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback the existence of solutions to the nonlinear system is obtained by using galerkin method combined with some energy estimates furthermore applying riemannian geometry method we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term

energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time varying delay in the boundary feedback abstract a variable coefficient wave equation with acoustic boundary conditions and a time varying delay in the boundary feedback is considered applying the riemannian geometry method we show that the decay rates of the nonlinear system with a time varying delay are described by solutions to a first order ode

controllability for transmission wave plate equations on riemannian manifolds abstract in this paper we consider controllability for transmission system of coupling wave equations with euler bernoulli equations on riemannian manifolds we show that such a system is exactly controllable by boundary controls only along the exterior boundary which means there is no control on the interface of the transmission system our proofs rely on the hilbert uniqueness method hum and geometric multiplier method

blow up problem for semilinear heat equation with absorption and a nonlocal boundary condition abstract in this paper we consider a semilinear parabolic equation u t u c x t u p for x t 0 with nonlinear and nonlocal boundary condition u 0 k x y t u l d y and nonnegative initial data where p 0 and l 0 we prove some global existence results criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given

on the observability inequalities for exact controllability of wave equations with variable coefficients this paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part subject to dirichlet or neumann boundary controls the observability inequalities are established by the riemannian geometry method under some geometric condition for the dirichlet problem and for the neumann problem respectively next a number of nontrivial examples are presented to verify the observability inequality in particular a counterexample is given without boundary exact controllability where the control is exerted on the whole boundary

nonlinear boundary stabilization of the wave equations with variable coefficients and time dependent delay abstract we study the stabilization the wave equations with variable coefficients in a bounded domain and nonlinear boundary feedback with time dependent delay by the riemannian geometry methods and a suitable assumption of delay we obtain the exponential decay when the nonlinear terms are linearly bounded and a general decay when the nonlinear terms have no upper growth bound

stabilization of transmission coupled wave and euler bernoulli equations on riemannian manifolds by nonlinear feedbacks abstract we consider the transmission system of coupling wave equations with euler bernoulli equations on riemannian manifolds by introducing nonlinear boundary feedback controls we establish the exponential and rational energy decay rate for the problem our proofs rely on the geometric multiplier method

not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks we present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls we show that in a general case when the controls are distributed stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks we also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems

stabilization of the euler bernoulli plate with variable coefficients by nonlinear internal feedback we consider the energy decay for solutions of the euler bernoulli plate equation with variable coefficients where a nonlinear internal feedback acts in a suitable subregion of the domain the riemannian geometric method is used to deal with variable coefficient problems when the feedback region has a structure similar to that for the wave equation with constant coefficients we establish the stabilization of the system in the case of fixed boundary conditions several energy decay rates are established according to various growth restrictions on the nonlinear feedback near the origin and at infinity we further show that unlike for the case of constant coefficients choices of such feedback regions depend not only on the type of boundary conditions but also on the curvature of a riemannian metric based on the coefficients of the system

on a structural acoustic model with interface a reissner mindlin plate or a timoshenko beam abstract this paper is concerned with a model which describes the interaction of sound and elastic waves in a structural acoustic chamber in which one wall is flexible and flat the model is new in the sense that the composite dynamics of the three dimensional structure is described by the linearized equations for a gas defined on the interior of the chamber and the reissner mindlin plate equations on the two dimensional flat wall of the chamber while if a two dimensional acoustic chamber is considered the timoshenko beam equations describe the deflections of the one dimensional wall with a view to achieving uniform stabilization of the structure linear feedback boundary damping is incorporated in the model viz in the wave equation for the gas and in the system of equations for the vibrations of the elastic medium we present the uniform stability result for the case of a two dimensional chamber and outline the method for the three dimensional model which shows strong resemblance with the system of dynamic plane elasticity

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

stabilization of euler bernoulli plate equation with variable coefficients by nonlinear boundary feedback abstract the aim of this paper is to investigate the uniform stabilization of euler bernoulli plate equation with variable coefficients in the principle part subject to nonlinear boundary feedback laws the exponential or rational energy decay rate is obtained by the multiplier method and the riemannian geometry method

observability inequalities for shallow shells we consider some observability inequalities from boundary for a general shallow shell with a middle surface of any shape at first an estimate is established by the geometric multiplier method in the case that no boundary conditions are imposed under some checkable geometric conditions then our results yield continuous observability estimates for two kinds of boundary conditions which have a physical meaning with an explicit observability time and hence by duality exact controllability results

an alternative choice of smoothing for kernel based density estimates in discrete discriminant analysis summary the kernel method of estimating the cell probabilities of a multivariate categorical distribution due to aitchison aitken 1976 depends crucially on an unknown smoothing parameter a a method of estimating a is introduced which is explicitly connected to multivariate discrimination the method based on maximization of the leaving one out estimator of the nonerror rate is shown to be bayes risk strongly consistent an example is given to illustrate the application