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Abstract: This study investigates the exact controllability problem for a vibrating non-classical Euler–Bernoulli micro-beam whose governing partial differential equation (PDE) of motion is derived based on the non-classical continuum mechanics. In this paper, it is proved that via boundary controls, it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time. This control objective is achieved based on the PDE model of the system which causes that spillover instabilities do not occur. [Copyright &y& Elsevier]

Abstract: In this paper we study the transmission for a partially viscoelastic beam, that is, a beam which is composed of two components, elastic and viscoelastic. In the rotation angle of the filaments of the beam, and , the dissipation is occasioned by the memory effect. In the transverse vibrations and we do not have dissipation and the system is purely elastic. For this type of beam we show the uniform stabilization, i.e., the rate of decay has directly relation with the velocity of the relaxation function. Display Omitted [Copyright &y& Elsevier]

Abstract: In this paper, we first present the Grammian determinant solutions to the non-isospectral and variable-coefficient Kadomtsev–Petviashvili (vcKP) equation. Then, by using the pfaffianization procedure of Hirota and Ohta, a new non-isospectral and variable-coefficient integrable coupled system is generated. Moreover, Gramm-type pfaffian solutions of the pfaffianized system are proposed. [Copyright &y& Elsevier]

Abstract: The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when , where , and are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension , the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if , and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown. [Copyright &y& Elsevier]

The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations (Trans. Amer. Math. Soc. 350 (1998) 4265–4311, Internat. J. Bifur. Chaos 8 (1998) 423–445, Internat. J. Bifur. Chaos 8 (1998) 447–470, J. Math. Phys. 39 (1998) 6459–6489, Internat. J. Bifur. Chaos 12 (2002) 535–559). In this paper, we characterize nonisotropic chaotic vibration by means of the total variation theory. We obtain the classification results on the growth of the total variation of the snapshots on the spatial interval in the long-time horizon with respect to two parameters entering different regimes in R2. [Copyright &y& Elsevier]

In continuation of Part (I) of this paper in issue of JMAA, properties of Spatial Vibration Methods (SVM) and corresponding solutions are subsequently presented. Additionally, SVM is qualitatively compared with the Finite Element Method and its applications to other problem areas are listed. [Copyright &y& Elsevier]