Asymptotic Analysis - Volume 87, issue 3-4

Authors: He, Qi | Yin, G.

Article Type: Research Article

Abstract: This work develops asymptotic properties of randomly switching and time inhomogeneous dynamic systems under Brownian perturbation with a small diffusion. The switching process is modeled by a continuous-time Markov chain, which portraits discrete events that cannot be modeled by a diffusion process. In the model, there are two small parameters. One of them is ε associated with the generator of the continuous-time, inhomogeneous Markov chain, and the other is δ=δε signifies the small intensity of the diffusion. Assume ε→0 and δε →0 as ε→0. This paper focuses on large deviations type of estimates for such Markovian switching systems with …small diffusions. The ratio ε/δε can be a nonzero constant, or equal to 0, or ∞. These three different cases yield three different outcomes. This paper analyzes the three cases and present the corresponding asymptotic properties. Show more

Keywords: large deviations, Markov chain, averaging principle

DOI: 10.3233/ASY-131198

Citation: Asymptotic Analysis, vol. 87, no. 3-4, pp. 123-145, 2014

Authors: Shubov, Marianna A.

Article Type: Research Article

Abstract: In this paper we investigate the Rayleigh beam model with non-dissipative boundary conditions recently considered in the literature. The beam is clamped at the left end and subject to a feedback control type boundary conditions at the right. The components of the 2-dim input vector are shear and moment at the right end and the components of the observation vector are time derivatives of displacement and slope at the right end. The input is related to the observation through a co-diagonal matrix depending on two non-negative control parameters. The paper contains two main results. First we derive the leading term, …the second order term, and the remainder in the asymptotic representation of the eigenmodes of the system. It follows from spectral asymptotics that the system has an infinite alternating sequence of stable and unstable eigenmodes. This type of instability has been recently obtained in the literature by a different method. The instability result might be of interest for energy harvesting models where destabilization of a system is desirable. The second result is the fact that the generalized eigenvectors of the dynamics generator of the system form a Riesz basis in the state space equipped with the energy metric. This result implies, in particular, that the system generates a C0 -semigroup in the state space. Show more

Keywords: discrete spectrum of normal eigenvalues, spectral asymptotics, non-selfadjoint differential operator, Riesz basis property of root vectors

DOI: 10.3233/ASY-131206

Citation: Asymptotic Analysis, vol. 87, no. 3-4, pp. 147-190, 2014

Authors: Savostianov, Anton | Zelik, Sergey

Article Type: Research Article

Abstract: Dissipative wave equations with critical quintic nonlinearity and damping term involving the fractional Laplacian are considered. The additional regularity of energy solutions is established by constructing the new Lyapunov-type functional and based on this, the global well-posedness and dissipativity of the energy solutions as well as the existence of a smooth global and exponential attractors of finite Hausdorff and fractal dimension is verified.

Keywords: damped wave equation, fractional damping, critical nonlinearity, global attractor, smoothness

DOI: 10.3233/ASY-131208

Citation: Asymptotic Analysis, vol. 87, no. 3-4, pp. 191-221, 2014

Authors: Heumann, Holger | Vogelius, Michael S.

Article Type: Research Article

Abstract: We extend and analyse an enhanced approximate cloaking scheme, which was recently introduced by Ammari, Kang, Lee and Lim [Comm. Math. Phys. 317(1) (2013), 253–266] to cloak a domain with a fixed, homogeneous Neumann boundary condition. Subject to the solvability of a finite set of algebraic equations we construct an approximate cloak for the two dimensional transmission case, which achieves invisibility of the order ρ2N+2 while maintaining the same level of local anisotropy as earlier schemes of order ρ2 [Inverse Problems 24(1) (2008), 015016]. The approximate cloak and the invisibility estimate is independent of the objects being cloaked. …Finally, we present analytical as well as numerical evidence for the solvability of the required algebraic equations. Show more

Keywords: enhanced cloaking, transmission setting, conductivity problem

DOI: 10.3233/ASY-131209

Citation: Asymptotic Analysis, vol. 87, no. 3-4, pp. 223-246, 2014

Article Type: Other

Citation: Asymptotic Analysis, vol. 87, no. 3-4, pp. 247-247, 2014