On the inverse problem for nonlinear strongly damped wave equations with discrete random noise

In this paper, we consider the existence of a solution u(x, t) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we use the trigonometric method in non-parametric regression associated with the truncated expansion method. We investigate the convergence rate under some a priori assumptions on an exact solution in both L 2 and H q (q > 0) norms. Moreover, a numerical example is given to illustrate our results.

Broadcast control of multi-robot systems with norm-limited update vector

This article addresses a problem in standard broadcast control framework which leads to an unstable solution in a certain motion-coordination task. First, the unstable phenomenon in a certain motion-coordination task is illustrated using standard broadcast control framework. This issue calls for modification to the standard broadcast control framework by limiting the norm of the update vector of robots’ positions into a constant value. Then, we demonstrate that the modified broadcast controller achieves the convergence with the probability of 1. Finally, we illustrate in numerical simulations that the modified broadcast controller can effectively solve the instability issue and also may improve the convergence time as compared to the standard broadcast controller.

A Mathematical Modeling for Stability of Ice Shelves

<p>Iceberg calving is the reason for more than half of mass loss in both Greenland and Antarctica. It also indirectly contributes to sea-level rise; changes in calving rate can shorten the ice shelves, speed up the grounded ice and increase changes in ice sheets. Therefore, having a better understanding of this phenomenon by a mathematical modeling seems essential.<br>Lacking of a precise representation of calving in ice sheet and glacier models may yield to nonphysical predictions.<br>We perform a parameter study to identify groups of key parameters. Here we use boundary element method and compare our result to works done by van der Veen (1998) and Nick et al. (2010). <br> A hydraulic crack propagation is assumed to happen vertically from both base and surface of the shelf. The solution for different scenarios is analysed in the form of stability of a dynamical system. An unstable solution results in an iceberg calving which leads us to a general calving law. </p>

On effect of surface tension in the Rayleigh–Taylor problem of stratified viscoelastic fluids

In this article, we investigate the effect of surface tension in the Rayleigh–Taylor (RT) problem of stratified incompressible viscoelastic fluids. We prove that there exists an unstable solution to the linearized stratified RT problem with a largest growth rate Λ under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold $\vartheta _{c}$ ϑ c ). Moreover, for this instability condition, the largest growth rate $\varLambda _{\vartheta }$ Λ ϑ decreases from a positive constant to 0, when ϑ increases from 0 to $\vartheta _{c}$ ϑ c , which mathematically verifies that the internal surface tension can constrain the growth of the RT instability during the linear stage.

Amplitude Incremental Method: A Novel Approach to Capture Stable and Unstable Solutions of Harmonically Excited Vibration Response of Functionally Graded Beams under Large Amplitude Motion

An interesting phenomenon is observed while conducting numerical simulation of non-linear dynamic response of FGM (functionally graded material) beam having large amplitude motion under harmonic excitation. Instead of providing a frequency sweep (forward or backward), if amplitude is incremented and response frequency is searched for a particular amplitude of vibration, solution domain can be enhanced and stable as well as unstable solution can be obtained. In the present work, first non-linear differential equations of motion for large amplitude vibration of a beam, which are obtained using Timoshenko beam theory, are converted into a set of non-linear algebraic equations using harmonic balance method. Subsequently an amplitude incremental iterative technique is imposed in order to obtain steady-state solution in frequency amplitude plane. It is observed that the method not only shows very good agreement with the available research but the domain of applicability of the method is enhanced up to a considerable extent as the stable and unstable solution can be captured. Subsequently forced vibration response of FGM beams are analysed.

Entropy production and multiple equilibria: the case of the ice-albedo feedback

Nonlinear feedbacks in the Earth System provide mechanisms that can prove very useful in understanding complex dynamics with relatively simple concepts. For example, the temperature and the ice cover of the planet are linked in a positive feedback which gives birth to multiple equilibria for some values of the solar constant: fully ice-covered Earth, ice-free Earth and an intermediate unstable solution. In this study, we show an analogy between a classical dynamical system approach to this problem and a Maximum Entropy Production (MEP) principle view, and we suggest a glimpse on how to reconcile MEP with the time evolution of a variable. It enables us in particular to resolve the question of the stability of the entropy production maxima. We also compare the surface heat flux obtained with MEP and with the bulk-aerodynamic formula.

Entropy production and multiple equilibria: the case of the ice-albedo feedback

Nonlinear feedbacks in the Earth System provide mechanisms that can prove very useful in understanding complex dynamics with relatively simple concepts. For example, the temperature and the ice cover of the planet are linked in a positive feedback which gives birth to multiple equilibria for some values of the solar constant: fully ice-covered Earth, ice-free Earth and an intermediate unstable solution. In this study, we show an analogy between a classical dynamical system approach to this problem and a Maximum Entropy Production (MEP) principle view, and we suggest a glimpse on how to reconcile MEP with the time evolution of a variable. It enables us in particular to resolve the question of the stability of the entropy production maxima. We also compare the surface heat flux obtained with MEP and with the bulk-aerodynamic formula.