Narrow Results

The purpose of this work is to study a class of singular elliptic system involving the $p(x)$-Laplace operator of the form

The purpose of this paper is to address the problem of assessing the stability of singular time-varying delay systems. In order to highlight the relations between the delay and the state, the singular system is transformed into a neutral form. Then, a model transformation using a three-terms approximation of the delayed state is exploited. Based on the lifting method and the Lyapunov–Krasovskii functional (LKF) method, a new linear matrix inequality (LMI) is obtained, allowing conclusions on stability to be drawn using the scaled small gain theorem (SSG). The use of SSG theorem for stability of singular systems with time-varying delay has not been investigated elsewhere in the literature. This represents the main novelty of this article. The result is applicable for assessing the stability of both singular systems and neutral systems with time-varying delays. The less conservativeness of the stability test is illustrated by comparison with recent literature results.

The dynamical model of a nonlinear wave is governed by a partial differential equation which is a special case of the $b$-family equation. Its traveling system is a singular system with a singular straight line. On this line, there exist two degenerate nodes of the associated regular system. By using the method of dynamical systems and the theory of singular traveling wave systems, in this paper we show that, corresponding to global level curves, this wave equation has global periodic wave solutions and anti-solitary wave solutions. We obtain their exact representations. Specially, we discover some new phenomena. (i) Infinitely many periodic orbits of the traveling wave system pass through the singular straight line. (ii) Inside some homoclinic orbits of the traveling wave system there is not any singular point. (iii) There exist periodic wave bifurcation and double anti-solitary waves bifurcation.

In this paper, we consider a simple equation which involves a parameter $k$, and its traveling wave system has a singular line.

Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties:

• (i)In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit.
• (ii)When $k<\frac{1}{2}$, in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave.
• (iii)When $k\ge \frac{1}{2}$, in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation.

Secondly, we perform numerical simulations to visualize the above properties.

Finally, when $k<\frac{1}{8}$ and the constant wave speed equals $\frac{1}{2}(1\pm \sqrt{1-8k})$, we give exact expressions to the above phenomena.

A technique for imperfection sensitivity analysis with reference to the geometrically nonlinear analysis of structures is presented. The paper discusses how detailed information on structural behavior can be obtained with less computational cost.

The imperfection sensitivity analysis of structures is carried out by detecting the critical states on the equilibrium path relating to the various imperfections. In fact we can investigate the behavior of imperfect structures without considering the imperfect equilibrium curve. In this work we obtained the fold line, the one-dimensional equilibrium subset of limit points relating to different values of imperfection, by asymptotic extrapolation from a known singular point. The evaluation of this singular point on the perfect equilibrium curve is carried out by a path-following algorithm. A procedure that overcomes the ill-conditioning of the systems defined in the critical points is described. This proves to be highly advantageous in terms of computational cost in comparison with classical methods of analysis.

The paper investigates the behavior of cylindrical shell. In particular the sensitivity analysis for load imperfections is carried out. The asymptotic extrapolation algorithm is also compared with continuation methods for the analysis of imperfect cylindrical shell.

In this paper, the problem of preview control for continuous-time singular stochastic systems is studied by the augmented error system approach. First, a static output feedback is introduced to make the closed-loop system impulse-free. And through the second restricted equivalent transformation, the continuous-time singular stochastic system is transformed into a normal stochastic system and some algebra equations. Then, an assistant system is designed to overcome the difficulty of trying to make derivation to a stochastic system like in deterministic system condition. And the stochastic system after the transformation is translated to the assistant system. In order to make the output of the singular system track the reference signal as accurately as possible without static error, an integrator is introduced. Based on the system after the translation and the error equation, an augmented error system is constructed with the state vector, the error vector and the reference signal. Finally, the tracking problem for the singular stochastic system is transformed into the optimal regulating problem for the augmented error system. By the dynamic programming method, the optimal preview controller for the original singular stochastic system is obtained. The existence and uniqueness solution of the Riccati equation is discussed and the numerical analysis shows the effectiveness for the preview controller designed in this paper.

Due to the complexity of interaction among constituents inside the whole system, it is difficult to establish accurate mathematics models to describe and analyze the complex systems exactly. There are few attempts concerning on the moving process of endocrine disruptor in human bodies, which have been the polluted material worldwide related to the reproduction, existence and development of human being. Focusing on such two challenging issues, a multi-compartment model of endocrine disruptor Benzene moving in the human body complex system is established in this paper. Furthermore, passivity of this model is described systematically. A feedback controller for this descriptor biological complex system is used under the station of strict passivity, and an example of the controller is given for a particular instantiation of the model.