Set stability of differential equations with time delay

Set stability and uniform set stability involve known specific bounds on the solutions of differential equations. These concepts are extended to include the case of differential equations with time delay. Liapunov-like theorems are presented which yield sufficient conditions for the set stability of such systems. Examples are given to demonstrate the method.     

Set stability of hereditary systems

For a large class of functional differential equations of the neutral type, sufficient conditions for set stability and uniform set stability are developed. The results arc obtained via Lyapunov function arguments and are extension of the recent work of Heinen and Dimino (1973).     

Lyapunov functions for quadratic differential equations with applications to adaptive control

Conditions are derived for the existence of quadratic Lyapunov functions for vector differential equations with quadratic terms. These conditions are used to establish a set of nonlinear equations for which the null solution is asymptotically stable in the whole. Many of the recent results in the stability of model reference adaptive systems are particular cases of these general conditions.     

粘弹性地基上损伤弹性Timoshenko梁动力学行为

建立了粘弹性地基上损伤弹性Timoshenko梁在有限变形情况下的运动微分方程,这是一组非线性偏微分方程.为了便于分析,首先利用Galerkin方法对该方程组进行简化,得到一组非线性常微分方程.然后利用Matlab软件进行数值模拟,考察了载荷参数、地基粘性参数和弹性参数、损伤对梁振动的影响.采用非线性动力学中的各种数值方法,如时程曲线、相平面图、Poincare截面和分叉图,发现增大地基的粘弹性参数,有利于增强结构运动的稳定性,而损伤会降低结构运动的稳定性.     

An alternative to liapunov's stability method

By introducing two new, basic conditions, finding appropriate Liapunov functions for determining the stability of singular points of a set of ordinary first-order differential equations becomes unnecessary. Also, these conditions based on the ‘flow’ concept, yield at least the same, in most cases even a more complete amount of information on stability as Liapunov's method does.     

Stability of a feedback controlled distributed system by modal representation

In this work a linear distributed parameter system controlled by a finite number of lumped linear feedback loops is considered. The spatial portion of the system is decomposed using generalized Fourier expansions, and a complex frequency domain transcendental characteristic determinant results.

The theory developed is then applied to the automatic regulation of water flow in a ‘ reach ’ of an open irrigation canal. The dynamics of water flow in the transient state are described by a set of two first-order non-linear partial differential equations. The boundary conditions, governed by the water-flow dynamics under the gate, are also non-linear equations. For this particular system the feedback controller signal originates at the downstream boundary and controls the gates at the ‘ head ’ of the canal.

The stability analysis of the complete closed-loop system is based on perturbation about some steady-state flow conditions which allows for the linearization of the system's non-linear partial differential equations and boundary conditions. Subsequent application of the Nyquist criterion to a modal representation of the distributed parameter portion of the control loop in cascade with the lumped parameter controller yields the stability conditions.     

On strong stability for linear integral equations

Summary The notion of strong or adjoint stability for linear ordinary differential equations is generalized to the theory of Volterra integral equations. It is found that this generalization is not unique in that equivalent definitions for differential equations lead to different stabilities for integral equations in general. Three types of stabilities arising naturally are introduced: strong stability, adjoint stability, and uniform adjoint stability. Necessary and sufficient conditions relative to the fundamental matrix for these stabilities are proved. Some lemmas dealing with non-oscillation of solutions and a semi-group property of the fundamental matrix are also given.     

Stability analysis of systems with impulse effects

The authors establish conditions for the uniform stability and the uniform asymptotic stability of equilibria of systems with impulse effects described by systems of nonlinear, time-varying ordinary differential equations. Under mild additional assumptions, these sufficient conditions turn out to be necessary as well. For the case of time-invariant systems with impulse effects described by nonlinear ordinary differential equations, the above results are used to establish conditions under which the uniform asymptotic stability of equilibria can be deduced from the linearizations of such systems     

基于LMI方法的时滞细胞神经网络稳定性分析

神经网络是一个复杂的大规模非线性系统，而时滞神经网络的动态行为更为丰富和复杂．现有的研究时滞神经网络稳定性的方法中最为流行的是Lyapunov方法．它把稳定性问题变为某些适当地定义在系统轨迹上的泛函，通过这些泛函相应的稳定性条件就可以获得．该文得到了时滞细胞神经网络渐近稳定性的一些充分条件．作者利用了泛函微分方程的Lyapunov—Krasovskii稳定性理论和线性矩阵不等式(LMI)方法，精炼和推广了一些已有的结果．它们比目前文献报道的结果更少保守．该文还给出了确定时滞细胞神经网络稳定性更多的判定准则．     

Research note System estimation by invariant imbedding with unknown inputs †

An equivalence relation, rim–similarity, is defined on the set of all sequences of square matrices of a given fixed dimension. For linear discrete – time systems, theorems are presented which show that certain stability properties are invariant under n∞ similarity. General linear systems as well as those of variational type are considered. Also, sufficient conditions are given which guarantee that the uniform stability of the null solution of a non– linear system carries over to the linear variational equation with respect to this solution. All of these results are analogous to known ones for ordinary differential equations.     

Exponential stability for second-order neutral stochastic differential equations with impulses

In this paper, we investigate the problem on the exponential stability of mild solution for the second-order neutral stochastic partial differential equations with impulses by utilising the cosine function theory. A set of novel sufficient conditions is derived by establishing an impulsive integral inequality. As a final point, an example is given to illustrate the effectiveness of the obtained theory.     

Connections between Razumikhin-type theorems and the ISS nonlinearsmall gain theorem

A Razumikhin-type theorem that guarantees input-to-state stability for functional differential equations with disturbances is established using the nonlinear small-gain theorem. The result is used to show that input-to-state stabilizability for nonlinear finite-dimensional control systems is robust, in an appropriate sense, to small time delays at the input. Also, relaxed Razumikhin-type conditions guaranteeing global asymptotic stability for differential difference equations are given     

Decomposition and stability of linear systems with multiplicative noise

This paper concerns a representation of solutions and the stability of linear systems with multiplicative white noise, which is described by a vector Ito stochastic differential equation. The solution can be represented as a finite product of exponential matrices if Lie algebra generated by system matrices is solvable. If Lie algebra is not solvable, it is shown by the decomposition principle of Lie algebra that the problem of solving an equation can be reduced to the problem of solving a set of equations, whose corresponding Lie algebra is simple. Noting the structure of the sample solution, we present a technique of obtaining asymptotic stability conditions of sample solutions w.p.1, in the pth-order moment and in the pth-mean moment. The necessary and/or sufficient conditions of stability in some stochastic sense are obtained under certain conditions.     

Dynamic modeling and vibration control for a nonlinear 3‐dimensional flexible manipulator

In this paper, the control design and stability analysis are presented for a 3‐dimensional flexible manipulator system with input disturbances. To provide an accurate and concise representation of the manipulator's dynamic behavior, the flexible manipulator is described by a distributed parameter system with a set of partial differential equations and ordinary differential equations. Boundary control laws with disturbance observers are proposed to regulate orientation and suppress elastic vibrations simultaneously. The closed‐loop stability is achieved through rigorous analysis without any simplification of the dynamics based on the Lyapunov direct method. Numerical simulations demonstrate the effectiveness of the proposed scheme.     

Strengthened theorems for set stability of differential equations

Set stability and uniform set stability involve known specific bounds on the solutions of differential equations. A result m set theory is employed in strengthening previously developed Liapunov-liko theorems for these forms of stability. An example illustrating the advantages of the new theorems is presented.     

Stability and control of nonlinear systems described by retarded functional equations: a review of recent results

This paper reports on recent results in a series of the work of the authors on the stability and nonlinear control for general dynamical systems described by retarded functional differential and difference equations. Both internal and external stability properties are studied. The corresponding Lyapunov and Razuminkhin characterizations for input-to-state and input-to-output stabilities are proposed. Necessary and sufficient Lyapunov-like conditions are derived for robust nonlinear stabilization. In particular, an explicit controller design procedure is developed for a new class of nonlinear time-delay systems. Lastly, sufficient assumptions, including a small-gain condition, are presented for guaranteeing the input-to-output stability of coupled systems comprised of retarded functional differential and difference equations.     

Exponential stability of linear equations arising in adaptive identification

The stability, properties are examined of various time-varying linear differential equations which arise in model reference adaptive identification schemes. Necessary and sufficient conditions for exponential stability are presented.     

Conditions for the moment stability of linear stochastic systems

Sufficient conditions for the moment stability of linear differential stochastic equations with Ornstein-Uhlenbeck parameters are obtained on the base of moment stability theory with white noise parameters.     

Moment exponential stability analysis of Markovian jump stochastic differential equations with uncertain transition jump rates

This paper is concerned with the moment exponential stability analysis of Markovian jump stochastic differential equations. The equations under consideration are more general, whose transition jump rates matrix Q is not precisely known. Sufficient conditions for testing the stability of such equations are established, and some numerical examples to illustrate the effectiveness of our results are presented.