Stability Of Uncertain Differential Equation Research Articles

Almost sure and pth moment stability of uncertain differential equations with time-varying delay

ABSTRACT Uncertain differential equations with time-varying delay are a special type of uncertain functional differential equation. Stability in measure and stability in mean for this type of differential equation have been investigated. For the sake of completeness, this article explores the other two types of stability. Firstly, the concepts of the almost sure stability and the pth moment stability of uncertain differential equations with time-varying delay are proposed. Next, sufficient conditions, in the form of theorems, for uncertain differential equations with time-varying delay to be stable almost surely and in pth moment are deduced. Finally, the relationships between several types of stability for uncertain differential equations with time-varying delay are discussed.

The pth moment exponential stability of uncertain differential equation with jumps

Under the axiom system of uncertainty theory, the paper mainly introduce the new definition of the pth moment exponential stability for uncertain differential equation with jumps. For illustrating the concept, some examples and counterexamples are given. Furthermore, we obtain a necessary and sufficient condition of stability in pth moment exponential for the linear uncertain differential equation with jumps. Also, the conclusion condition is illustrated very clearly by two examples.

Stability analysis for uncertain differential equation by Lyapunov’s second method

Uncertain differential equation is a type of differential equation driven by Liu process that is the counterpart of Wiener process in the framework of uncertainty theory. The stability theory is of particular interest among the properties of the solutions to uncertain differential equations. In this paper, we introduce the Lyapunov’s second method to study stability in measure and asymptotic stability of uncertain differential equation. Different from the existing results, we present two sufficient conditions in sense of Lyapunov stability, where the strong Lipschitz condition of the drift is no longer indispensable. Finally, illustrative examples are examined to certify the effectiveness of our theoretical findings.

Exponential stability of uncertain differential equation with jumps

Exponential stability of uncertain differential equation with jumps

Some results about uncertain differential equations with time-dependent delay

Uncertain differential equations with time-dependent delay are a type of differential equations driven by Liu process. This paper mainly proves that this type of uncertain differential equations have unique solutions in the infinite domain under some conditions. In addition, we also focus on the stability of uncertain differential equations with time-dependent delay, and some correspondence results are shown in the form of theorems.

The pth moment exponential stability of uncertain differential equation

Uncertain variable and uncertain process are often employed to describe uncertainty phenomena. Uncertain variable is a measurable function from an uncertainty space to the real number set and uncertain process is essentially a sequence of uncertain variables indexed by time. Uncertain differential equation is a type of differential equation involving uncertain processes. Stability of an uncertain differential equation plays a very important role in differential equation which means insensitivity of the state of a system to small changes in the initial state. In this paper, we focus on pth moment exponential stability of uncertain differential equation which is regarded as an extension of pth moment stability and exponential stability. Some stability theorems are given for uncertain differential equation, in which a sufficient and necessary condition for a linear uncertain differential equation being pth moment exponentially stable is provided.

Exponential stability of uncertain differential equation

Uncertain differential equation is a type of differential equations involving uncertain processes. So far, the concepts of stability in measure and stability in mean have been proposed for uncertain differential equations. This paper proposes a new type of stability for uncertain differential equation, named exponential stability. Some examples are given to illustrate the concept, and the relationships between exponential stability, stability in measure and stability in mean are discussed. Besides, a sufficient and necessary condition for a linear uncertain differential equation being exponentially stable is derived.