New Results on Finite-Time Stability and Instability Theorems for Stochastic Nonlinear Time-Varying Systems

1. Research Background and Significance

Stability theory is a central topic in systems theory and engineering applications, serving as the fundamental consideration in system analysis and synthesis. In stability theory, the two most commonly used concepts are asymptotic stability and finite-time stability. Asymptotic stability describes the behavior of a system’s state when time tends to infinity, while finite-time stability focuses on the transient performance within a finite time period.

In many engineering problems, finite-time stability becomes more critical than asymptotic stability. For example, in trajectory control of robotic manipulators and attitude control of underwater vehicles, greater emphasis is placed on the system’s ability to achieve desired states within a finite time. Finite-time stable systems are not only more robust but also exhibit faster convergence rates. However, research on finite-time stability still has gaps. In particular, existing finite-time stability theories generally require the infinitesimal generator of the Lyapunov function (denoted as L_v) to be negative definite or non-positive. These requirements are overly stringent and are challenging to apply to complex stochastic nonlinear systems.

Against this background, Weihai Zhang from Shandong University of Science and Technology and Liqiang Yao from Yantai University conducted an in-depth study of stochastic nonlinear time-varying systems. The research aims to overcome the limitations of existing finite-time stability results and propose weaker sufficient conditions to ensure the stability or instability of stochastic systems.

2. Paper Details

This study, titled “New Results on Finite-Time Stability and Instability Theorems for Stochastic Nonlinear Time-Varying Systems,” was published in Science China Information Sciences in the February 2025 issue (Vol. 68, No. 2). Its DOI is: 10.1007/s11432-024-4118-x. The lead authors are Weihai Zhang and Liqiang Yao, affiliated with Shandong University of Science and Technology and Yantai University, respectively.

The paper primarily investigates new theorems on finite-time stability and instability in the probabilistic sense for stochastic nonlinear time-varying systems, providing novel perspectives to address theoretical shortcomings and enhance the applicability of system analysis.

3. Research Content and Methods

This paper is an original study that focuses on proposing new, more relaxed constraints for stochastic Lyapunov functions, along with establishing new criteria for finite-time stability and instability. The main research process includes the following parts:

3.1 Mathematical Formulation of the System Model and Problem

The authors study a class of stochastic nonlinear Itô systems, modeled as:

$$ dx(t) = f(t, x(t))dt + g(t, x(t))dW(t), $$

where ( x(t) \in \mathbb{R}^r ) is the system state, and ( W(t) ) is a standard Wiener process (of dimension ( d )). The functions ( f ) and ( g ) represent time-varying drift and diffusion terms, respectively, satisfying corresponding continuity and local Lipschitz conditions.

Based on the model’s characteristics, the authors develop their research on the following core issues: 1. The existence of solutions for stochastic systems, i.e., providing weaker conditions to ensure the existence of global solutions. 2. Proposing entirely new finite-time stability and instability theorems that allow ( L_v ) to be indefinite (either positive or negative) instead of strictly negative definite.

3.2 Conditions for the Existence of Global Solutions for Stochastic Systems

By applying Skorokhod’s stochastic process theory and related lemmas, the authors present two new lemmas that provide sufficient conditions for the global existence of solutions for stochastic systems. Specifically, they relax the strict constraints in the existing literature [23], where the drift and diffusion terms of stochastic systems are required to satisfy strict boundary conditions, making these lemmas applicable to a broader range of stochastic nonlinear systems.

Lemma 1: If there exists a constant ( h > 0 ) such that the stochastic system satisfies:

$$ |f(t, x)|^2 + ||g(t, x)||^2 \leq h(1 + |x|^2), $$

then the stochastic process has a continuous solution, which is defined on ( [t_0, \infty) ).

3.3 Finite-Time Stability Theorems

To overcome the strict constraints on Lyapunov functions in existing stochastic stability theories, the authors propose a class of finite-time stability criteria associated with Uniformly Asymptotically Stable Functions (UASF). The core theoretical results are as follows:

Theorem 1:

If there exists a global solution ( x(t) ), and the following conditions are satisfied: 1. For any closed set ( U \subseteq \mathbb{R}^r ), the function ( v(t, x) ) satisfies the positivity constraint: ( \gamma(|x|) \leq v(t, x) \leq \bar{\gamma}(|x|) ). 2. The Lyapunov generator ( L_v ) satisfies: ( L_v(t, x) \leq \mu(t) \cdot [v(t, x)]^\kappa ), where ( \kappa \in [0, 1) ), and ( \mu(t) ) is a UASF.

Under these conditions, the system solution is finite-time stable in the probabilistic sense.

Through this theorem, the authors successfully generalize existing theories by relaxing the requirement that ( L_v ) must be negative definite. By allowing a broad expression for ( \kappa ), the criteria retain general applicability.

Theorem 2:

When ( \kappa = 0 ), Theorem 1 reduces to Theorem 2. The conditions are simplified while still encompassing special results in the existing literature, such as the finite-time asymptotic stability conclusions in [21] and [23].

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4. Major Results and Case Studies

4.1 Case 1: Stability Analysis of a Non-Stationary Stochastic System

Consider the stochastic system:

$$ dx(t) = \frac{1}{2} \mu(t)x^{¹⁄₃}(t)dt - \frac{1}{2}x(t)dt + x(t)\cos(x(t))dW(t), $$

where ( \mu(t) ) is a piecewise continuous function that can take positive or negative values. Traditional finite-time stability criteria cannot directly analyze this system.

• Method: Choose the Lyapunov function ( v(x) = x^2 ).
• Results: By computing ( L_v = \mu(t) v^{²⁄₃}(x) ), it is shown that the system is finite-time stable under specific constraints on ( \mu(t) ).

4.2 Case 2: Finite-Time Stability of a Controlled System

A stochastic nonlinear system achieves finite-time stability through a feedback controller:

$$ dx(t) = \mu(t)x^{¹⁄₅}(t)dt + x^{³⁄₅}(t)dW(t). $$

The controller ( u(t) ) is designed as a nonlinear function. Numerical simulations confirm finite-time convergence to the equilibrium point.

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5. Research Significance and Highlights

The key contributions of this paper include: 1. Theoretical Innovation: Proposing new criteria that relax the constraints on Lyapunov generators, providing more flexibility for the stability analysis of stochastic dynamical systems. 2. Simplified Framework: Introducing UASF simplifies the theoretical analysis framework and guides the stability design for practical engineering problems. 3. Broad Applicability: Numerical experiments verify the method’s effectiveness in complex non-stationary systems and lay a theoretical foundation for further development of practical control algorithms.

This study significantly advances the analysis and control of finite-time stability in complex stochastic systems. It has potential applications in various fields such as robotics, signal processing, and modeling/optimization of complex networks.