This paper investigates the problems of finite-time stability and finite-time stabilization for nonlinear quadratic systems with jumps. The jump time sequences here are assumed to satisfy some given constraints. Based on Lyapunov function and a particular presentation of the quadratic terms, sufficient conditions for finite-time stability and finite-time stabilization are developed to a set containing bilinear matrix inequalities (BLIMs) and linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed methodology.

Most practical systems, such as missile systems and satellite systems, possess a typical characterization that their operating times always have a finite duration. In this case, the main concern for the researchers is the stability over a fixed finite-time interval rather than the classical Lyapunov asymptotic stability, although the Lyapunov theory is pervasive in control fields from linear methods to nonlinear systems. Usually, a system is finite-time stability (FTS), if, given a finite duration at first, its state is contained within some prescribed bound during this finite duration. The problem of finite-time stability analysis for the control plants has been extensively investigated and a great number of results on this topic have been reported, for example, see [

At the same time, there are a number of real evolution processes in which the states are subjected to rapid changes at certain time instants. It has been shown that the finite-time stability of a continuous dynamical system could be destroyed by such rapid changes. Therefore, it is important to study the jump’s influence on FTS of the control systems. There are several research works that appeared in the literature on control systems with jumps. In [

Recently, the issue of analysis and design of nonlinear quadratic systems has received increasing interest. The Lyapunov asymptotic stability of quadratic systems has achieved great success both in theory and in practice, for example, see [

In this paper, we are interested in FTS and finite-time stabilization for nonlinear quadratic systems with jumps. Based on the Lyapunov function and a particular presentation of the quadratic terms, sufficient conditions for FTS and finite-time stabilization are presented for such quadratic systems in terms of LMIs. Furthermore, two examples are presented to illustrate its effectiveness.

Throughout this paper, standard notation is adopted. Given a scalar

Consider a class of nonlinear quadratic systems with finite jumps described as

Let us define matrices

In what follows, we introduce two lemmas, which are essential for the developments in the next section.

For any matrices

if

Consider matrix

In this section, we establish sufficient conditions of FTS and finite-time stabilization for the nonlinear quadratic system (

First, we introduce the definition of FTS for nonlinear quadratic system (

Given a scalar

Now, we provide a sufficient condition for FTS of the system (

For a prescribed scalar

Note (

When

When

When

Compared with the method presented in [

Now, we extend Theorem

Then, the corresponding closed-loop system is defined by

The following theorem presents a sufficient condition of finite-time stabilization for the closed-loop system (

For a prescribed scalar

Then

if

if

Since the proof of Theorem

Due to Remark 8 in [

In this section, we will present two examples to illustrate the effectiveness of our method. In Theorem

Consider the system (

Through Theorem

Consider the system (

It is noticed that the jump system is helpful to FTS of closed-loop system by the discrete state feedback control law. Our goal is to solve the finite-time stabilization problem by using nonlinear state feedback with the same

Based on Theorem

In this paper, the problems of FTS and finite-time stabilization for nonlinear quadratic systems with jumps are investigated. Sufficient conditions of FTS and finite-time stabilization based on the quadratic Lyapunov function and a particular presentation for the quadratic terms are established in terms of BLMIs and LMIs. And two examples have been provided to explain the effectiveness of our methodology. In the future, we will continue the study of nonlinear quadratic systems with jumps, such as the systems subject to jumps and input saturation or time delay.

The authors declare that there is no conflict of interests regarding the publication of this paper.