# Finite-Time Stability Analysis: A Tutorial Survey

Authors: Honglei Xu
Journal: Complexity
 (2020)

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## Abstract

In the past decades, there has been a growing research interest in the field of finite-time stability and stabilization. This paper aims to provide a self-contained tutorial review in the field. After a brief introduction to notations and two distinct finite-time stability concepts, dynamical system models, particularly in the form of linear time-varying systems and impulsive linear systems, are studied. The finite-time stability analysis in a quantitative sense is reviewed, and a variety of stability results including state transition matrix conditions, the piecewise continuous Lyapunov-like function theory, and the converse Lyapunov-like theorem are investigated. Then, robustness and time delay issues are studied. Finally, fundamental finite-time stability results in a qualitative sense are briefly reviewed.


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## Body

## 1. Introduction

Finite-time stability was first introduced in a Russian journal [1] and later appeared in the western literature [2–4]. The term short-time stability is another name for it [5]. In the current literature, there are two different concepts of finite-time stability. The first is the traditional finite-time stability concept which concerns the restrained system behavior during a specified interval of time. The initial and trajectory domains and the time interval need to be specified in advance, so the traditional concept is a quantitative one. We call it finite-time stability in a quantitative sense. The second one characterizes an asymptotically stable system whose state reaches zero in a finite time, called a settling time. Similar to the Lyapunov stability, it is a qualitative concept, and hence, we call the second concept finite-time stability in the qualitative sense. The analysis and synthesis results of both finite-time stability concepts can be applied to many practical applications such as ATM networks [6], car suspension systems [7], and robot manipulators [8].Finite-time stability in a quantitative sense emphasizes the following characteristics: the system restrains its trajectory to a predefined time-varying domain over a finite time interval for a bounded initial condition. Even though it mimics the Lyapunov stability, it is quite different from the classical one due to its finite time interval and specified domains for initial conditions and system trajectories, i.e., a system is finite-time stable for some chosen initial and trajectory domains and time intervals but not finite-time stable for different ones. In the past few decades, many finite-time stability analysis and control design problems have been investigated and a variety of stability criteria have been obtained, see, for example, [3, 4, 9] and the references therein. Recently, computationally tractable finite-time stability criteria with less conservatism have been established under the help of new tools such as linear matrix inequalities [10], Lyapunov matrix equations [11], and differential linear matrix inequalities [12]. More recently, studies on the finite-time stability and stabilization have been extended from linear time-varying systems to complex dynamical systems such as switched systems [13–15] and stochastic systems [16].On the other hand, finite-time stability in a qualitative sense has attracted much attention in recent years and become a growing interdisciplinary research area. It focuses on asymptotical stability analysis for dynamical systems whose trajectories reach an equilibrium point in a finite time. It is a stronger concept than asymptotical stability and has the settling-time characteristic. Relevant results on autonomous and nonautonomous nonlinear systems have been discussed in [17–20]. Later, switched versions and time-delay versions appeared in the literature, e.g., [21–23]. Recently, relevant issues of underactuated systems with disturbance have been considered in [24]. In the context of this paper, the readers should be not hard to distinguish whether the concept of finite-time stability is quantitative or qualitative, so we can use the term “finite-time stability” in most places without causing confusion.In this paper, we will summarize the results of finite-time stability from both quantitative and qualitative aspects. Several excellent surveys on finite-time stability have been found, see, for example, the review papers [25–27], the books [7, 28, 29], and the references therein. These publications report and survey finite-time stability on one aspect or another. This paper aims to provide a unified self-contained tutorial review of finite-time stability to introduce the recent discoveries in the field.The remainder of this paper is as follows. Section2 gives some basic mathematical preliminaries including two finite-time stability concepts. Section 3 reviews finite-time stability results in a quantitative sense, mostly for linear time-varying systems. Results involving time-dependent and state-dependent impulses, time delays, and uncertainty are also investigated. In Section 4, we briefly overview some results on finite-time stability in a qualitative sense. Finally, a conclusion is drawn in Section 5.


## 2. Mathematical Preliminaries

### 2.1. Notations and Definitions

Letℝ+ denote a set of nonnegative real numbers and ℝn the n-dimensional Euclidean space, and consider the time interval Ω=0,T,T>0. Let AT be the transpose of A and I be the identity matrix with an appropriate dimension. For a square matrix A, we denote by λA, λmaxA, and λminA the set of eigenvalues, the maximum eigenvalues, and the minimum eigenvalues of A, respectively. The symmetric components in a matrix are represented by ∗. A≥0A>0, called to be positive semidefinite (positive definite), means x⊤Ax≥0x⊤Ax>0 for all x∈Rn.A≥B is equivalent to A−B≥0. Let C−h,0,Rn denote the set of all vector-valued continuous functions on −h,0. For xt∈C−h,0,ℝn, it is represented by xt=xt+s:s∈−h,0 with the norm xt=sups∈−h,0xt+s. Let Sρ and S¯ρ denote the open and closed sets of the allowable system states defined as Sρ=x∈ℝnxtQ2<ρ and S¯ρ=x∈ℝnxtQ2≤ρ, respectively, where ρ>0 and Q is a symmetric positive definite real matrix. We denote Vminρt=minx=ρVx,t and Vmaxρt=maxx=ρVx,t. For a set Sp=x1,x2,…,xp⊆ℝn, the conical hull of Sp is the set of all conical combinations, i.e., coneSp=xx=∑i=1paixi,ai≥0. The set of normalized extremal rays generating Sq, denoted by extrS=x¯1,…,x¯q with x¯i2=1, i=1,…,q≤p, is the minimal set of unit vectors such that S=conex¯1,…,x¯q. Given a piecewise continuous matrix-valued (or vector-valued) function H⋅ over Ω and a positive real number ɛ, we denote H−t=limɛ⟶0Ht−ɛ and H+t=limɛ⟶0Ht+ɛ, i.e., H−t and H+t are the left and right limits, respectively. Let the set C be an open set having the origin and a boundary ∂C.We first look at the basic definition of finite-time stability for a dynamical system(1)x˙t=ft,xt,where t∈ℝ+ is the time variable, x∈ℝn is the state variable, and f⋅ is a ℝn-valued function. Suppose that system (1) has a unique solution. We first introduce the concepts of “finite-time stability” in a quantitative sense and in a qualitative sense, respectively.Definition 1 (Finite-Time Stability in a Quantitative Sense).
Given two setsX0 and Xt, 0∈X0, system (1) is said to be finite-time stable with respect to Ω,X0,Xt if(2)x0∈X0impliesxt∈Xtfort∈Ω.
In Figure1, we can see a graphical explanation of Definition 1, where the initial set X0 and the time-varying set Xt can be in various forms. (1)
When they are ellipsoids, which are the most common forms existing in the literature (see, e.g., [7, 30–32]), they can be formulated as X0=x0∈ℝnx0⊤Rx0≤1 and Xt=x∈ℝnx⊤Qtx≤1, where Qt is a bounded and piecewise continuous matrix-valued function of time and Q0<R. In many cases, X0 and Xt can also be expressed as X0=x0∈ℝnx0Q2<c1 and Xt=x∈ℝnxtQ2<c2 for c2>c1>0 [33].(2)
When they are in the form of polytopes, they can be described byX0=convx10,x20,…,xp0 and Xt=convx1t,x2t,…,xqt=x∈ℝnai⊤x≤1,i=1,2,…,q, where p and q are the number of vertices of the polytopes X0 and Xt, and xi0 and xjt are the i-th vertex of the polytope X0 and the j-th vertex of the polytope Xt [34].(3)
They can be formulated as much generalized piecewise quadratic domains over conical partitionsP0=U10,U20,…,Uu0 and Pt=U1t,U2t,…,Uvt. Then, X0=x0∈ℝnx0⊤Rix0≤1,x0∈Ui0,i=1,2,…,u and Xt=x∈ℝnx⊤Qitx≤1,x∈Uit,i=1,2,…,v. It is obvious to see that the set of piecewise quadratic domains is a generalized set of ellipsoidal domains since an ellipsoidal domain is indeed a piecewise quadratic domain choosing Qi=Q for all i=1,2,…,v. It can also represent the set of polytopic domains whose boundary is a polyhedral function’s level curve.Figure 1
Schematic illustration of finite-time stability in a quantitative sense.Definition 2 (Finite-Time Stability in a Qualitative Sense).
System (1) is said to be finite-time stable if for any x0∈C⊂ℝn, t≥0 and ɛ>0, there exist δt,x0>0 and Tx0>0 such that x0≤δ implies xt,x0≤ɛ, limt⟶∞xt,x0=0 and xt=0 for all t>Tx0. Here, Tx0=infT≥0xt,x0=0,∀t≥T is called the settling-time function of system (1), and the set C is called the domain of attraction. Moreover, if C=ℝn, system (1) is globally finite-time stable.
Similarly, a schematic illustration of Definition2 is given in Figure 2.Figure 2
Schematic illustration of finite-time stability in a qualitative sense.


### 2.2. Mathematical Formulations

System (1) is a general model for both linear and nonlinear systems depending on the choice of the function f⋅. In the first part of this paper, we will focus on finite-time stability issues in a quantitative sense for continuous-time linear time-varying systems with and without finite jumps. In the second part of this paper, we will analyze finite-time stability in a qualitative sense for continuous-time nonlinear systems.First, we introduce a linear time-varying system described as(3)x˙t=Atxt,for a given initial condition x0=x0, which has been considered in many papers, see, e.g., [7, 30]. Here, A⋅:Ω↦ℝn×n is a continuous matrix-valued function.In many practical scenarios, abrupt state changes and system jumping behaviors are commonly existing, and these finite jumps occur when the time points and/or the system states satisfy a certain triggering condition, sayt,xt∈S⊂Ω×ℝn. When the impulses are triggered, the impulsive mappings can be described by(4)x+t=Btx−t=Btxt,t,xt∈S,where B⋅:Ω↦ℝn×n is a matrix-valued function, which describes the jumping behavior of system (3) with left continuity over the triggering set S⊂Ω×ℝn. We call a dynamical system modeled by (3) and (4) to be an impulsive linear system. According to the triggering set S, impulsive linear systems expressed by (3) and (4) can be categorized into two main types: the time-dependent impulsive linear systems and the state-dependent impulsive linear systems, which can be described by(5)x˙t=Atxt,xt0=x0,t∉ℐ=t1,t2,…⊂Ω,xt+=Btxt,t∈ℐ,k=1,2,…,(6)x˙t=Atxt,xt0=x0,xt∈ℝn\∪k=1NSk,xt+=Btxt,xt∈Sk,k=1,2,…,respectively [31, 32, 35]. For a time-dependent impulsive linear system, the impulses occur at the given time points, t∈ℐ=t1,t2,…⊂Ω, so the triggering set S can be written as S=ℐ×Dx0,ℐ, where Dx0,ℐ=xt:t∈ℐ⊂ℝn. For a state-dependent impulsive linear system, the impulses happen when the system state reaches a preassigned set D⊂ℝn, and then, the triggering set S is written as ℐx0,D=t∈Ω:xt∈D⊂Ω. It is worth pointing out that the well-posedness of the triggering times should be guaranteed and the Zeno phenomena need to be avoided in this paper.


## 2.1. Notations and Definitions

Letℝ+ denote a set of nonnegative real numbers and ℝn the n-dimensional Euclidean space, and consider the time interval Ω=0,T,T>0. Let AT be the transpose of A and I be the identity matrix with an appropriate dimension. For a square matrix A, we denote by λA, λmaxA, and λminA the set of eigenvalues, the maximum eigenvalues, and the minimum eigenvalues of A, respectively. The symmetric components in a matrix are represented by ∗. A≥0A>0, called to be positive semidefinite (positive definite), means x⊤Ax≥0x⊤Ax>0 for all x∈Rn.A≥B is equivalent to A−B≥0. Let C−h,0,Rn denote the set of all vector-valued continuous functions on −h,0. For xt∈C−h,0,ℝn, it is represented by xt=xt+s:s∈−h,0 with the norm xt=sups∈−h,0xt+s. Let Sρ and S¯ρ denote the open and closed sets of the allowable system states defined as Sρ=x∈ℝnxtQ2<ρ and S¯ρ=x∈ℝnxtQ2≤ρ, respectively, where ρ>0 and Q is a symmetric positive definite real matrix. We denote Vminρt=minx=ρVx,t and Vmaxρt=maxx=ρVx,t. For a set Sp=x1,x2,…,xp⊆ℝn, the conical hull of Sp is the set of all conical combinations, i.e., coneSp=xx=∑i=1paixi,ai≥0. The set of normalized extremal rays generating Sq, denoted by extrS=x¯1,…,x¯q with x¯i2=1, i=1,…,q≤p, is the minimal set of unit vectors such that S=conex¯1,…,x¯q. Given a piecewise continuous matrix-valued (or vector-valued) function H⋅ over Ω and a positive real number ɛ, we denote H−t=limɛ⟶0Ht−ɛ and H+t=limɛ⟶0Ht+ɛ, i.e., H−t and H+t are the left and right limits, respectively. Let the set C be an open set having the origin and a boundary ∂C.We first look at the basic definition of finite-time stability for a dynamical system(1)x˙t=ft,xt,where t∈ℝ+ is the time variable, x∈ℝn is the state variable, and f⋅ is a ℝn-valued function. Suppose that system (1) has a unique solution. We first introduce the concepts of “finite-time stability” in a quantitative sense and in a qualitative sense, respectively.Definition 1 (Finite-Time Stability in a Quantitative Sense).
Given two setsX0 and Xt, 0∈X0, system (1) is said to be finite-time stable with respect to Ω,X0,Xt if(2)x0∈X0impliesxt∈Xtfort∈Ω.
In Figure1, we can see a graphical explanation of Definition 1, where the initial set X0 and the time-varying set Xt can be in various forms. (1)
When they are ellipsoids, which are the most common forms existing in the literature (see, e.g., [7, 30–32]), they can be formulated as X0=x0∈ℝnx0⊤Rx0≤1 and Xt=x∈ℝnx⊤Qtx≤1, where Qt is a bounded and piecewise continuous matrix-valued function of time and Q0<R. In many cases, X0 and Xt can also be expressed as X0=x0∈ℝnx0Q2<c1 and Xt=x∈ℝnxtQ2<c2 for c2>c1>0 [33].(2)
When they are in the form of polytopes, they can be described byX0=convx10,x20,…,xp0 and Xt=convx1t,x2t,…,xqt=x∈ℝnai⊤x≤1,i=1,2,…,q, where p and q are the number of vertices of the polytopes X0 and Xt, and xi0 and xjt are the i-th vertex of the polytope X0 and the j-th vertex of the polytope Xt [34].(3)
They can be formulated as much generalized piecewise quadratic domains over conical partitionsP0=U10,U20,…,Uu0 and Pt=U1t,U2t,…,Uvt. Then, X0=x0∈ℝnx0⊤Rix0≤1,x0∈Ui0,i=1,2,…,u and Xt=x∈ℝnx⊤Qitx≤1,x∈Uit,i=1,2,…,v. It is obvious to see that the set of piecewise quadratic domains is a generalized set of ellipsoidal domains since an ellipsoidal domain is indeed a piecewise quadratic domain choosing Qi=Q for all i=1,2,…,v. It can also represent the set of polytopic domains whose boundary is a polyhedral function’s level curve.Figure 1
Schematic illustration of finite-time stability in a quantitative sense.Definition 2 (Finite-Time Stability in a Qualitative Sense).
System (1) is said to be finite-time stable if for any x0∈C⊂ℝn, t≥0 and ɛ>0, there exist δt,x0>0 and Tx0>0 such that x0≤δ implies xt,x0≤ɛ, limt⟶∞xt,x0=0 and xt=0 for all t>Tx0. Here, Tx0=infT≥0xt,x0=0,∀t≥T is called the settling-time function of system (1), and the set C is called the domain of attraction. Moreover, if C=ℝn, system (1) is globally finite-time stable.
Similarly, a schematic illustration of Definition2 is given in Figure 2.Figure 2
Schematic illustration of finite-time stability in a qualitative sense.


## 2.2. Mathematical Formulations

System (1) is a general model for both linear and nonlinear systems depending on the choice of the function f⋅. In the first part of this paper, we will focus on finite-time stability issues in a quantitative sense for continuous-time linear time-varying systems with and without finite jumps. In the second part of this paper, we will analyze finite-time stability in a qualitative sense for continuous-time nonlinear systems.First, we introduce a linear time-varying system described as(3)x˙t=Atxt,for a given initial condition x0=x0, which has been considered in many papers, see, e.g., [7, 30]. Here, A⋅:Ω↦ℝn×n is a continuous matrix-valued function.In many practical scenarios, abrupt state changes and system jumping behaviors are commonly existing, and these finite jumps occur when the time points and/or the system states satisfy a certain triggering condition, sayt,xt∈S⊂Ω×ℝn. When the impulses are triggered, the impulsive mappings can be described by(4)x+t=Btx−t=Btxt,t,xt∈S,where B⋅:Ω↦ℝn×n is a matrix-valued function, which describes the jumping behavior of system (3) with left continuity over the triggering set S⊂Ω×ℝn. We call a dynamical system modeled by (3) and (4) to be an impulsive linear system. According to the triggering set S, impulsive linear systems expressed by (3) and (4) can be categorized into two main types: the time-dependent impulsive linear systems and the state-dependent impulsive linear systems, which can be described by(5)x˙t=Atxt,xt0=x0,t∉ℐ=t1,t2,…⊂Ω,xt+=Btxt,t∈ℐ,k=1,2,…,(6)x˙t=Atxt,xt0=x0,xt∈ℝn\∪k=1NSk,xt+=Btxt,xt∈Sk,k=1,2,…,respectively [31, 32, 35]. For a time-dependent impulsive linear system, the impulses occur at the given time points, t∈ℐ=t1,t2,…⊂Ω, so the triggering set S can be written as S=ℐ×Dx0,ℐ, where Dx0,ℐ=xt:t∈ℐ⊂ℝn. For a state-dependent impulsive linear system, the impulses happen when the system state reaches a preassigned set D⊂ℝn, and then, the triggering set S is written as ℐx0,D=t∈Ω:xt∈D⊂Ω. It is worth pointing out that the well-posedness of the triggering times should be guaranteed and the Zeno phenomena need to be avoided in this paper.


## 3. Finite-Time Stability in a Quantitative Sense

In this section, we will provide some results on finite-time stability in a quantitative sense for linear time-varying system (3) and its variants with impulses (4).


### 3.1. State Transition Matrix

In linear system theory, we know that the solution of (3) can be described as xt=Φt,0x0, where the matrix-valued function Φ⋅,⋅ has the following basic properties:(7)Φ˙t,0=AtΦt,0,Φ0,0=I.This matrix-valued functionΦ⋅,⋅ is called to be the state transition matrix, and an appropriate assumption on the nature of At can ensure the existence and uniqueness of the state transition matrix. For system (3), it is stable in the sense of Lyapunov if there exists a positive constant M such that Φt,0<M for all t≥0. Moreover, system (3) is asymptotically stable if it is stable and limt⟶∞Φt,0=0. As for finite-time stability of the system (3), a necessary and sufficient stability condition will be provided in the following theorem [30].Theorem 1.
System (3) is finite-time stable with respect to Ω,X0,Xt, where X0=x∈ℝnx0⊤Rx0≤1 and Xt=x∈ℝnx⊤Qtx≤1 if and only if the state transition matrix of system (3) satisfies(8)Φt,0⊤QtΦt,0<R.The state transition matrix approach has been extended to impulsive linear system (5) in [31, 32, 36]. Letting T∈tl,tl+1, the solution of the impulsive linear system (5) will be xt=Φ^t,0x0,t∈Ω, where Φ^t,0, called the state transition matrix of (5), is a piecewise continuous matrix-valued function with discontinuous right-hand sides at the time instants tk,k=1,2,…,l. In detail, when t∈0,t1, Φ^t,0 is the solution of the following matrix differential equation:(9)∂∂tΦ^t,0=AtΦt,0,Φ^0,0=I,Φ^t1+,0=Bt1Φ^t1,0.In the sequel intervals fork=1,2,…,l−1, Φ^t,tk should satisfy(10)∂∂tΦ^t,tk+=AtΦ^t,tk+,t∈tk,tk+1,Φ^tk+1+,tk+=Btk+1Φ^tk+1,tk+.In the end, whent∈tl,tl+1, we have(11)∂∂tΦ^t,tl+=AtΦ^t,tl+,t∈tl,T.Theorem 2.
Impulsive linear system (5) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈0,T, the following is satisfied:(12)Φ^t,0⊤QtΦ^t,0<R.The conditions in the form of state transition matrices (8) and (12) in Theorems 1 and 2 are valuable for theoretical analysis but hard to apply due to the high computational difficulty, particularly for the time-varying case.To obtain computational conditions for finite-time stability, some Lyapunov-like functions are needed to establish conditions in the form of linear matrix equalities or Lyapunov matrix inequalities. In some early work such as [3], it concludes that system (1) is finite-time stable with respect to Ω,Sα,Sβ if and only if there exists a real-valued Lipschitz function Vt,x, continuous on Ω×S¯β, and a real-valued integrable function φt such that, for t∈Ω, we have V˙0t,x≤φt for all x∈Sβ−S¯α and ∫t1t2φtdt<Vminβt2−Vmaxαt1 for t1<t2 and t1,t2∈Ω. A piecewise continuous Lyapunov-like function is the most common one in the literature, and relevant results will be presented in the following subsection.


### 3.2. Piecewise Continuous Lyapunov-Like Functions

To obtain computationally tractable finite-time stability conditions, we choose a quadratic piecewise continuous Lyapunov-like functionVt,x=x⊤Ptx and establish the following conditions containing coupled differential Lyapunov matrix equations and differential linear matrix inequalities.Theorem 3 (see [7, 30]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈Ω, there exists a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving the differential matrix equation with boundary conditions are satisfied:(13)P˙τ=−Aτ⊤Pτ−PτAτ−ɛI,τ∈0,t,ɛ>0,Pt=Qt,P0<R.We see that Theorem3 provides a necessary and sufficient condition for finite-stability of system (3). However, it is not practicable to verify the differential matrix equation in (13) for every τ∈0,t, and hence, Theorem 3 is not suitable for the computational purpose. Using Theorem 3, we can obtain a sufficient condition with computational tractability for finite-stability of system (3).Theorem 4 (see [7, 30]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω there exists a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving the differential matrix equation with boundary conditions are satisfied:(14)P˙t=−At⊤Pt−PtAt−ɛI,ɛ>0,Pt=Qt,P0<R.Nowadays, computational tools in the convex optimization framework such as linear matrix inequalities or differential linear matrix inequalities are very efficient, and we can obtain the following necessary and sufficient conditions including differential linear matrix inequalities, equivalent to (13):(15)P˙τ<−Aτ⊤Pτ−PτAτ,Qt≤Pt,P0<R.The piecewise continuous Lyapunov-like function is also applied to the finite-time stability problem of impulsive linear systems. For linear time-varying systems with time-dependent impulses, we have the following finite-time stability results.Theorem 5 (see [36]).
System (5) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈Ω there exists a piecewise differentiable positive definite matrix-valued function P⋅ such that the following conditions involving differential linear matrix inequalities are satisfied:(16)P˙t+At⊤Pt+PtAt<0,t∉ℐ,Ptk>Btk⊤Ptk+Btk,k=1,2,…,Pt≥Qt,P0<R.Moreover, it is finite-time stable with respect toΩ,X0,Xt if and only if there exists a piecewise continuous positive definite matrix-valued solution Z⋅:Ω↦ℝn×n such that the following conditions involving differential/difference Lyapunov equations are satisfied:(17)Z˙t−AtZt−ZtAt⊤=0,t∉ℐ,Z+tk=BtkZ−tkBtk⊤,t∈ℐ,Z0=R−1,GtWtG⊤t<I,∀t∈Ω,where G⋅ is a nonsingular matrix-valued function satisfying Qt=G⊤tGt in Ω.As for a linear time-varying system with state-dependent impulses (6), we have the following theorem to provide a sufficient condition for its finite-time stability.Theorem 6 (see [35]).
System (6) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈Ω there exists a piecewise differentiable positive definite matrix-valued function P⋅ such that the following conditions involving linear differential/difference matrix inequalities are satisfied:(18)P˙t+At⊤Pt+PtAt<0,xt∈ℝn\∪k=1NSk,x⊤Bt⊤PtBt−Ptx<0,xt∈Sk,k=1,…,N,Pt≥Gt,P0<R.When the initial setX0 and the time-varying set Xt are two given polytopes and a quadratic function Vx=x⊤Px is chosen, we have following sufficient conditions of finite-time stability for a quadratic system:(19)x˙=Ax+x⊤C1⊤x,x⊤C2⊤x,…,x⊤Cn⊤x⊤,where Ci∈ℝn×n,i=1,…,n.Theorem 7 (see [34]).
System (19) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are two given polytopes, if there exists a positive definite symmetric matrix P∈ℝn×n such that(20)λmaxPmaxixX0i2eαT≤1,α>0,1ak⊤akP≥0,k=1,2,…,q¯,xX0i⊤PxX0i≤1,i=1,2,…,q,A⊤+B1⊤xXtiB2⊤xXti…Bn⊤xXtiP+PA+B1⊤xXtiB2⊤xXti…Bn⊤xXti⊤−αP≤0.The initial setX0 and the time-varying set Xt can also be piecewise quadratic domains over conical partitions P0=U10,U20,…,Uu0 and Pt=U1t,U2t,…,Uvt, say X0=x∈ℝnx0⊤Rix0≤1,x0∈Ui0,i=1,2,…,u and Xt=x⊤Qitx≤1,x∈Uit,i=1,2,…,v [37, 38]. In such cases, we choose a time-varying piecewise quadratic Lyapunov-like function defined over the abovementioned conical partition as(21)VXtt,x=x⊤Pitx,∀x∈Siwithi=1,…,v,where Pi∈ℝn×n,i=1,…,v, are symmetric matrices. Then, a sufficient condition for the finite-time stability of linear-varying system (1) can be presented as follows.Theorem 8 (see [37]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the given piecewise quadratic domains, if there exist positive definite symmetric matrices Pi∈ℝn×n such that(22)x⊤P˙it+At⊤Pit+PitAtx<0,x⊤Pit−Qitx≥0,x⊤Pi0−Rix<0.
Fort∈Ω and x∈Si with i=1,…,v.The sufficient conditions in (22) are not applicable due to the infinite number of matrix inequalities. Applying S-procedure arguments and considering the conical partition, a computationally tractable sufficient condition is also obtained in [37].Theorem 9 (see [37]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the given piecewise quadratic domains, if there exist positive numbers bi,k, positive real-valued functions ci,kt, zi,kt and matrices Hi,k, i=1,…,v and k=1,…,s, and positive definite symmetric matrices Pi∈ℝn×n such that x⊤Hi,kx≤0,∀x∈Si and there exist positive piecewise continuously differentiable matrix-valued functions Pit∈ℝn×n, such that the following conditions involving differential linear matrix inequalities and linear matrix inequalities are satisfied:(23)P˙it+At⊤Pit+PitAt−∑k=1sci,ktHi,k<0,Pit−Qit+∑k=1szi,ktHi,k≥0,Pi0−Ri−∑k=1sbi,kHi,k<0,x^k⊤Pitx^k=x^k⊤Pjtx^k,∀x^k∈extrSi∩Sj,x^h⊤Pitx^k=x^h⊤Pjtx^k,∀x^h,x^k∈extrSi∩Sj.


### 3.3. Converse Lyapunov-Like Theorem

In [9], a converse Lyapunov-like theorem is established for finite-time uniformly stable continuous-time nonautonomous system (1). It provides the characterization of finite-time stability with regards to the existence of Lyapunov-like functions.Theorem 10 (see [9]).
If system (1) is uniformly stable with respect to Ω,Sα,Sβ, α<β, then there exists a real-valued Lyapunov-like function Vt,x satisfying(24)V˙0t,x≤φt,for all x∈Sβ−S¯α, and(25)∫t1t2φtdt<Vminβt2−Vmaxαt1,for t1<t2 and t1,t2∈Ω.Moreover, two necessary and sufficient conditions for the existence of a Lyapunov-like functionVt,x for finite-time stability are given as follows.Theorem 11 (see [9]).
System (1) is finite-time stable with respect to Ω,Sα,Sβ, α<β, if and only if there exist a Lipschitz continuous real-valued Lyapunov-like function Vt,x and a continuous real-valued function μ⋅ such that the following conditions are satisfied: (i) Vt,x≥μx for all, (ii) V˙t,x≤0 for all x∈Sβ−S¯α, and (iii) Vmaxαt<μβ.Theorem 12 (see [39]).
System (1) is finite-time stable with respect to Ω,Sα,Sβ, α<β, if and only if there exist a continuous real-valued Lyapunov-like function Vt,x such that(26)V˙t,x≤0,for all x∈S¯β, and(27)Vminβt2<Vmaxαt1,for all t2>t1 and δ<α with t1,t2∈Ω.


### 3.4. Miscellaneous Issues

Time delays are often encountered in many practical systems such as chemical processes, electric circuits, and networked systems, leading to unsatisfactory system behaviours and even instability. So, various stability problems for delayed systems have attracted much attention to lots of researchers. Among them, finite-time stability analysis has been of particular interest bringing forth many papers such as [40–42].A linear time-invariant delayed system can be represented by(28)x˙t=A0xt+A1xt−h,h>0,with an associated initial state function(29)xt=ψt,−h≤t≤0.To proceed, we need the following definitions.Definition 3.
System (28) associated with initial condition (29) is said to be finite-time stable with respect to ζ⋅,β,h if ψt⊤ψt<ζt,∀t∈−τ,0, which implies xt⊤xt<β,∀t∈Ω, where ζ⋅ is a positive scalar-valued function satisfying ζt≤α for −h≤t≤0, and β>α>0.Definition 4.
System (28) associated with initial condition (29) is said to be finite-time stable with respect to Ω,α,β if sup−h≤θ≤0ψθ2<α implies xt2<β,∀t∈Ω.
Next, we introduce two sufficient conditions for finite-time stability of linear delayed system (28) in the following two theorems.Theorem 13 (see [41]).
System (28) associated with the initial condition (29) is finite-time stable with respect to α,β,h if for all t∈Ω, we have(30)Φt<β/α1+A1,where Φt is the fundamental matrix of linear delayed system (28).Theorem 14 (see [41]).
System (28) associated with initial condition (29) is finite-time stable with respect to Ω,α,β if(31)1+σmt2e2σmt<βα,where(32)σm=σmaxA0+σmaxA1,with σm⋅ being the largest singular value of the corresponding matrix.In [40], the authors construct a delay-dependent Lyapunov-like function(33)Vt=xt+∫0hJθxt−θdθ⊤xt+∫0hJθxt−θdθ,where Jt∈ℝn×n is a differentiable matrix-valued function on 0,h such that the following differential matrix equation is satisfied:(34)J˙θ=A0+J0Jθ,θ∈0,h,with the initial condition Jh=A1. Then, the following result based on Lyapunov function (33) can be given.Theorem 15 (see [40]).
System (28) associated with initial condition (29) is finite-time stable with respect to (α, β, h) if there exists a positive real number γ such that the following conditions are satisfied:(35)xt−θ⊤xt−θ<axt⊤xt,a>0,θ∈−τ,0,t∈Ω,1+h1+ψ1−γψ−ahγ−1eλmaxR⊤+RT<βα,γ∈maxγ1,0,γ2,γ1,2=1±1−4ψha2ψ,4ψha<1,ψ=λmaxJ0J⊤0e2μ1A¯0h−12μ1A¯0,A¯0=A0+J0,where μ1⋅ is a matrix measure of the given matrix and J0 is the solution of the following transcendental matrix equation:(36)eA0+J0hJ0=A1.Next, consider a singular linear delayed system(37)Ex˙t=A0xt+A1xt−h,h>0,where E∈ℝn×n is a singular matrix with rank r<n. There exist two nonsingular matrices M=M1M2 and G such that Gsuch thatIr000=MEG. Let(38)MA0G=A11A12A21A22,0M2A0G=00A21A22,MA2G=D11D12D21D22,W1=PA0+A0⊤P⊤+Q1+Q2M¯A0+A0⊤M¯⊤Q2⊤−ηPE,W2=PA1+Q2M¯A1,M¯=0M2,G⊤PM−1=P11P12P21P22,M−1⊤M−1=R11R12∗R22,α1=λminP11λmaxR11,α2=λmaxP11λminG⊤G+hλmaxQ1,α3=α2αα1,α4=∑i=0⊤/h−1A22−1D22i,η=A22−1D21+A22−1D22,α6=λmaxG⊤G.Then, a sufficient condition for the finite-time stability of singular linear delayed system (37) was established in [42].Theorem 16 (see [42]).
System (37) associated with initial condition (29) is finite-time stable with respect to α,β,h if there exists a positive number γ, a symmetric positive definite matrix Q1∈ℝn×n, a nonsingular matrix P∈ℝn×n, and a matrix Q2∈ℝn×n such that(39)PE=E⊤P≥0,W1W2∗−Q1<0,eγTα3+α5+ηα4eγTα32≤βα6.More recently, the Lyapunov–Razumikhin approach is extended to finite-time stability for a nonlinear delayed system(40)x˙t=ft,xt−h,in [43]. Sufficient conditions can be illustrated through the following theorem.Theorem 17 (see [43]).
System (40) associated with initial condition (29) is finite-time stable with respect to α,β,h if there exists positive scalars α,β,η,σ,T with η<α<β and σ∈0,T, integrable real-valued function c⋅:ℝ+↦ℝ, class K functions γ1, γ2, and a differentiable function V:−h,T×ℝn↦ℝ+ such that (i) γ1x≤Vt,x≤γ2x,∀t,x∈−h,T×ℝn, (ii) V˙t,ψ0≤ctVt,ψ0, whenever Vt+s,ψs≤Θt,sVt,ψ0 for all t∈0,T, s∈−h,0, where(41)Θt,s=exp−∫maxt+s,0tcudu,and (iii)(42)∫0tcudu≤lnγ1ηγ2α,∀t∈0,T.Besides time delays, uncertainty is another important phenomenon commonly encountered in practical systems. The existence of uncertainty causes the poor performance and even instability. The finite-stability concept has been extended to uncertain linear systems [44–46]. The uncertainty can be expressed as norm-bounded uncertainty and structured uncertainty. Consider an uncertain linear system(43)x˙=At+F1tI−ΔtHt−1ΔtF2txt,where xt∈ℝn, Δ⋅ is a norm-bounded uncertainty function such as Δt≤1 and F1⋅,F2⋅ are known matrices of appropriate dimensions. The following theorem will present a necessary and sufficient condition for finite-time stability of system (43).Theorem 18 (see [46]).
System (43) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω, there exist a piecewise continuous function γ and a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving differential linear matrix inequalities are satisfied:(44)P˙τ+A⊤τPτ+PτAτ+γτF2⊤τF2τPτF1τ+γτF2⊤τHτF1⊤τPτ+γτHτF2τ−γτI−H⊤τHτ<0,Pt≥Qt,P0<R,γt>0,τ∈0,t.Similar to Theorem3, necessary and sufficient conditions in Theorem 18 are not computationally tractable. For the computational purposes, we can see the following sufficient condition for finite-time stability of system (43).Theorem 19 (see [46]).
System (43) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω, there exist a piecewise continuous function γ and a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving differential linear matrix inequalities are satisfied:(45)P˙t+A⊤tPt+PtAt+γtF2⊤tF2tPtF1t+γtF2⊤tHtF1⊤tPt+γtHtF2t−γtI−H⊤tHt<0,Pt≥Qt,P0<R,γt>0,t∈Ω.Robustness analysis for a linear delayed system with structured uncertainty was conducted in [45], where the uncertain system is described by(46)x˙t=A0+D0FtE0xt+A1+D1FtE1xt−h,where D0,E0,D1,E1 are known matrices of appropriate dimension, and Ft is the uncertain time-varying matrix with FtFt⊤≤I,∀t∈Ω.Theorem 20.
(see [45]). System (46) associated with initial condition (29) is finite-time stable with respect to α,β,h if there exists positive real numbers γ,δ,β0,β1,β2,β3 and symmetric positive definite matrices P1 and P2 such that the following conditions are satisfied:(47)Ω11P1A1P1D0P1D1∗−P2+δE1⊤E100∗∗−γI0∗∗∗−δI<0,Ω11=A0⊤P1+P1A0+P2−β0P1+γE0⊤E0,β1I<P1<β2I,0<P2<β3I,−βe−β0Tβ1αβ2αhβ3∗−β20∗∗−β3<0.If uncertain linear system (43) is affected by finite impulses (4), the system state will undergo abrupt changes at discrete time instants, which leads to more difficulty to analyze its stability performance. The following theorem from [46] gives a sufficient condition for finite-time stability of the linear time-varying system with both time-dependent impulses and uncertainty. More complex cases with a norm-bounded uncertainty on the impulsive matrix-valued function and state-dependent impulses were also provided in [46] as well.Theorem 21 (see [44]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω, there exists positive real number γ and a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions are satisfied:(48)P˙t+A⊤tPt+PtAt+γF2⊤F2PtF1+γF2⊤HF1⊤Pt+γH⊤F2−γI−H⊤H<0,t∉ℐ,Bt⊤P+tBt−Pt≤0,t∈ℐ,All results mentioned above have illustrated the finite-time stability conditions in a quantitative sense, and we continue to introduce more results in a qualitative sense.


## 3.1. State Transition Matrix

In linear system theory, we know that the solution of (3) can be described as xt=Φt,0x0, where the matrix-valued function Φ⋅,⋅ has the following basic properties:(7)Φ˙t,0=AtΦt,0,Φ0,0=I.This matrix-valued functionΦ⋅,⋅ is called to be the state transition matrix, and an appropriate assumption on the nature of At can ensure the existence and uniqueness of the state transition matrix. For system (3), it is stable in the sense of Lyapunov if there exists a positive constant M such that Φt,0<M for all t≥0. Moreover, system (3) is asymptotically stable if it is stable and limt⟶∞Φt,0=0. As for finite-time stability of the system (3), a necessary and sufficient stability condition will be provided in the following theorem [30].Theorem 1.
System (3) is finite-time stable with respect to Ω,X0,Xt, where X0=x∈ℝnx0⊤Rx0≤1 and Xt=x∈ℝnx⊤Qtx≤1 if and only if the state transition matrix of system (3) satisfies(8)Φt,0⊤QtΦt,0<R.The state transition matrix approach has been extended to impulsive linear system (5) in [31, 32, 36]. Letting T∈tl,tl+1, the solution of the impulsive linear system (5) will be xt=Φ^t,0x0,t∈Ω, where Φ^t,0, called the state transition matrix of (5), is a piecewise continuous matrix-valued function with discontinuous right-hand sides at the time instants tk,k=1,2,…,l. In detail, when t∈0,t1, Φ^t,0 is the solution of the following matrix differential equation:(9)∂∂tΦ^t,0=AtΦt,0,Φ^0,0=I,Φ^t1+,0=Bt1Φ^t1,0.In the sequel intervals fork=1,2,…,l−1, Φ^t,tk should satisfy(10)∂∂tΦ^t,tk+=AtΦ^t,tk+,t∈tk,tk+1,Φ^tk+1+,tk+=Btk+1Φ^tk+1,tk+.In the end, whent∈tl,tl+1, we have(11)∂∂tΦ^t,tl+=AtΦ^t,tl+,t∈tl,T.Theorem 2.
Impulsive linear system (5) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈0,T, the following is satisfied:(12)Φ^t,0⊤QtΦ^t,0<R.The conditions in the form of state transition matrices (8) and (12) in Theorems 1 and 2 are valuable for theoretical analysis but hard to apply due to the high computational difficulty, particularly for the time-varying case.To obtain computational conditions for finite-time stability, some Lyapunov-like functions are needed to establish conditions in the form of linear matrix equalities or Lyapunov matrix inequalities. In some early work such as [3], it concludes that system (1) is finite-time stable with respect to Ω,Sα,Sβ if and only if there exists a real-valued Lipschitz function Vt,x, continuous on Ω×S¯β, and a real-valued integrable function φt such that, for t∈Ω, we have V˙0t,x≤φt for all x∈Sβ−S¯α and ∫t1t2φtdt<Vminβt2−Vmaxαt1 for t1<t2 and t1,t2∈Ω. A piecewise continuous Lyapunov-like function is the most common one in the literature, and relevant results will be presented in the following subsection.


## 3.2. Piecewise Continuous Lyapunov-Like Functions

To obtain computationally tractable finite-time stability conditions, we choose a quadratic piecewise continuous Lyapunov-like functionVt,x=x⊤Ptx and establish the following conditions containing coupled differential Lyapunov matrix equations and differential linear matrix inequalities.Theorem 3 (see [7, 30]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈Ω, there exists a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving the differential matrix equation with boundary conditions are satisfied:(13)P˙τ=−Aτ⊤Pτ−PτAτ−ɛI,τ∈0,t,ɛ>0,Pt=Qt,P0<R.We see that Theorem3 provides a necessary and sufficient condition for finite-stability of system (3). However, it is not practicable to verify the differential matrix equation in (13) for every τ∈0,t, and hence, Theorem 3 is not suitable for the computational purpose. Using Theorem 3, we can obtain a sufficient condition with computational tractability for finite-stability of system (3).Theorem 4 (see [7, 30]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω there exists a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving the differential matrix equation with boundary conditions are satisfied:(14)P˙t=−At⊤Pt−PtAt−ɛI,ɛ>0,Pt=Qt,P0<R.Nowadays, computational tools in the convex optimization framework such as linear matrix inequalities or differential linear matrix inequalities are very efficient, and we can obtain the following necessary and sufficient conditions including differential linear matrix inequalities, equivalent to (13):(15)P˙τ<−Aτ⊤Pτ−PτAτ,Qt≤Pt,P0<R.The piecewise continuous Lyapunov-like function is also applied to the finite-time stability problem of impulsive linear systems. For linear time-varying systems with time-dependent impulses, we have the following finite-time stability results.Theorem 5 (see [36]).
System (5) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈Ω there exists a piecewise differentiable positive definite matrix-valued function P⋅ such that the following conditions involving differential linear matrix inequalities are satisfied:(16)P˙t+At⊤Pt+PtAt<0,t∉ℐ,Ptk>Btk⊤Ptk+Btk,k=1,2,…,Pt≥Qt,P0<R.Moreover, it is finite-time stable with respect toΩ,X0,Xt if and only if there exists a piecewise continuous positive definite matrix-valued solution Z⋅:Ω↦ℝn×n such that the following conditions involving differential/difference Lyapunov equations are satisfied:(17)Z˙t−AtZt−ZtAt⊤=0,t∉ℐ,Z+tk=BtkZ−tkBtk⊤,t∈ℐ,Z0=R−1,GtWtG⊤t<I,∀t∈Ω,where G⋅ is a nonsingular matrix-valued function satisfying Qt=G⊤tGt in Ω.As for a linear time-varying system with state-dependent impulses (6), we have the following theorem to provide a sufficient condition for its finite-time stability.Theorem 6 (see [35]).
System (6) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same with those in Theorem 1, if and only if for all t∈Ω there exists a piecewise differentiable positive definite matrix-valued function P⋅ such that the following conditions involving linear differential/difference matrix inequalities are satisfied:(18)P˙t+At⊤Pt+PtAt<0,xt∈ℝn\∪k=1NSk,x⊤Bt⊤PtBt−Ptx<0,xt∈Sk,k=1,…,N,Pt≥Gt,P0<R.When the initial setX0 and the time-varying set Xt are two given polytopes and a quadratic function Vx=x⊤Px is chosen, we have following sufficient conditions of finite-time stability for a quadratic system:(19)x˙=Ax+x⊤C1⊤x,x⊤C2⊤x,…,x⊤Cn⊤x⊤,where Ci∈ℝn×n,i=1,…,n.Theorem 7 (see [34]).
System (19) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are two given polytopes, if there exists a positive definite symmetric matrix P∈ℝn×n such that(20)λmaxPmaxixX0i2eαT≤1,α>0,1ak⊤akP≥0,k=1,2,…,q¯,xX0i⊤PxX0i≤1,i=1,2,…,q,A⊤+B1⊤xXtiB2⊤xXti…Bn⊤xXtiP+PA+B1⊤xXtiB2⊤xXti…Bn⊤xXti⊤−αP≤0.The initial setX0 and the time-varying set Xt can also be piecewise quadratic domains over conical partitions P0=U10,U20,…,Uu0 and Pt=U1t,U2t,…,Uvt, say X0=x∈ℝnx0⊤Rix0≤1,x0∈Ui0,i=1,2,…,u and Xt=x⊤Qitx≤1,x∈Uit,i=1,2,…,v [37, 38]. In such cases, we choose a time-varying piecewise quadratic Lyapunov-like function defined over the abovementioned conical partition as(21)VXtt,x=x⊤Pitx,∀x∈Siwithi=1,…,v,where Pi∈ℝn×n,i=1,…,v, are symmetric matrices. Then, a sufficient condition for the finite-time stability of linear-varying system (1) can be presented as follows.Theorem 8 (see [37]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the given piecewise quadratic domains, if there exist positive definite symmetric matrices Pi∈ℝn×n such that(22)x⊤P˙it+At⊤Pit+PitAtx<0,x⊤Pit−Qitx≥0,x⊤Pi0−Rix<0.
Fort∈Ω and x∈Si with i=1,…,v.The sufficient conditions in (22) are not applicable due to the infinite number of matrix inequalities. Applying S-procedure arguments and considering the conical partition, a computationally tractable sufficient condition is also obtained in [37].Theorem 9 (see [37]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the given piecewise quadratic domains, if there exist positive numbers bi,k, positive real-valued functions ci,kt, zi,kt and matrices Hi,k, i=1,…,v and k=1,…,s, and positive definite symmetric matrices Pi∈ℝn×n such that x⊤Hi,kx≤0,∀x∈Si and there exist positive piecewise continuously differentiable matrix-valued functions Pit∈ℝn×n, such that the following conditions involving differential linear matrix inequalities and linear matrix inequalities are satisfied:(23)P˙it+At⊤Pit+PitAt−∑k=1sci,ktHi,k<0,Pit−Qit+∑k=1szi,ktHi,k≥0,Pi0−Ri−∑k=1sbi,kHi,k<0,x^k⊤Pitx^k=x^k⊤Pjtx^k,∀x^k∈extrSi∩Sj,x^h⊤Pitx^k=x^h⊤Pjtx^k,∀x^h,x^k∈extrSi∩Sj.


## 3.3. Converse Lyapunov-Like Theorem

In [9], a converse Lyapunov-like theorem is established for finite-time uniformly stable continuous-time nonautonomous system (1). It provides the characterization of finite-time stability with regards to the existence of Lyapunov-like functions.Theorem 10 (see [9]).
If system (1) is uniformly stable with respect to Ω,Sα,Sβ, α<β, then there exists a real-valued Lyapunov-like function Vt,x satisfying(24)V˙0t,x≤φt,for all x∈Sβ−S¯α, and(25)∫t1t2φtdt<Vminβt2−Vmaxαt1,for t1<t2 and t1,t2∈Ω.Moreover, two necessary and sufficient conditions for the existence of a Lyapunov-like functionVt,x for finite-time stability are given as follows.Theorem 11 (see [9]).
System (1) is finite-time stable with respect to Ω,Sα,Sβ, α<β, if and only if there exist a Lipschitz continuous real-valued Lyapunov-like function Vt,x and a continuous real-valued function μ⋅ such that the following conditions are satisfied: (i) Vt,x≥μx for all, (ii) V˙t,x≤0 for all x∈Sβ−S¯α, and (iii) Vmaxαt<μβ.Theorem 12 (see [39]).
System (1) is finite-time stable with respect to Ω,Sα,Sβ, α<β, if and only if there exist a continuous real-valued Lyapunov-like function Vt,x such that(26)V˙t,x≤0,for all x∈S¯β, and(27)Vminβt2<Vmaxαt1,for all t2>t1 and δ<α with t1,t2∈Ω.


## 3.4. Miscellaneous Issues

Time delays are often encountered in many practical systems such as chemical processes, electric circuits, and networked systems, leading to unsatisfactory system behaviours and even instability. So, various stability problems for delayed systems have attracted much attention to lots of researchers. Among them, finite-time stability analysis has been of particular interest bringing forth many papers such as [40–42].A linear time-invariant delayed system can be represented by(28)x˙t=A0xt+A1xt−h,h>0,with an associated initial state function(29)xt=ψt,−h≤t≤0.To proceed, we need the following definitions.Definition 3.
System (28) associated with initial condition (29) is said to be finite-time stable with respect to ζ⋅,β,h if ψt⊤ψt<ζt,∀t∈−τ,0, which implies xt⊤xt<β,∀t∈Ω, where ζ⋅ is a positive scalar-valued function satisfying ζt≤α for −h≤t≤0, and β>α>0.Definition 4.
System (28) associated with initial condition (29) is said to be finite-time stable with respect to Ω,α,β if sup−h≤θ≤0ψθ2<α implies xt2<β,∀t∈Ω.
Next, we introduce two sufficient conditions for finite-time stability of linear delayed system (28) in the following two theorems.Theorem 13 (see [41]).
System (28) associated with the initial condition (29) is finite-time stable with respect to α,β,h if for all t∈Ω, we have(30)Φt<β/α1+A1,where Φt is the fundamental matrix of linear delayed system (28).Theorem 14 (see [41]).
System (28) associated with initial condition (29) is finite-time stable with respect to Ω,α,β if(31)1+σmt2e2σmt<βα,where(32)σm=σmaxA0+σmaxA1,with σm⋅ being the largest singular value of the corresponding matrix.In [40], the authors construct a delay-dependent Lyapunov-like function(33)Vt=xt+∫0hJθxt−θdθ⊤xt+∫0hJθxt−θdθ,where Jt∈ℝn×n is a differentiable matrix-valued function on 0,h such that the following differential matrix equation is satisfied:(34)J˙θ=A0+J0Jθ,θ∈0,h,with the initial condition Jh=A1. Then, the following result based on Lyapunov function (33) can be given.Theorem 15 (see [40]).
System (28) associated with initial condition (29) is finite-time stable with respect to (α, β, h) if there exists a positive real number γ such that the following conditions are satisfied:(35)xt−θ⊤xt−θ<axt⊤xt,a>0,θ∈−τ,0,t∈Ω,1+h1+ψ1−γψ−ahγ−1eλmaxR⊤+RT<βα,γ∈maxγ1,0,γ2,γ1,2=1±1−4ψha2ψ,4ψha<1,ψ=λmaxJ0J⊤0e2μ1A¯0h−12μ1A¯0,A¯0=A0+J0,where μ1⋅ is a matrix measure of the given matrix and J0 is the solution of the following transcendental matrix equation:(36)eA0+J0hJ0=A1.Next, consider a singular linear delayed system(37)Ex˙t=A0xt+A1xt−h,h>0,where E∈ℝn×n is a singular matrix with rank r<n. There exist two nonsingular matrices M=M1M2 and G such that Gsuch thatIr000=MEG. Let(38)MA0G=A11A12A21A22,0M2A0G=00A21A22,MA2G=D11D12D21D22,W1=PA0+A0⊤P⊤+Q1+Q2M¯A0+A0⊤M¯⊤Q2⊤−ηPE,W2=PA1+Q2M¯A1,M¯=0M2,G⊤PM−1=P11P12P21P22,M−1⊤M−1=R11R12∗R22,α1=λminP11λmaxR11,α2=λmaxP11λminG⊤G+hλmaxQ1,α3=α2αα1,α4=∑i=0⊤/h−1A22−1D22i,η=A22−1D21+A22−1D22,α6=λmaxG⊤G.Then, a sufficient condition for the finite-time stability of singular linear delayed system (37) was established in [42].Theorem 16 (see [42]).
System (37) associated with initial condition (29) is finite-time stable with respect to α,β,h if there exists a positive number γ, a symmetric positive definite matrix Q1∈ℝn×n, a nonsingular matrix P∈ℝn×n, and a matrix Q2∈ℝn×n such that(39)PE=E⊤P≥0,W1W2∗−Q1<0,eγTα3+α5+ηα4eγTα32≤βα6.More recently, the Lyapunov–Razumikhin approach is extended to finite-time stability for a nonlinear delayed system(40)x˙t=ft,xt−h,in [43]. Sufficient conditions can be illustrated through the following theorem.Theorem 17 (see [43]).
System (40) associated with initial condition (29) is finite-time stable with respect to α,β,h if there exists positive scalars α,β,η,σ,T with η<α<β and σ∈0,T, integrable real-valued function c⋅:ℝ+↦ℝ, class K functions γ1, γ2, and a differentiable function V:−h,T×ℝn↦ℝ+ such that (i) γ1x≤Vt,x≤γ2x,∀t,x∈−h,T×ℝn, (ii) V˙t,ψ0≤ctVt,ψ0, whenever Vt+s,ψs≤Θt,sVt,ψ0 for all t∈0,T, s∈−h,0, where(41)Θt,s=exp−∫maxt+s,0tcudu,and (iii)(42)∫0tcudu≤lnγ1ηγ2α,∀t∈0,T.Besides time delays, uncertainty is another important phenomenon commonly encountered in practical systems. The existence of uncertainty causes the poor performance and even instability. The finite-stability concept has been extended to uncertain linear systems [44–46]. The uncertainty can be expressed as norm-bounded uncertainty and structured uncertainty. Consider an uncertain linear system(43)x˙=At+F1tI−ΔtHt−1ΔtF2txt,where xt∈ℝn, Δ⋅ is a norm-bounded uncertainty function such as Δt≤1 and F1⋅,F2⋅ are known matrices of appropriate dimensions. The following theorem will present a necessary and sufficient condition for finite-time stability of system (43).Theorem 18 (see [46]).
System (43) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω, there exist a piecewise continuous function γ and a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving differential linear matrix inequalities are satisfied:(44)P˙τ+A⊤τPτ+PτAτ+γτF2⊤τF2τPτF1τ+γτF2⊤τHτF1⊤τPτ+γτHτF2τ−γτI−H⊤τHτ<0,Pt≥Qt,P0<R,γt>0,τ∈0,t.Similar to Theorem3, necessary and sufficient conditions in Theorem 18 are not computationally tractable. For the computational purposes, we can see the following sufficient condition for finite-time stability of system (43).Theorem 19 (see [46]).
System (43) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω, there exist a piecewise continuous function γ and a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions involving differential linear matrix inequalities are satisfied:(45)P˙t+A⊤tPt+PtAt+γtF2⊤tF2tPtF1t+γtF2⊤tHtF1⊤tPt+γtHtF2t−γtI−H⊤tHt<0,Pt≥Qt,P0<R,γt>0,t∈Ω.Robustness analysis for a linear delayed system with structured uncertainty was conducted in [45], where the uncertain system is described by(46)x˙t=A0+D0FtE0xt+A1+D1FtE1xt−h,where D0,E0,D1,E1 are known matrices of appropriate dimension, and Ft is the uncertain time-varying matrix with FtFt⊤≤I,∀t∈Ω.Theorem 20.
(see [45]). System (46) associated with initial condition (29) is finite-time stable with respect to α,β,h if there exists positive real numbers γ,δ,β0,β1,β2,β3 and symmetric positive definite matrices P1 and P2 such that the following conditions are satisfied:(47)Ω11P1A1P1D0P1D1∗−P2+δE1⊤E100∗∗−γI0∗∗∗−δI<0,Ω11=A0⊤P1+P1A0+P2−β0P1+γE0⊤E0,β1I<P1<β2I,0<P2<β3I,−βe−β0Tβ1αβ2αhβ3∗−β20∗∗−β3<0.If uncertain linear system (43) is affected by finite impulses (4), the system state will undergo abrupt changes at discrete time instants, which leads to more difficulty to analyze its stability performance. The following theorem from [46] gives a sufficient condition for finite-time stability of the linear time-varying system with both time-dependent impulses and uncertainty. More complex cases with a norm-bounded uncertainty on the impulsive matrix-valued function and state-dependent impulses were also provided in [46] as well.Theorem 21 (see [44]).
System (3) is finite-time stable with respect to Ω,X0,Xt, where the sets X0 and Xt are the same as those in Theorem 1, if and only if for all t∈Ω, there exists positive real number γ and a symmetric piecewise differentiable matrix-valued function P⋅ such that the following conditions are satisfied:(48)P˙t+A⊤tPt+PtAt+γF2⊤F2PtF1+γF2⊤HF1⊤Pt+γH⊤F2−γI−H⊤H<0,t∉ℐ,Bt⊤P+tBt−Pt≤0,t∈ℐ,All results mentioned above have illustrated the finite-time stability conditions in a quantitative sense, and we continue to introduce more results in a qualitative sense.


## 4. Finite-Time Stability in a Qualitative Sense

It is well known that a radially unbounded positive definite functionV:C⊂ℝn⟶ℝ+ with the property V˙x<0 is a Lyapunov function. Lyapunov’s second method demonstrates that the existence of the Lyapunov function is also equivalent to the asymptotical stability of system (1), which provides the foundation for the following necessary and sufficient conditions for finite-time stability results.Theorem 22 (see [27, 47]).
Consider an autonomous nonlinear system(49)V˙x≤−cVxη,for allx∈C.
System (49) is finite-time stable if and only if there exists a smooth Lyapunov function (equivalently, all smooth Lyapunov functions) V:C⟶ℝ+ such that for all x∈C,(50)Tx=∫Vx0dsV˙xθxs,x<+∞,where the map θx is the inverse of t↦Vxt,x.Theorem 23 (see [27, 47]).
System (49) is finite-time stable with a continuous settling-time function at the origin if and only if there exist a scalar η∈0,1, a positive scalar c, and a smooth Lyapunov function V:C⟶ℝ+ such that(51)V˙x≤−cVxη,for allx∈C.
Moreover, the settling-time functionTx should satisfy(52)Tx≤Vx1−ηc1−η.A converse Lyapunov theorem was obtained for finite-time stability of nonlinear system (49) in [18]. As for the case that settling-time is continuous, we have the following converse theorem.Theorem 24 (see [18]).
If system (49) is finite-time stable with a continuous settling-time function at the origin and η∈0,1, then there exists a continuous function V:C⟶ℝ+ such that(53)V˙x≤−cVxη,for allx∈C.The abovementioned Lyapunov-based methods for analysis of finite-time stability may not be suitable for constructive design. Recently, an implicit Lyapunov function method to solve an algebraic equation was derived in [48], which provides a design method for a robust controller for the closed-loop systems to handle exogenous disturbances. The implicit Lyapunov function theorem only verifies stability conditions in an implicit way and does not need to solve the equation.Theorem 25 (see [48]).
System (1) is finite-time stable with a settling-time function Tx0≤V0μ/cμ, where QV0,x0=0 if there exists a continuously differentiable function Z:ℝ+×ℝn⟶ℝ such that for any x∈C, there exists a radially unbounded function V∈ℝ+ such that(54)ZV,x=0,∂ZV,x∂V<0,supt∈ℝ+∂ZV,x∂xf<0,for all V,x∈Ω, where Ω=V,x∈ℝ+×ℝn:ZV,x=0 and limx⟶0V=0.Recently, the notion of finite-time stability for nonlinear autonomous system (49) was extended to nonautonomous nonlinear system (1). These Lyapunov and converse Lyapunov results are derived and introduced in the following theorems.Theorem 26.
(see [17]). System (1) is finite-time stable if there exist a scalar η∈0,1, a positive function c(t), a class K function γ1⋅, and a continuously differentiable function V:ℝ+×C⟶ℝ+ such that Vt,0=0, Vt,x≥γ1⋅, and(55)V˙x≤−ctVxη,for allx∈C.
Moreover, for the caseC=ℝn, then system (1) is be globally finite-time stable. If there exists a class K function γ2⋅ such that Vt,x≤γ2⋅, then system (1) is uniformly finite-time stable.Theorem 27 (see [17]).
Letη∈0,1 and there exists a class K function φ:0,r⟶ℝ+, where r>0 such that ℬr0⊆C and(56)ft,x≤φx,t∈0,∞,x∈ℬr0.
If system (1) is uniformly finite-time stable and the settling-time function T⋅,⋅ is jointly continuous at t,0, t≥0, then there exist a positive scalar c, a class K function γ⋅, and a continuously differentiable function V:ℝ+×C⟶ℝ+ such that Vt,0=0, Vt,x≥γ⋅, and(57)V˙x≤−cVxη,for allx∈C.


## 5. Conclusions

This paper has overviewed the fundamental results of the finite-time stability analysis of dynamical systems. The concepts of finite-time stability are classified into those in the quantitative and qualitative senses. Finite-time stability in a quantitative sense is firstly investigated. Then, finite-time stability results in a qualitative sense are outlined. This review paper is far from complete due to our limitations and nonawareness. We hope that this paper can be a useful resource for practitioners, researchers, and graduate students working in this field.



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