Necessary and sufﬁcient conditions for stabilisability of discrete-time time-varying switched systems

This paper is concerned with necessary and sufﬁcient conditions for stabilisability of time-varying discrete-time switched systems. Starting with an asymptotically stable function, an exponentially stable function and a uniformly exponentially stable function, we succes-sively propose necessary and sufﬁcient conditions for asymptotic stabilisability, exponential stabilisability and uniform exponential stabilisability of time-varying switched linear systems. Further, considering the broad applications of ﬁnite-time stability in practical systems, based on an additionally introduced concept of ﬁnite-time stable functions, we derive a necessary and sufﬁcient condition for ﬁnite-time stabilisability of time-varying switched linear systems. Afterwards, three illustrative examples are given to show the applicability of our theoretical results. In the end, we further discuss the necessary and sufﬁcient conditions for global exponential stabilisability and global uniform exponential stabilisability


INTRODUCTION
As an important class of hybrid systems, switched systems consist of several subsystems along with a switching signal that orchestrates the switching among these subsystems (see [1,2]). Numerous researchers focus on such systems due to their broad applications in practical multiple-mode systems such as constrained robotics, chemical processes, multi-agent systems, manufacturing systems, and traffic control (see [3][4][5][6][7]). There are many interesting issues on stability analysis of switched systems, for example, the issue on proposing necessary and sufficient conditions to guarantee the stability under arbitrary switching signals (see [8,9]), the issue on demonstrating various stability under certain constrained switching signals (see [10][11][12][13]), and the issue on analysing the existence of switching signals such that switched systems can be stable (i.e. the issue on stabilisability analysis [3,14]). Among these issues, stabilisability analysis is an essential and significant one. However, owing to the multiple subsystems and various types of switching signals, stabilisability analysis of switched systems is complicated. certain switching signals such that switched systems be divergent [1]. Besides, there may also exist switching signals such that switched systems with unstable subsystems become stable. Therefore, the issue of stabilisability analysis is interesting and worthy to be paid more attention to.
Fortunately, the stabilisability problem of systems has been widely concerned and many significant contributions on this topic have been given (see [15][16][17][18][19][20][21][22][23][24][25][26]). Especially, with respect to the stabilisability of time-invariant switched systems, several valuable results are made based on Lyapunov functions method and dwell time method. For example, Fiacchini et al. [18] provided a geometrical and numerical insight on the stabilisability criteria proposed in [16,17]; based on multiple discontinuous Lyapunov functions and mode-dependent average dwell time, Zhao et al. [25] investigated the stabilisation problem of switched systems with unstable subsystems; several equivalent stability conditions for switched linear systems (SLSs) were presented in [26]. However, the stabilisability analysis of timevarying switched systems has been less extensively studied (see [11,27,28]). For instance, Chen et al. [28] researched the sufficient conditions of the asymptotic stability for positive linear time-varying switched systems according to time-varying copositive Lyapunov functions; Liu et al. [11] provided stabilisability criteria for switched systems with both stable and unstable subsystems based on multiple Lyapunov functions and dwell time. In particular, stabilisability of discrete-time time-varying switched systems has been even less studied.
It is known that the above-mentioned results mainly focused on the Lyapunov stability, which describes the behaviours of a system over an infinite-time interval. However, in many practical systems, it is necessary to consider their finite-time stability. Note that there are mainly two kinds of concepts of finitetime stability. One, proposed by Bhat et al., refers that the system state converges to the equilibrium point within a relatively finite time interval [29,30]. Another, proposed by Amato et al., means that when pre-given a bound on the initial condition, states do not exceed a certain threshold during a finite-time interval [31,32]. Many attentions have been paid on these two finite-time stability and numerous significant results emerged (see [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]). For example, Lu and She [42] proposed a sufficient condition of finite-time stability based on a piecewise timevarying Lyapunov-like function; Zhao et al. [43] considered the finite-time boundedness of uncertain switched linear systems based on average dwell time method; Chen and Yang [29] investigated the sufficient conditions of the finite-time stability for time-varying continuous-time switched non-linear systems via indefinite common Lyapunov functions and indefinite multiple Lyapunov functions; for time-invariant impulsive discrete-time SLSs with and without perturbation, [36] and [33], respectively, established the corresponding sufficient conditions of the finitetime stability. We choose to investigate the finite-time stability, proposed by Amato et al., since it mainly focuses on the stability of the system during a specified interval, and the system dynamic quality can be better described when the initial state is bounded.
Inspired by the idea of using scalar functions in [24] and [41] and the time-varying Lyapunov functions in [14] and [27], we are dedicated to analysing the stabilisability of time-varying discrete-time switched systems. Firstly, resorting to asymptotically (exponentially, uniformly exponentially) stable scalar functions, we explore necessary and sufficient conditions for the asymptotic (exponential, uniform exponential) stabilisability of time-varying SLSs, respectively. Note that Zhou and Zhao [24] had considered the stability of the time-varying linear dynamical systems, which can be regarded as a special case of timevarying SLSs with a single subsystem. Then, considering the broad applications of finite-time stability in practical systems, based on an additionally introduced concept of finite-time stable functions, a necessary and sufficient condition for the finitetime stabilisability of time-varying discrete-time SLSs is presented, where the constraints are all independent with the initial state. Note that Chen and Yang [41] had proposed a sufficient and necessary condition for the finite-time stability of continuous-time time-varying impulsive SLSs based on indefinite multiple Lyapunov functions, where one of the constraints depended on initial state. Afterwards, three examples are given to illustrate the applicability of our results. Finally, we give further discussions on the necessary and sufficient conditions of the global exponential stabilisability and global uniform exponential stabilisability of switched non-linear systems.
The main contributions can be summarised as follows.
(1) Combining time-varying Lyapunov functions with asymptotically (exponentially, uniformly exponentially) stable scalar functions, we propose alternative necessary and sufficient conditions for the asymptotic (exponential, uniform exponential) stabilisability of time-varying discretetime SLSs. Compared to traditional difference Lyapunov inequalities [45], we release the requirement on negative definiteness of the time-difference of time-varying Lyapunov functions, which can be seen in Example 1. (2) Considering the broad applications of finite-time stability in practical systems, we introduce a finite-time stable function and then attain a necessary and sufficient condition for the finite-time stabilisability of time-varying discrete-time SLSs, where the constraints release the traditional difference Lyapunov inequalities [39] (see Example 3). Moreover, compared to [41], our finite-time stable function (see Lemma 4) is independent with the initial state. (3) We further attain a necessary and sufficient condition for the global exponential stabilisability and global uniform exponential stabilisability of switched non-linear systems.
This paper is organised as follows. In Section 2, several relevant definitions are introduced. Specially, we additionally introduce a concept of finite-time stable functions. In Section 3, several necessary and sufficient conditions for stabilisability of discrete-time time-varying switched linear systems are proposed. Three illustrative examples are presented in Section 4. Moreover, further discussions on the stabilisability of switched non-linear systems are presented in Section 5. Finally, Section 6 concludes the paper.

PRELIMINARIES
Notions: Let ℤ, ℤ ≥0 and ℤ >0 be the set of all integers, nonnegative integers and positive integers. Denote ℝ, ℝ ≥0 and ℝ >0 as the set of all real numbers, non-negative real numbers and positive real numbers. ℝ n denotes the space of n-dimensional real vectors, ‖ ⋅ ‖ is its Euclidean norm and for x ∈ ℝ n , x T denotes its transpose. ℝ n×n stands for the space of n × n real matrices; I n is an n × n identity matrix; and for A ∈ ℝ n×n , A T denotes its transpose and tr(A) denotes its trace. Moreover, we can use |A| to represent any norm of the matrix A due to the equivalence between different norms, while |A| F denotes the Frobenius norm of the matrix A.
A time-varying switched non-linear system which contains N subsystems can be formed as where t, t 0 ∈ ℤ ≥0 , x(t ) ∈ ℝ n , (t ) : ℤ ≥0 → Γ = {1, 2, … , N } is a piecewise constant map defined as the switching signal, and N with N ≥ 1 represents the number of subsystems of the switched system (1). (t ) = i means that the ith subsystem is active at time t . Indeed, Equation (1) models a system that can switch between N subsystems of the form where f i : ℤ ≥0 × ℝ n → ℝ n is continuous and satisfies f i (t, 0) = 0 for each i ∈ Γ and all t ∈ ℤ ≥0 . Moreover, a time-varying switched linear system can be described by we assume that A i (t ) is non-singular for all t ≥ t 0 , all t 0 ∈ ℤ ≥0 and all i ∈ Γ. Note that any entry in matrix A i (t ) changes with t and can be a non-linear function with respect to t . The aim of this paper is to investigate the stabilisability of discrete-time switched systems. To this purpose, we first need the following definitions selected from [1,39].

Definition 1.
A function x(t ) : ℤ ≥0 → ℝ n is called the trajectory (or solution) of system (1) if there is a switching signal (t ) such that Under a given switching signal (t ), we usually use x (t ) (t, t 0 , x 0 ) to represent the trajectory of the system (1) starting from the initial state x 0 at initial instant t 0 ∈ ℤ ≥0 . For simplicity, we can use x (t, t 0 , x 0 ) to denote the trajectory x (t ) (t, t 0 , x 0 ). Without confusion, x (t, t 0 , x 0 ) can be expressed as x(t, t 0 , x 0 ) or even x(t ).

Definition 2. System (1) is called
(i) stabilisable with respect to the origin if there is a switching signal satisfying that for any t 0 ∈ ℤ ≥0 and any > 0, there exists a constant ( , t 0 , ) > 0 such that ‖x (t, t 0 , x 0 )‖ < for all t ≥ t 0 when ‖x 0 ‖ < ( , t 0 , ); (ii) globally asymptotically stabilisable with respect to the origin if there is a switching signal such that system (1) is both stable and lim and all x 0 ∈ ℝ n ; (iii) globally exponentially stabilisable with respect to the origin if for any t 0 ∈ ℤ ≥0 there is a switching signal , constants (t 0 ) ≥ 1 and > 0 such that ‖x (t, t 0 , x 0 )‖ ≤ (t 0 )e − (t −t 0 ) ‖x 0 ‖ for all t ≥ t 0 and all x 0 ∈ ℝ n ; (iv) globally uniformly exponentially stabilisable with respect to the origin if (t 0 ) in Item (iii) is independent of t 0 .  → Γ such that system (1) is finite-time stable with respect to [t 0 , F, Λ, Υ(t ), ], that is, Moreover, for system (2) under a given switching signal (t ), we know that x (t, t 0 , . Thus, we can alternatively use the state transition matrix Φ (t, t 0 ) of system (2) to obtain its asymptotic stabilisability, exponential stabilisability, uniform exponential stabilisability and finite-time stabilisability as follows according to the works in [39,44,45].

globally asymptotically stabilisable with respect to the origin if and only
if there is a switching signal such that system (2) becomes stable and lim t →+∞ (iii) globally exponentially stabilisable with respect to the origin if and only if there is a switching signal such that for any t 0 ∈ ℤ ≥0 , there exist a scalar (t 0 ) ≥ 1 and a constant > 0 such that Afterwards, we introduce a scalar time-varying linear system of form For system (3), we firstly introduce the following definition [24]. (iii) a globally uniformly exponentially stable function if system (3) is globally uniformly exponentially stable (GUES).
Then, inspired by the above stable functions, we additionally introduce the concept of finite-time stable functions as follows. (3) with the pre-given initial time t 0 and a positive constant F , the scalar function (t ) : ℤ ≥0 → ℝ >0 is called a finite-time stable function if for arbitrary two positive constants Λ 0 and Υ 0 with

Definition 5. For system
Remark 1. Note that in Definition 5, when positive constants Λ 0 and Υ 0 are pre-given, the scalar function (t ) : ℤ ≥0 → ℝ >0 can be called a finite-time stable function with respect to Based on Definition 4, we can easily obtain the following lemma, which has been studied in [24] and will be used in Sections 3 and 5.

ii) a globally exponentially stable function if and only if for any t
(iii) a globally uniformly exponentially stable function if and only if there exist a ≥ 1 and a constant > 0 such that Further, based on our additionally introduced Definition 5, we can derive the following lemma, which will also be used in Section 3.

STABILIZABILITY ANALYSIS OF TIME-VARYING SWITCHED LINEAR SYSTEMS
Inspired by the idea of using scalar functions in [24] and [41] and the idea of using time-varying Lyapunov functions in [14] and [27], we in this section are dedicated to utilise an asymptotically stable function, an exponentially stable function, an uniformly exponentially stable function and a finite-time stable function to establish necessary and sufficient conditions for the global asymptotic stabilisability, global exponential stabilisability, global uniform exponential stabilisability and finite-time stabilisability of time-varying discrete-time SLSs, respectively. Note that Zhou and Zhao [24] considered the stability of the timevarying linear dynamical systems, which can be regarded as a special case of time-varying switched linear systems with a single subsystem. Firstly, for system (2), we obtain the following necessary and sufficient condition of its global asymptotic stabilisability. (2) is globally asymptotically stabilisable if and only if there exist a GAS function (t ) : ℤ ≥0 → ℝ >0 and a non-singular matrix P (t ) : ℤ ≥0 → ℝ n×n with P T (t ) = P (t ) such that (i) for all t ∈ ℤ ≥0 , I n ≤ P (t ); and (ii) for all t ∈ ℤ ≥0 , an i t ∈ Γ can be found to satisfy that A T i t (t )P (t + 1)A i t (t ) ≤ 2 (t )P (t ). (2) is globally exponentially stabilisable if and only if there exist a GES function (t ) : ℤ ≥0 → ℝ >0 and a non-singular matrix P (t ) : ℤ ≥0 → ℝ n×n with P T (t ) = P (t ) such that conditions (i) and (ii) are satisfied.

Moreover, system
Proof. (1) Sufficiency: For proving the global asymptotic (exponential) stabilisability of system (2), we aim to construct a switching signal such that system (2) is GA(E)S under it. For this, define the switching signal (t ) as (t ) = i t for all t ∈ ℤ ≥0 . Then, define timevarying Lyapunov function V (t, x(t )) = x T (t )P (t )x(t ), where x(t ) denotes the trajectory x (t ) (t, t 0 , x 0 ) of system (2) under the above well-defined switching signal.
Clearly, due to condition (i), for all t ≥ t 0 , all t 0 ∈ ℤ ≥0 and any initial state x 0 . Further, for any initial state x 0 , all t ≥ t 0 and all t 0 ∈ ℤ ≥0 , we can derive from condition (ii) that where |P (t 0 )| represents the spectral norm of matrix P (t 0 ). Therefore, it follows from inequalities (4) and (5) that Due to the arbitrariness of x 0 and the definition of the norm of a operator defined by a matrix, we have Since (t ) is a GAS function, from inequality (6) and Lemma 3, we can conclude that the system (2) is GAS under the above constructed switching signal , that is, the sufficiency of the global asymptotic stabilisability is proved due to Lemma 1.
(2) Necessity: Since system (2) is globally asymptotically stabilisable, there exists a switching signal such that system (2) is GAS under it. Thus, for all t ∈ ℤ ≥0 , there exists an index i t ∈ Γ such that (t ) = i t . Then, define (t ) as where |Φ (t, t 0 )| F denotes the Frobenius norm of the state transform matrix Φ (t, t 0 ) of system (2) under the above welldefined switching signal. Moreover, (t ) is a well-defined scalar function for all t ≥ t 0 , all t 0 ∈ ℤ ≥0 since A i (t ) is non-singular for all t ∈ ℤ ≥0 and all i ∈ Γ. Then, for all t ≥ t 0 and all t 0 ∈ ℤ ≥0 , let Clearly, lim t →∞ (t, t 0 ) = 0. Thus, (t ) is a GAS function due to Lemma 3. Define P (t ) as where . Then, P T (t ) = P (t ) and P (t ) can be rewritten as Further, it follows from the positive definition of Therefore, Moreover, we can derive from Equation (8) that Thus, condition (i) is satisfied.
Next, due to the above well-defined switching signal and we have that, for all t ≥ t 0 and all t 0 ∈ ℤ ≥0 , an index i t ∈ Γ can be found such that which indicates that condition (ii) is satisfied. Thus, the necessity of the global asymptotic stabilisability is proved.
We in the following prove the necessity of the global exponential stabilisability. Similar with the above proof process, for all t ∈ ℤ ≥0 , an index i t ∈ Γ can be found such that (t ) = i t . Then, we define (t ) = Moreover, according to Lemma 1, there exist a scalar (t 0 ) ≥ 1 and a positive constant such that for all t ≥ t 0 and all t 0 ∈ ℤ ≥0 , (t, t 0 ) ≤ n − 1 2 (t 0 )e − (t −t 0 ) . Obviously, due to Lemma 3, (t ) is a GES function. From the rest proof of the necessity of the global asymptotic stabilisability, the necessity of the global exponential stabilisability can be obtained. □ Note that it is difficult to directly solve the necessary and sufficient condition of Theorem 1. We can combine Theorem 1 and the average dwell time method to construct computable sufficient conditions for the asymptotic and exponential stability analysis, respectively. Then, based on the S-procedure, semidefinite program and sum of squares decomposition, which is well used in [14,46], the computation of P (t ) can be similarly solved if replacing t ∈ ℤ ≥0 with t ∈ ℝ ≥0 .
Next, combining the above result with a special form of P (t ), a necessary and sufficient condition for the global uniform exponential stabilisability of system (2) can be proposed via a GUES function as follows.

Theorem 2. System (2) is globally uniformly exponentially stabilisable if and only if there exist a GUES function (t ) :
ℤ ≥0 → ℝ >0 , a positive constant c ≥ 1 and a non-singular matrix P (t ) : ℤ ≥0 → ℝ n×n with P T (t ) = P (t ) such that (i) for all t ∈ ℤ ≥0 , I n ≤ P (t ) ≤ cI n ; and (ii) for all t ∈ ℤ ≥0 , an i t ∈ Γ can be found to satisfy that
(2) Necessity: We can derive from the global uniform exponential stabilisability of system (2) that there exists a switching signal such that system (2) is GUES under it. Thus, for all t ∈ ℤ ≥0 , there exists an index i t ∈ Γ such that (t ) = i t . Then, according to Lemma 1, there exist positive constants ≥ 1 and such that for all t ≥ t 0 and all t 0 ∈ ℤ ≥0 , Next, for all t ∈ ℤ ≥0 , define P (t ) as where 0 < < 2 . Clearly, P T (t ) = P (t ) for all t ∈ ℤ ≥0 . Then, owing to inequalities (11) and (12), we have for all t ∈ ℤ ≥0 . Moreover, we can derive from Thus, I n ≤ P (t ) ≤ cI n , where c = 2 1−e −2 ≥ 1, that is, condition (i) is satisfied. Next, due to the above well-defined switching signal , we obtain that for all t ∈ ℤ ≥0 , an index i t can be found such that Clearly, (t ) = e − 2 is a GUES function due to Lemma 3. Thus, the necessity is proved. □ In Theorem 2, we have proposed a necessary and sufficient condition for the global uniform exponential stabilisability of system (2). Clearly, we can similarly utilise the average dwell time method to construct a computable sufficient condition for the uniform exponential stability of system (2). Then, the computation issue can be solved based on the S-procedure, semi-definite program and sum of squares decomposition [14,46]. Besides, [47,48] have proposed sufficient conditions for the uniform exponential stability of periodic time-varying systems based on a time-varying periodic Lyapunov matrix P (t ). Motivated by this, we can combine Theorem 2, the periodic Lyapunov matrix P (t ) and dwell time method to construct another tractable sufficient condition formed by time-invariant matrix inequalities. Then, this tractable sufficient condition can be solved by LMI Matlab toolbox.
Moreover, Theorems 1 and 2 deal with the behaviour of a system within a sufficiently long time interval, while, in practical systems, the behaviour of a system on a finite time interval is also worthy to be concentrated on. Therefore, we in the following are dedicated to proposing a necessary and sufficient condition for the finite-time stabilisability of system (2).

Proof. (1) Sufficiency:
We aim to prove that system (2) is finite-time stabilisable via constructing a switching signal such that system (2) is finitetime stable under it. For this, define a finite-time switching signal (t ) as (t ) = i t for all t ∈  and define time-varying Lyapunov function V (t, x(t )) = x T (t )P (t )x(t ). Then, according to the well-defined switching signal, condition (ii) and P (t 0 ) < Λ, we have for all t ∈  ⧵ {t 0 + F }. Thus, we can further derive from inequality (14), condition (i) and Lemma 4 that, when x T (t 0 )Λx(t 0 ) < 1 and t ∈  ⧵ {t 0 }, Clearly, system (2) is finite-time stable under the above constructed finite-time switching signal, which indicates that the sufficiency is proved.
(2) Necessity: Since system (2) is finite-time stabilisable with respect to [t 0 , F, Λ, Υ(t )], there exists a finite-time switching signal (t ) defined over  such that system (2) is finite-time stable with respect to [t 0 , F, Λ, Υ(t ), (t )]. Thus, for all t ∈  , there exists an index i t ∈ Γ such that (t ) = i t . Moreover, we can derive from the continuity arguments that there exists a small enough positive constant such that the following system is also finitetime stable with respect to [t 0 , F, Λ, Υ(t ), (t )], Note that, for any given positive definite matrix-valued sequence Υ(t ) defined in  , we can find a positive definite nonsingular matrix-valued sequenceΥ(t ) defined in  such that Similarly, there exists a positive definite non-singular matrixΛ such that Λ =Λ TΛ . Thus, it follows from Lemma 2 that for all t ∈  ⧵ {t 0 }, where Φ z (t, t 0 ) represents the state transition matrix of system (15) under the switching signal . Moreover, we can derive from inequality (16) that for all t ∈  ⧵ {t 0 }, which implies that for all t ∈  ⧵ {t 0 }. Then, due to inequality (17), we can obtain for all t ∈  ⧵ {t 0 }, we can directly attain that P (t 0 ) = cΛ < Λ and Υ(t ) < P (t ) for all t ∈  ⧵ {t 0 }, that is, condition (i) is obtained.
Further, due to the above well-defined finite-time switching signal and Φ z (t + 1, t 0 ) = (1 + )A (t ) (t )Φ z (t, t 0 ), we obtain that for all t ∈  ⧵ {t 0 + F }, an index i t ∈ Γ can be found such that Thus, we can further attain that A T i t (t )P (t + 1)A i t (t ) < P (t ) for all t ∈  ⧵ {t 0 + F }. Moreover, due to the property of positive definite matrix, there definitely exists a constant 0 with 0 < all t ∈  ⧵ {t 0 }, which indicates that (t ) is a finite-time stable function and condition (ii) is satisfied. Thus, the necessity is proved. □ In Theorem 3, resorting to the additionally proposed finitetime stable function, we propose a necessary and sufficient condition for the finite-time stabilisability of discrete-time timevarying SLSs, which releases the requirement of the nonincreasing of Lyapunov functions. Note that [38] obtained necessary and sufficient conditions for single continuoustime time-varying linear systems based on a Lyapunov function non-increasing on a series of subregions. Besides, from Lemma 4, the constraint of our proposed finite-time stable function (t ) is independent with initial state, Λ and P (t 0 ) in Theorem 3, while the necessary and sufficient condition of the finite-time stability for continuous-time impulsive time-varying SLSs proposed in [41] requires the corresponding constraint of (t ) (t ) that depends on initial state, Λ and P (t 0 ). Moreover, we can combine Theorem 3, average dwell time method and periodic time-varying matrix P (t ) to obtain tractable sufficient conditions in our future work inspired by [14,[46][47][48].

ILLUSTRATIVE EXAMPLES
In this section, three illustrative examples will be presented to manifest the applicability and validity of our theoretical results obtained above.

DISCUSSIONS ON SWITCHED NON-LINEAR SYSTEMS
Necessary and sufficient conditions for the stabilisability and finite-time stabilisability of time-varying switched linear systems have been adequately discussed. Naturally, the question ] T arises whether it is possible to derive similar necessary and sufficient conditions for the time-varying switched non-linear systems. Since there has been no common approach to construct proper Lyapunov functions for the necessary condition of the asymptotic stabilisability and finite-time stabilisability, we here only consider necessary and sufficient conditions for the global exponential stabilisability and global uniform exponential stabilisability of switched non-linear system (1). (ii) for all t ∈ ℤ ≥0 and all x ∈ ℝ n , an i t ∈ Γ can be found to satisfy that V (t + 1, f i t (t, x)) ≤ 2p (t )V (t, x). (1)  Proof. (1) Sufficiency: For proving the global exponential stabilisability of system (1), we aim to construct a switching signal such that system (1) is GES under it. For this, define the switching signal (t ) as (t ) = i t for all t ∈ ℤ ≥0 . Then, for any initial state x 0 , all t ≥ t 0 and all t 0 ∈ ℤ ≥0 , we can derive from condition (ii) that

Moreover, system
Further, it follows from inequality (21) and condition (i) that Then, since (t ) is a GES function, we can derive from inequality (22) and Lemma 3 that system (1) is GES under the above constructed switching signal , that is, the sufficiency of the global exponential stabilisability of system (1) is proved due to Definition 2. Obviously, if (t ) is a GUES function, we can derive from inequality (22) and Lemma 3 that system (1) is GUES under the above constructed switching signal , that is, the sufficiency of the global uniform exponential stabilisability of system (1) is proved.
Remark 2. Note that the function formed by (23) cannot qualify as a Lyapunov function for proving the necessity of the global asymptotic stabilisability since the convergence of series ∞ ∑ k=t e (k−t ) (x T (k, t, x)x(k, t, x)) p cannot be guaranteed.
Likewise, there is still no proper Lyapunov function that can be chosen to derive the necessity of the finite-time stabilisability for general time-varying single non-linear systems not even to say switched non-linear systems [38]. Thus, more relaxed necessary and sufficient conditions for the asymptotic stabilisability and finite-time stabilisability of system (1) are still required to explore.

CONCLUSION
This paper has provided several necessary and sufficient conditions for the stabilisability of time-varying discrete-time switched systems. Specifically, necessary and sufficient conditions for the asymptotic (exponential, uniform exponential) stabilisability of time-varying SLSs are proposed by resorting to asymptotically (exponentially, uniformly exponentially) stable functions. Moreover, a necessary and sufficient condition for finite-time stabilisability of time-varying discrete-time SLSs is further obtained via an additionally introduced finite-time stable function. Specially, we further attain the necessary and sufficient conditions for global (uniform) exponential stabilisability of time-varying switched non-linear systems. Based on these stable functions, we had released the requirement on negative definiteness of the time-difference of Lyapunov functions, compared to traditional difference Lyapunov inequalities. Our possible future work is to investigate the computation mechanism of the required P (t ) (or V (t, x)) and stable function (t ) based on [14,[46][47][48]. Besides, motivated by the work in [11, 19 21, 46 49, 50], we would like to further research on constructing switching signals to stabilising a time-varying switched system via utilising the idea of multiple Lyapunov functions and average dwell time method. Moreover, based on [51][52][53][54], it merits further investigation on the computation of basin of attraction via stable functions for locally asymptotically stable time-varying switched systems.