# Finite-time stability of linear non-autonomous systems with time-varying delays

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## Abstract

In this paper, we investigate the problem of finite-time stability (*FTS*) of linear non-autonomous systems with time-varying delays. By constructing an appropriated function, we derive some explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Finally, two examples are given to show the effectiveness of the main results.

## Keywords

Finite-time stability Linear non-autonomous systems Time-varying delay Metzler matrix## 1 Introduction

During the past decades, finite time stability (*FTS*) in linear systems has received considerable attention since it was first introduced in 1960s. *FTS* is a system property which concerns the quantitative behavior of the state variables over an assigned finite-time interval. A system is *FTS* if, given a bound on the initial condition, its state trajectories do not exceed a certain threshold during a pre-specified time interval. Hence, *FTS* enables us to specify quantitative bounds on the state of a linear system and plays an important role in addressing transient performances of the systems. Therefore, in recent years, many interesting results for *FTS* have been proposed, see [1, 2, 3, 4, 5] for instances. It should be noticed that *FTS* and Lyapunov asymptotic stability (LAS) are completely independent concepts. Indeed, a system can be *FTS* but not LAS, and vice versa [6, 7, 8]. Asymptotic stability in dynamical systems implies convergence of the system trajectories to an equilibrium state over the infinite horizon. However, in practice, it is desirable that a dynamical system possesses *FTS*, that is, its state norm does not exceed a certain threshold in finite time. Furthermore, LAS is concerned with the qualitative behavior of a system and it does not involve quantitative information (e.g., specific estimates of trajectory bounds), whereas *FTS* involves specific quantitative information.

In the process of investigating linear systems, time delays are frequently encountered [9, 10, 11, 12]. And in hardware implementation, time delays usually cause oscillation, instability, divergence, chaos, or other bad performances of neural networks. In recent years, various interesting results have been obtained for the *FTS* of linear autonomous systems. For linear time-invariant systems with constant delay, some finite-time stability conditions have been derived in terms of feasible linear matrix inequalities based on the Lyapunov–Krasovskii functional methods [6, 13, 14, 15, 16, 17]. It is worth noting that non-autonomous phenomena often occur in many realistic systems; for instance, when considering a long-term dynamical behavior of the system, the parameters of the system usually change along with time [18, 19, 20, 21, 22]. Moreover, stability analysis for non-autonomous systems usually requires specific and quite different tools from the autonomous ones (systems with constant coefficients). To our knowledge, there are a few results concerned with the *FTS* of non-autonomous systems with time-varying delays. In addition, it should be noted that the conditions for *FTS* of the time-varying system are usually based on the Lyapunov or Riccati matrix differential equation [7, 23, 24], which leads to indefinite matrix inequalities and lacks effective computational tools to solve them. Therefore, when dealing with the *FTS* of time-varying systems with delays, an alternative approach is clearly needed, which motivates our present investigation.

In present paper, the problems of *FTS* are investigated for linear non-autonomous systems with discrete and distributed time-varying delays. By constructing an appropriated function, some sufficient conditions are derived to guarantee the *FTS* of the addressed linear non-autonomous systems. We do not impose any restriction on the states of the system in this sense, which is better than the results in [25]. The rest of this paper is organized as follows. In Sect. 2, some notations, definitions, and a lemma are given. In Sect. 3, we present the main results. Two examples are provided in Sect. 4 to demonstrate the effectiveness of the proposed criteria. Section 5 shows the summary of this paper.

## 2 Preliminaries

## Notations

Let \(\mathbb{R}\) denote the set of real numbers, \(\mathbb{R}_{+}\) the set of positive numbers, \(\mathbb{R}^{n}\) the *n*-dimensional real spaces equipped with the norm \(\|x\|_{\infty}=\max_{i\in\underline{n}}|x_{i}|\) and \(\mathbb{R}^{n\times m}\) the \(n\times m\)-dimensional real spaces. *I* denotes the identity matrix with appropriate dimensions and \(\Lambda=\{1,2,\ldots,n\}\). For any interval \(J\subseteq\mathbb {R}\), set \(S\subseteq\mathbb{R}^{k}\) (\(1\leq k\leq n\)), \(C(J, S) =\{\varphi:J\rightarrow S\mbox{ is continuous}\}\). \(\mathscr{F}=\{\mu:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}\mbox{ is continuously differentiable}\}\) and \(A\vee B=\max\{A, B\}\) for constants *A* and *B*. \(u=(u_{i})\), \(v=(v_{i})\) in \(\mathbb{R}^{n}\), \(u\geq v\) iff \(u_{i}\geq v_{i}\), \(\forall i\in\Lambda\); \(u\gg v\) iff \(u_{i}>v_{i}\), \(i\in\Lambda\).

## Definition 1

(Amato et al. [7])

*T*with \(r_{1}< r_{2}\), linear non-autonomous system (1) is said to be

*FTS*with respect to \((T, r_{1}, r_{2})\) if

## Definition 2

(Liao et al. [26])

- (A
_{1}) -
\(a_{ii}(t)\leq\bar{a}_{ii}\), \(i\in\Lambda\), \(|a_{ij}(t)|\leq \bar{a}_{ij}\), \(i\neq j\), \(i,j\in\Lambda\), \(t\in[0,T]\).

- (A
_{2}) -
\(|d_{ij}(t)|\leq\bar{d}_{ij}\), \(|g_{ij}(t)|\leq\bar{g}_{ij}\), \(i,j\in\Lambda\), \(t\in[0,T]\).

*A*are positive. The following lemma is used in our main results.

## Lemma 1

(Hien et al. [29])

*Let*\(A=(a_{ij})\)

*be an off*-

*diagonal non*-

*positive matrix*, \(a_{ii}>0\), \(i\in\Lambda\).

*Then the following statements are equivalent*:

- (i)
*A**is a nonsingular**M*-*matrix*. - (ii)
\(\operatorname{Re}\lambda_{k}(A)>0\)

*for all eigenvalues*\(\lambda_{k}(A)\)*of**A*. - (iii)
*There exist a matrix*\(B\geq0\)*and a scalar*\(s>\rho(B)\)*such that*\(A=sI_{n}-B\),*where*\(\rho(B)=\max\{|\lambda_{k}(A)|\}\)*denotes the spectral radius of**B*. - (iv)
*There exist a vector*\(\xi\in\mathbb{R}^{n}\)*and*\(\xi\gg0\)*such that*\(A\xi\gg0\). - (v)
*There exist a vector*\(\eta\in\mathbb{R}^{n}\)*and*\(\eta\gg0\)*such that*\(A^{T}\eta\gg0\).

## 3 Main results

We are now in a position to state our main result as follows. In this section, we shall investigate the *FTS* of the linear non-autonomous system by constructing an appropriated function and using the Metzler matrix method.

## Theorem 1

*Under assumptions*(A

_{1})

*and*(A

_{2}),

*linear non*-

*autonomous system*(1)

*is*FTS

*with respect to*\((T, r_{1}, r_{2})\),

*if there exist three positive scalars*\(r_{1}\), \(r_{2} \),

*and*

*T*

*with*\(r_{1}< r_{2}\),

*a function*\(\mu(t)\in\mathscr{F}\),

*and three constants*\(\beta_{i}\), \(i=1,2,3\),

*satisfying*

*Moreover*,

*there exists a vector*\(\xi\in\mathbb{R}^{n}\)

*such that*

*where*\(\mathcal{M}^{0}=\mathcal{A}+\beta_{1}\mathcal{D}+\beta _{2}\mathcal{G}-\beta_{3}I\), \(m=\min_{i\in\Lambda}\xi_{i}\), \(M=\max_{i\in\Lambda}\xi_{i}\).

## Proof

*FTS*with respect to \((T, r_{1}, r_{2})\). The proof is complete. □

## Corollary 1

*Under assumptions*(A

_{1})

*and*(A

_{2}),

*linear non*-

*autonomous system*(1)

*is*FTS

*with respect to*\((T, r_{1}, r_{2})\)

*if there exist three positive scalars*\(r_{1}\), \(r_{2} \),

*and*

*T*

*with*\(r_{1}< r_{2}\),

*a function*\(\mu(t)\)

*that is monotonous increasing and*\(\mu(t)\geq1\), \(t\in[0, T]\),

*and three constants*\(\beta_{i}\), \(i=1,2,3\),

*satisfying*

*Moreover*,

*there exists a vector*\(\xi\in\mathbb{R}^{n}\)

*such that*

*where*\(\mathcal{M}^{0}=\mathcal{A}+\beta_{1}\mathcal{D}+\bar{\kappa}\mathcal{G}-\beta_{3}I\), \(m=\min_{i\in\Lambda}\xi_{i}\), \(M=\max_{i\in\Lambda}\xi_{i}\).

## Corollary 2

*Under assumptions*(A

_{1})

*and*(A

_{2}),

*linear non*-

*autonomous system*(1)

*is*FTS

*with respect to*\((T, r_{1}, r_{2})\),

*if there exist three positive scalars*\(r_{1}\), \(r_{2} \),

*and*

*T*

*with*\(r_{1}< r_{2}\),

*a function*\(\mu(t)\equiv\mu>0\),

*and three scalars*\(\beta_{i}\), \(i=1,2,3\),

*satisfying*

*Moreover*,

*there exists a vector*\(\xi\in\mathbb{R}^{n}\)

*such that*

*where*\(\mathcal{M}^{0}=\mathcal{A}+\beta_{1}\mathcal{D}+\beta _{2}\mathcal{G}-\beta_{3}I\), \(m=\min_{i\in\Lambda}\xi_{i}\), \(M=\max_{i\in\Lambda}\xi_{i}\).

## Remark 1

In [25], Hien considered the *FTS* of system (1) and derived some conditions for exponential estimation. In this paper, we study the *FTS* of system (1) via the auxiliary function *μ*, and some new sufficient conditions for *FTS*, which are different from the results in [25], are derived. In other words, our development result is more general than the result in [25].

## 4 Example

In this section, we present two numerical examples to illustrate the effectiveness of the proposed results.

## Example 1

_{1}) and (A

_{2}) hold, and then we have

*Case* I. Let us take \(r_{1}=1\), \(r_{2}=1.25\), and then system (8) is *FTS* with respect to \((T, r_{1}, r_{2})\) for any finite time \(0< T\leq T_{\mathrm{max}}=25\), and in this case \(\beta_{3}=0.08\). Note that in [25], the maximum value of *T* is \(T_{\mathrm{max}}=21.3144\). Hence, our result is more general than [25].

*Case* II. Let us take \(r_{1}=1\), when \(T=21.3144\), we obtain \(\beta _{3}=0.8243\) and \(r_{2}=1.213144\).

## Example 2

_{1}) and (A

_{2}) hold, and we have

*FTS*with respect to \((T, r_{1}, r_{2})\) for any finite time \(0< T\leq T_{\mathrm{max}}=15\), and in this case \(\beta_{3}=0.04\), see Fig. 2.

## 5 Conclusion

In the present paper, we have investigated the *FTS* of a class of non-autonomous systems with time-varying delays. Some new sufficient conditions for *FTS* have been derived in terms of inequalities for a type of Metzler matrixes. Finally, two examples were provided to show the effectiveness of the proposed method.

## Notes

### Acknowledgements

This work was supported by the National Natural Science Foundation of China (11301308, 61673247), and the Research Fund for Distinguished Young Scholars and Excellent Young Scholars of Shandong Province (JQ201719, ZR2016JL024). The paper has not been presented at any conference.

### Authors’ contributions

The main idea of this paper was proposed by YX and LX. YX prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

## Competing interests

The authors declare that none of the authors have any competing interests in the manuscript.

## References

- 1.Stojanovic, S.: Further improvement in delay-dependent finite-time stability criteria for uncertain continuous-time systems with time-varying delays. IET Control Theory Appl.
**10**, 926–938 (2016) MathSciNetCrossRefGoogle Scholar - 2.Amato, F., Carannante, G., Tommasi, G., Pironti, A.: Input–output finite-time stability of linear systems: necessary and sufficient conditions. IEEE Trans. Autom. Control
**57**, 3051–3063 (2012) MathSciNetCrossRefMATHGoogle Scholar - 3.Pang, D., Jiang, W.: Finite-time stability analysis of fractional singular time-delay systems. Adv. Differ. Equ.
**2014**(1), 259 (2014) MathSciNetCrossRefMATHGoogle Scholar - 4.Yang, W., Sun, J.: Finite-time stability of quantum systems with impulses. IET Control Theory Appl.
**8**, 641–646 (2014) MathSciNetCrossRefGoogle Scholar - 5.Kang, W., Zhong, S., Shi, K., Cheng, J.: Finite-time stability for discrete-time system with time-varying delay and nonlinear perturbations. ISA Trans.
**60**, 67–73 (2016) CrossRefGoogle Scholar - 6.Amato, F., Ariola, M., Cosentino, C.: Finite-time stabilization via dynamic output feedback. Automatica
**42**, 337–342 (2006) MathSciNetCrossRefMATHGoogle Scholar - 7.Amato, F., Ambrosino, R., Ariola, M., Cosentino, C.: Finite-time stability of linear time-varying systems with jumps. Automatica
**45**, 1354–1358 (2009) MathSciNetCrossRefMATHGoogle Scholar - 8.Lin, X., Du, H., Li, S., Zou, Y.: Finite-time boundedness and finite-time \(l_{2}\) gain analysis of discrete-time switched linear systems with average dwell time. J. Franklin Inst.
**350**, 911–928 (2013) MathSciNetCrossRefMATHGoogle Scholar - 9.Li, X., Wu, J.: Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica
**64**, 63–69 (2016) MathSciNetCrossRefMATHGoogle Scholar - 10.Rakkiyappan, R., Udhayakumar, K., Velmurugan, G., et al.: Stability and Hopf bifurcation analysis of fractional-order complex-valued neural networks with time delays. Adv. Differ. Equ.
**2017**(1), 225 (2017) MathSciNetCrossRefGoogle Scholar - 11.Li, X., Song, S.: Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Autom. Control
**62**, 406–411 (2017) MathSciNetCrossRefMATHGoogle Scholar - 12.Li, X., Cao, J.: An impulsive delay inequality involving unbounded time-varying delay and applications. IEEE Trans. Autom. Control
**62**, 3618–3625 (2017) MathSciNetCrossRefMATHGoogle Scholar - 13.He, S., Liu, F.: Observer-based finite-time control of time-delayed jump systems. Appl. Math. Comput.
**217**, 2327–2338 (2010) MathSciNetMATHGoogle Scholar - 14.Xiang, W., Xiao, J., Iqbal, M.: Robust finite-time bounded observer design for a class of uncertain non-linear Markovian jump systems. IMA J. Math. Control Inf.
**29**, 551–572 (2012) MathSciNetCrossRefMATHGoogle Scholar - 15.Zhang, Y., Liu, C., Mu, X.: Robust finite-time stabilization of uncertain singular Markovian jump systems. Appl. Math. Model.
**36**, 5109–5121 (2012) MathSciNetCrossRefMATHGoogle Scholar - 16.Lin, X., Du, H., Li, S., Zou, Y.: Finite-time boundedness and finite-time \(l_{2}\) gain analysis of discrete-time switched linear systems with average dwell time. J. Franklin Inst.
**350**, 911–928 (2013) MathSciNetCrossRefMATHGoogle Scholar - 17.Hou, L., Zong, G., Wu, Y.: Observer-based finite-time exponential \(l_{2}\)-\(l_{\infty}\) control for discrete-time switched delay systems with uncertainties. Trans. Inst. Meas. Control
**35**, 310–320 (2013) CrossRefGoogle Scholar - 18.Stamova, I., Stamov, T., Li, X.: Global exponential stability of a class of impulsive cellular neural networks with supremums. Int. J. Adapt. Control Signal Process.
**28**(11), 1227–1239 (2014) MathSciNetCrossRefMATHGoogle Scholar - 19.Li, X., Song, S.: Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans. Neural Netw. Learn. Syst.
**24**, 868–877 (2013) CrossRefGoogle Scholar - 20.Wang, X., Jiang, M., Fang, S.: Stability analysis in Lagrange sense for a non-autonomous Cohen–Grossberg neural network with mixed delays. Nonlinear Anal., Theory Methods Appl.
**70**, 4294–4306 (2009) MathSciNetCrossRefMATHGoogle Scholar - 21.Li, X., Bohner, M., Wang, C.: Impulsive differential equations: periodic solutions and applications. Automatica
**52**, 173–178 (2015) MathSciNetCrossRefMATHGoogle Scholar - 22.Thuan, M., Hien, L., Phat, V.: New results on exponential stabilization of time-varying delay neural networks via Riccati equations. Appl. Math. Comput.
**246**, 533–545 (2014) MathSciNetMATHGoogle Scholar - 23.Yang, X., Li, X.: Robust finite-time stability of singular nonlinear systems with interval time-varying delay. J. Franklin Inst.
**355**, 1241–1258 (2018) MathSciNetCrossRefGoogle Scholar - 24.Amato, F., Ariola, M., Cosentino, C.: Finite-time control of discrete-time linear systems: analysis and design conditions. Automatica
**46**, 919–924 (2010) MathSciNetCrossRefMATHGoogle Scholar - 25.Hien, L.: An explicit criterion for finite-time stability of linear non-autonomous systems with delays. Appl. Math. Lett.
**30**, 12–18 (2014) MathSciNetCrossRefMATHGoogle Scholar - 26.Liao, X., Wang, L., Yu, P.: Stability of Dynamical Systems. Elsevier, Amsterdam (2007) CrossRefMATHGoogle Scholar
- 27.Ngoc, P.: On exponential stability of nonlinear differential systems with time-varying delay. Appl. Math. Lett.
**25**, 1208–1213 (2012) MathSciNetCrossRefMATHGoogle Scholar - 28.Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1987) MATHGoogle Scholar
- 29.Hien, L., Son, D.: Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl. Math. Comput.
**251**, 14–23 (2015) MathSciNetMATHGoogle Scholar

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