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Finitetime stability of linear nonautonomous systems with timevarying delays
Advances in Difference Equations volume 2018, Article number: 101 (2018)
Abstract
In this paper, we investigate the problem of finitetime stability (FTS) of linear nonautonomous systems with timevarying 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 prespecified finite time interval. Finally, two examples are given to show the effectiveness of the main results.
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 finitetime interval. A system is FTS if, given a bound on the initial condition, its state trajectories do not exceed a certain threshold during a prespecified 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 [1–5] 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 [6–8]. 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 [9–12]. 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 timeinvariant systems with constant delay, some finitetime stability conditions have been derived in terms of feasible linear matrix inequalities based on the Lyapunov–Krasovskii functional methods [6, 13–17]. It is worth noting that nonautonomous phenomena often occur in many realistic systems; for instance, when considering a longterm dynamical behavior of the system, the parameters of the system usually change along with time [18–22]. Moreover, stability analysis for nonautonomous 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 nonautonomous systems with timevarying delays. In addition, it should be noted that the conditions for FTS of the timevarying 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 timevarying 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 nonautonomous systems with discrete and distributed timevarying delays. By constructing an appropriated function, some sufficient conditions are derived to guarantee the FTS of the addressed linear nonautonomous 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 ndimensional 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 \(u_{i}>v_{i}\), \(i\in\Lambda\).
Consider the following linear nonautonomous system with timevarying delays:
where \(x(t)\in\mathbb{R}^{n}\) is the state; \(A(t)=(a_{ij}(t))\in \mathbb{R}^{n\times n}\), \(D(t)=(d_{ij}(t))\in\mathbb{R}^{n\times n}\), and \(G(t)=(g_{ij}(t))\in\mathbb{R}^{n\times n}\) are the system matrices; \(\tau(t)\) and \(\kappa(t)\) are timevarying delays satisfying \(0\leq\underline{\tau}\leq\tau(t)\leq\bar{\tau}\), \(0\leq\kappa(t)\leq\bar{\kappa}\), \(t\geq0\); \(\phi(t)=(\phi _{i}(t))\in C([d, 0], \mathbb{R}^{n})\) is the initial condition, where \(d=\bar{\tau}\vee\bar{\kappa}\). Denote \(\phi_{i}=\sup_{d\leq t\leq0}\phi_{i}(t)\) and \(\\phi\_{\infty}=\max_{i\in\Lambda}\phi_{i}\).
Definition 1
(Amato et al. [7])
Assume that \(x(t, \phi )=x(t,0,\phi)\) is the solution of system (1) through \((0,\phi )\). Given three positive constants \(r_{1}\), \(r_{2}\), T with \(r_{1}< r_{2}\), linear nonautonomous system (1) is said to be FTS with respect to \((T, r_{1}, r_{2})\) if
implies that
Definition 2
(Liao et al. [26])
Let \(I:=[0, +\infty)\), \(f(t)\in C(I, \mathbb{R})\). For any \(t\in I\), the following derivative
is called rightupper derivative of \(f(t)\).
Let \(A(t)=(a_{ij}(t))\), \(D(t)=(d_{ij}(t))\), and \(G(t)=(g_{ij}(t))\) be given matrices with continuous elements. We make the following assumptions which are usually used for a timevarying system (also see [27]). For given \(T>0\), assume that:
 (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}):

\(d_{ij}(t)\leq\bar{d}_{ij}\), \(g_{ij}(t)\leq\bar{g}_{ij}\), \(i,j\in\Lambda\), \(t\in[0,T]\).
We denote \(\mathcal{A}=(\bar{a}_{ij})\), \(\mathcal{D}=(\bar{d}_{ij})\), \(\mathcal{G}=(\bar{g}_{ij})\). Next, we recall here some properties of a Metzler matrix. For more details, one can refer to [28]. A matrix \(A=(a_{ij})\) is called a Metzler matrix if \(a_{ij}\leq0\) whenever \(i\neq j\) and all principal minors of A are positive. The following lemma is used in our main results.
Lemma 1
(Hien et al. [29])
Let \(A=(a_{ij})\) be an offdiagonal nonpositive matrix, \(a_{ii}>0\), \(i\in\Lambda\). Then the following statements are equivalent:

(i)
A is a nonsingular Mmatrix.

(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 nonautonomous system by constructing an appropriated function and using the Metzler matrix method.
Theorem 1
Under assumptions (A_{1}) and (A_{2}), linear nonautonomous 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
If there exists \(\xi\in\mathbb{R}^{n}\) satisfying (3), then we have
that is,
For convenience, let \(x(t)=x(t,0,\phi)\) be the solution of (1) through \((0,\phi)\). It follows from (1) that
where \(D^{+}\) denotes the Dini upperright derivative.
Denote the functions \(V_{i}(t)\), \(i\in\Lambda\), as follows:
we have
Thus, it follows from (6) that
We claim that
Let
Then we have
and hence
Next, we claim
If not, assume that there exist an index \(i\in\Lambda\) and \(t_{1}\in(0,T]\) such that
and
Then
However, it follows from (5) and (6) that for \(t\in[0,t_{1}]\),
therefore,
which yields a contradiction. This shows that
thus, we obtain
Consequently,
If \(\\phi\_{\infty}\leq r_{1}\), then it follows from (2) and (7) that
This shows that system (1) is FTS with respect to \((T, r_{1}, r_{2})\). The proof is complete. □
Corollary 1
Under assumptions (A_{1}) and (A_{2}), linear nonautonomous 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 nonautonomous 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
Consider the following system:
where
and \(\tau(t)=\sin4t\), \(\kappa(t)=\cos t\).
It is easy to see that (A_{1}) and (A_{2}) hold, and then we have
Let
Thus, we have
Let
It should be noted that system (8) does not satisfy the Lyapunov stability conditions proposed in [27]. More precisely, in this case the matrix \(\mathcal{M}=\mathcal{A}+\mathcal{D}+\bar{\kappa}\mathcal{G}\) is not invertible, and hence it does not satisfy conditions of Theorem 2.5 in [27]. However, \(\mathcal {M}^{0}=\mathcal{M}\beta_{3}I\) satisfies (3) and the domain of the solution \(\xi\in R^{2}\) of (3) is defined by \(\frac {2}{2+\beta_{3}}\xi_{1}<\xi_{2}<\frac{2+\beta_{3}}{2}\xi_{1}\).
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 \(r_{2}=1.213144\).
It should be mentioned that the simulation in Fig. 1 of Example 1 is FTS with respect to \((T, r_{1}, r_{2})\), but not LAS.
Example 2
Consider system (8) with parameters as follows:
and \(\tau(t)=\cos2t\), \(\kappa(t)=\cos3t\).
It is easy to see that (A_{1}) and (A_{2}) hold, and we have
Let
Thus, we have
Let
In this case the matrix \(\mathcal{M}=\mathcal{A}+\mathcal{D}+\bar{\kappa}\mathcal{G}\) is not invertible, and hence it does not satisfy conditions of Theorem 2.5 in [27]. However, \(\mathcal {M}^{0}=\mathcal{M}\beta_{3}I\) satisfies (3) and the domain of the solution \(\xi\in R^{3}\) of (3) is defined by
Let us take \(r_{1}=1\), \(r_{1}=2.5\), and then system (8) is 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 nonautonomous systems with timevarying 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.
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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.
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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.
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Yang, X., Li, X. Finitetime stability of linear nonautonomous systems with timevarying delays. Adv Differ Equ 2018, 101 (2018). https://doi.org/10.1186/s1366201815573
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DOI: https://doi.org/10.1186/s1366201815573