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 Open Access
On the stability of a class of slowly varying systems
 M. F. M. Naser^{1}Email author,
 G. N. Gumah^{1},
 S. K. AlOmari^{1} and
 O. M. Bdair^{1}
https://doi.org/10.1186/s1366001819341
© The Author(s) 2018
 Received: 17 July 2018
 Accepted: 3 December 2018
 Published: 11 December 2018
Abstract
Slowly varying systems are common in physics and control engineering and thus stability analysis for those systems has drawn considerable attention in the literature. This paper uses the “frozen time approach” to derive Lyapunov inequality conditions for the stability of a wide class of slowly varying systems. These conditions refine those developed in (Khalil in Nonlinear Systems, 2002) and display generality and effectiveness for both linear and nonlinear systems. To illustrate the utility of the proposed results, an example has been included.
Keywords
 Slowly varying system
 Stability
 Lyapunov function
1 Introduction
Slowly varying systems were first introduced in the 1960s by Desoer [5] in a one page article where he investigated conditions that ensure the exponential stability of an unforced linear system using the socalled “frozen time approach”. This approach draws conclusions on the stability of systems for any frozen time of an input function, a timevarying parameter or an internal/external disturbance. For instance, the input/output system \(\dot{x}=f(x,u)\) is expected to possess stability results that are similar to the frozen system (i.e. when the input u is treated as a constant). Numerous techniques for solving slowly varying systems with parameters influenced by environmental conditions have been developed in [6, 9, 15, 18, 22]. The aforementioned parameters are typically smooth and involve sufficiently small derivatives; see [5], because otherwise the stability of the system is hard to guarantee [16].
Stability analysis of slowly varying systems can be simplified using the frozen time approach by approximating timevarying systems with slowly varying inputs or parameters by timeinvariant ones. To this end, the system under study is required to be attractive or even asymptotically stable as well; see [4, 10, 14] and [20] for further details. This makes the Lyapunov analysis quite involved in studying such systems. For instance, [11] gives a method for constructing strict Lyapunov functions for the class of systems under study. Furthermore, the frozen parameter approach is used in the field of stabilizing feedback systems [8]. Alternatively, the stability of slowly varying systems can be described by eigenvaluebased methods as in [19].
Many references have been devoted to the study of the linear case as in [7] where a Popov criterion is given. The exponential stability and instability of continuous linear systems on time scales are studied in [2] and [3], respectively. In Ref. [21], the author investigates the stability conditions for certain continuous linear slowly varying system, while in Theorem 9.3 of [10], the author provides Lyapunovbased sufficient conditions for the stability of slowly varying systems in some detail.
The main contribution of this paper is new as far as we are aware. We establish a generalization of [10, Theorem 9.3] in a different perspective. We claim the generality, with additional implementations, that our results can be extended to both linear and nonlinear models and are highly suitable for the nonlinear case. To illustrate the described results, an example is given.
The present paper has the following structure. Section 2 presents some results and definitions that are used throughout the paper. In Sect. 3, we establish the main result of this article with the proposed conditions. Simulations are provided in Sect. 4. A brief conclusion part is added at the end of the paper.
2 Background results and definitions
This section states some results and definitions that are needed in the paper.
Lemma 2.1
 (a)
The function z is absolutely continuous on each compact interval of \([t_{0},\omega)\).
 (b)
There exist \(z_{1}\geq0\) and \(z_{2}>0\) such that \(z_{1}< z_{2}\), \(z (t_{0} )< z_{2}\) and \(\dot{z} (t )\leq0\) for almost all \(t\in (t_{0},\omega )\) that satisfy \(z_{1}< z (t )< z_{2}\).
Lemma 2.2
([12, Corollary 2.6])
Definition 2.1
([1, p. 79])
 (i)stable if for any \(t_{0}\in\mathbb{R}_{+}\) and any \(\varepsilon>0\), there is \(c>0\) such that if \(\vert x_{0} \vert < c\) then each solution x of \(\dot{x} (t )=F (t,x (t ) )\), \(x (t_{0} )=x_{0}\) is continuable on \([t_{0},\infty)\) and$$\bigl\vert x (t ) \bigr\vert < \varepsilon,\quad \mbox{for all } t\geq t_{0}, $$
 (ii)uniformly stable if for any \(\varepsilon>0\), there is \(c>0\) such that, for each \(t_{0}\in\mathbb{R}_{+}\) and each \(\vert x_{0} \vert < c\), every solution x of system \(\dot{x} (t )=F (t,x (t ) )\), \(x (t_{0} )=x_{0}\) is continuable on \([t_{0},\infty)\) and$$\bigl\vert x (t ) \bigr\vert < \varepsilon,\quad \mbox{for all } t\geq t_{0}, $$
 (iii)
globally attractive if for all \(t_{0}\in\mathbb{R}_{+}\) and all \(x_{0}\in\mathbb{R}^{m}\), each solution x of system \(\dot{x} (t )=F (t,x (t ) )\), \(x (t_{0} )=x_{0}\) is continuable on \([t_{0},\infty)\) with \(\lim_{t\rightarrow\infty}x (t )=0\),
 (iv)
globally asymptotically stable if it is stable and globally attractive.
3 Main results
Let \(u\in\varGamma\), then the righthand side of Eq. (2) is continuous. This implies that, for any \(x_{0}\in\mathbb{R}^{m}\), Eq. (2) admits a continuous solution that is defined on a maximal interval of existence \([t_{0},\omega)\) where \(\omega\in(t_{0},\infty]\). Moreover, each solution of Eq. (2) is continuously differentiable because of the continuity of the righthand side of Eq. (2) (see [17]).
 (A1):

There exists \(h\in C^{1} (\mathbb{R}^{n},\mathbb{R}^{m} )\) such that \(f (h (v ),v )=0\), for all \(v\in\mathbb{R}^{n}\). Additionally, there exists some \(L>0\) such thatwhere \(\vert \cdot \vert _{2}\) is the induced 2norm for matrices.$$ \biggl\vert \frac{dh (v )}{dv} \biggr\vert _{2}\leq L, \quad \mbox{for all } v\in\mathbb{R}^{n}, $$(4)
 (A2):

For the change of variables \(y (\cdot )=x (\cdot )h (u (\cdot ) )\), there exists a Lyapunov function \(V_{*}\in C^{2} (\mathbb{R}^{m}\times\mathbb{R}^{n},\mathbb{R}_{+} )\) with a finite third derivative such that for all \(\alpha_{1}\in\mathbb{R}^{m}\) and all \(\alpha_{2}\in\mathbb{R}^{n}\) there exist some strictly positive numbers \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), and \(c_{5}\) satisfying$$\begin{aligned} &c_{1} \vert \alpha_{1} \vert ^{2}\leq V_{*} (\alpha_{1},\alpha_{2} )\leq c_{2} \vert \alpha_{1} \vert ^{2}, \end{aligned}$$(5)$$\begin{aligned} &\biggl\vert \frac{\partial V_{*} (\alpha_{1},\alpha_{2} )}{\partial\alpha_{2}} \biggr\vert \leq c_{5} \vert \alpha_{1} \vert ^{2}, \end{aligned}$$(6)$$\begin{aligned} &\biggl\vert \frac{\partial V_{*} (\alpha_{1},\alpha_{2} )}{\partial\alpha_{1}} \biggr\vert \leq c_{4} \vert \alpha_{1} \vert , \end{aligned}$$(7)for all \(t\geq t_{0}\) and all \(u\in\varGamma\).$$\begin{aligned} &\frac{\partial V_{*} (\alpha_{1},u (t ) )}{\partial\alpha_{1}} \biggm_{\alpha_{1}=y (t )} {}\cdot f \bigl(y (t )+h \bigl(u (t ) \bigr),u (t ) \bigr)\leqc_{3} \bigl\vert y (t ) \bigr\vert ^{2}, \end{aligned}$$(8)
 (A3):

The quantities ϵ and \(y(t_{0})\) are less than some number that depends on L and \(c_{i}\); \(i=1,2,\ldots,5\).
 (A4):

One has \(\lim_{t \to \infty} \dot{u}(t)=0\).
In the following theorem we relax Assumption (A2) of [10, Theorem 9.3] where we prove that the Lyapunov function \(V_{*}\) needs only to be continuously differentiable (instead of being \(C^{2}\) with a finite third derivative in [10, Theorem 9.3]). Moreover, in inequalities (5) and (8), we replace the functions \(c_{i} \vert \cdot \vert ^{2}\), \(i=1,2,3\), by class \(\mathcal{K}_{\infty}\) functions (a continuous function β is of class \(\mathcal{K}_{\infty}\) if it is strictly increasing with \(\beta (0 )=0\) and \(\lim_{t\rightarrow\infty}{\beta (t )}=\infty\)). Furthermore, in inequalities (6) and (7), we replace the functions \(c_{4} \vert \cdot \vert \) and \(c_{5} \vert \cdot \vert ^{2}\) by a continuous function H.
Theorem 3.1
 (a)
Assumption (A1) of [10, Theorem 9.3] is satisfied.
 (b)For each solution \(x(t)\) of the system (2) with maximal interval of existence \([t_{0},\omega)\), suppose that there exist \(\delta>0\), a function \(V\in C^{1} (\mathbb{R}^{m}\times\mathbb{R}^{n},\mathbb{R}_{+} )\) and class \(\mathcal{K}_{\infty}\) functions \(\beta_{1}\), \(\beta_{2}\) and \(\beta_{3}\) satisfying$$\begin{aligned} &\beta_{1} \bigl( \vert \alpha_{1} \vert \bigr)\leq V (\alpha_{1},\alpha_{2} )\leq\beta_{2} \bigl( \vert \alpha_{1} \vert \bigr),\quad \textit{for all } ( \alpha_{1},\alpha_{2} )\in\mathbb{R}^{m}\times \mathbb{R}^{n}, \end{aligned}$$(9)for all \(u\in\varGamma\) and all \(t\in(t_{0},\omega)\) that satisfy \(\vert y (t ) \vert <\delta\) where \(y=xh\circ u\).$$\begin{aligned} &\frac{\partial V (\alpha_{1},u (t ) )}{\partial\alpha_{1}} \biggm_{\alpha_{1}=y (t )}{}\cdot f \bigl(y (t )+h \bigl(u (t ) \bigr), u (t ) \bigr)\leq\beta_{3} \bigl( \bigl\vert y (t ) \bigr\vert \bigr), \end{aligned}$$(10)
 (c)There exists a nondecreasing function \(H\in C^{0} (\mathbb{R}_{+},\mathbb{R}_{+} )\) such that \(H (v )>0\), for all \(v>0\) andfor all \((\alpha_{1},\alpha_{2} )\in\mathbb{R}^{m}\times\mathbb{R}^{n}\) that satisfy \(\vert \alpha_{1} \vert <\delta\).$$ \max{ \biggl( \biggl\vert \frac{\partial V (\alpha_{1},\alpha_{2} )}{\partial\alpha_{2}} \biggr\vert , \biggl\vert \frac{\partial V (\alpha_{1},\alpha_{2} )}{\partial\alpha_{1}} \biggr\vert \biggr)}\leq H \bigl( \vert \alpha_{1} \vert \bigr), $$(11)
 (d)One has \(\vert y (t_{0} ) \vert <\beta_{2}^{1} (\beta_{1} (\delta ) )\) and$$ \epsilon< \frac{\beta_{3} (\beta_{2}^{1} (\beta_{1} (\delta ) ) )}{ (L+1 )H (\delta )}. $$(12)
 (i)
For any \(u\in\varGamma\), each solution \(x(t)\) of the system (2) is continuable on \([t_{0},\infty)\).
 (ii)
If Assumption (A4) of [10, Theorem 9.3] is satisfied (that is, \(\dot{u} (t )\rightarrow 0\) as \(t\rightarrow\infty\)), then we have \(\lim_{t\rightarrow\infty}y (t )=0\). (This implies that \(\lim_{t\rightarrow\infty}x (t )=\lim_{t\rightarrow\infty}h (u (t ) )\) whenever \(\lim_{t\rightarrow\infty}h (u (t ) )\) exists^{1}).
 (iii)
If \(h (\cdot )\) is the zero function and \(V (\cdot,\cdot )\) is independent of its second component (i.e. for every \(\alpha\in\mathbb{R}^{m}\), \(V (\alpha,\cdot )\) is a constant function), then, for any \(u\in\varGamma\), the origin \(x=0\) is uniformly stable and is globally asymptotically stable.
We prove Results (i), (ii) and (iii) separately as follows.
Claim 1
\(\omega=\infty\) and \(\Vert y \Vert _{\infty}<\infty\).
Proof
Claim 1 proves that each solution \(x(t)\) of the system (2) is continuable on \([t_{0},\infty)\), which completes the proof of Result (i).
Claim 2
Each solution \(W (\cdot )\) of the system (18)–(19) is nonnegative, continuable on \([t_{0},\infty)\), and satisfies \(\lim_{t\rightarrow\infty}{W (t )}=0\).
Proof
 (i)
If \(W(t_{0})>0\), the property \(\lim_{t\rightarrow\infty}{W (t )}=0\) follows from Lemma 2.2.
 (ii)
If \(W(t_{0})=0\), then either W is the zero function or can be strictly positive at some element in its domain. When z is the zero function, the property \(\lim_{t\rightarrow\infty}{W (t )}=0\) is trivially valid. Otherwise, there exists some \(t_{4}>t_{0}\) such that \(z(t_{4})>0\); then, by seeing the number \(t_{4}\) as a new initial time, one can simply deduce by Lemma 2.2 that \(\lim_{t\rightarrow\infty}{W (t )}=0\). This completes the proof of the claim.
By Claim 2, one can use the comparison lemma [10, p. 102] to deduce that \(z (t )\leq W (t )\), for all \(t\geq t_{0}\). Hence the fact that \(\lim_{t\rightarrow\infty}W (t )=0\) leads to \(\lim_{t\rightarrow\infty}{z (t )}=0\). Thus inequality (9) implies that \(\lim_{t\rightarrow\infty}y (t )=0\) and Result (ii) is seen to be true.
Corollary 3.1
Proof
It follows by Eq. (2), the result \(\lim_{t\rightarrow\infty} (x (t )h (u (t ) ) )=0\) of Theorem 3.1 and the continuity of the function h. □
We observe that the special cases \(\beta_{i} (\cdot )=c_{i} \vert \cdot \vert ^{2}\), \(i=1,2,3\), and \(H (\cdot )=\max{ (c_{4} \vert \cdot \vert ^{2},c_{5} \vert \cdot \vert )}\) makes Items (b) and (c) of Theorem 3.1 reduced to Assumption (A2) of [10, Theorem 9.3]. Moreover, Assumptions (A1), (A3) and (A4) of [10, Theorem 9.3] are already assumed in Theorem 3.1 (see Items (a) and (d)).
4 Example and simulations
In this example, we have proved that the quadratic Lyapunov function satisfies all conditions of Theorem 3.1 with \(\beta_{3} (\cdot )=\frac{1}{\delta^{2}} (\cdot )^{4}\) (i.e. \(\nabla V\cdot f\leq\beta_{3} ( \vert y \vert )=\frac{1}{\delta^{2}} \vert y \vert ^{4}\) for small \(\vert y \vert \)). This implies that there is no guarantee on the inequality \(\nabla V\cdot f\leqc_{3} \vert y \vert ^{2}\) to be valid for small \(\vert y \vert \) and so is inequality (8).
5 Conclusion
We have provided sufficient conditions that ensure the stability of the slowly varying system \(\dot{x} (t )=f (x (t ),u(t) )\) where u is treated as a “frozen parameter”. These conditions open the routes to further knowledge on the stability of more generic classes of systems. Numerical simulations for the nonlinear case have been carried out to illustrate the results.
Observe that the converse of Barbalat’s lemma is not true; that is, the fact that \(\lim_{t\rightarrow\infty}\dot{u} (t )=0\) does not necessarily imply that \(\lim_{t\rightarrow\infty}u (t )\) exists. To see this take \(t_{0}\) positive, \(n=1\) and for all \(t\geq t_{0}\) one has \(u (t )=\sin{ (\ln{t} )}\).
Declarations
Acknowledgements
The authors are grateful to the responsible editor and the anonymous referees for their valuable efforts.
Funding
There are no funding sources to be declared.
Authors’ contributions
All the authors worked jointly. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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