On the stability of a class of slowly varying systems

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.


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 so-called "frozen time approach". This approach draws conclusions on the stability of systems for any frozen time of an input function, a time-varying parameter or an internal/external disturbance. For instance, the input/output systemẋ = 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 time-varying systems with slowly varying inputs or parameters by time-invariant 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 eigenvalue-based 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 Lyapunov-based 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.

Background results and definitions
This section states some results and definitions that are needed in the paper. Lemma 2.1 ([13, 14]) Consider a function z : [t 0 , ω) → R + where t 0 ∈ R and t 0 < ω ≤ ∞. Assume that: (a) The function z is absolutely continuous on each compact interval of [t 0 , ω).
where t ≥ t 0 , W (t) ∈ R, β(·) is of class K (that is, continuous, strictly increasing, and β(0) = 0), and e(·) is a positive continuous function that goes to 0 as t → ∞. Then each global solution W (t) of Eq. (1); with a strictly positive initial value, goes to 0 as t → ∞.

Main results
This section derives sufficient conditions for the stability of the following slowly varying system: where for some positive integers m and n. The origin x = 0 is an equilibrium point for Eq. (2). One considers Eq. (2) to be slowly varying because all the elements of the set Γ are continuously differentiable with "sufficiently" small derivative (see [10]). The idea is to derive some stability properties that are uniformly valid in u ∈ Γ (when u is treated as a frozen parameter). Let u ∈ Γ , then the right-hand side of Eq. (2) is continuous. This implies that, for any x 0 ∈ R m , Eq. (2) admits a continuous solution that is defined on a maximal interval of existence [t 0 , ω) where ω ∈ (t 0 , ∞]. Moreover, each solution of Eq. (2) is continuously differentiable because of the continuity of the right-hand side of Eq. (2) (see [17]). Theorem 9.3 in [10] studies the stability of the slowly varying system (2) under the as- where | · | 2 is the induced 2-norm for matrices. (A2) For the change of variables y(·) = x(·)h(u(·)), there exists a Lyapunov function V * ∈ C 2 (R m × R n , R + ) with a finite third derivative such that for all α 1 ∈ R m and all α 2 ∈ R n there exist some strictly positive numbers c 1 , c 2 , c 3 , c 4 , and c 5 satisfying for all t ≥ t 0 and all u ∈ Γ . (A3) The quantities and |y(t 0 )| are less than some number that depends on L and c i ; i = 1, 2, . . . , 5. (A4) One has lim t→∞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 | · | 2 , i = 1, 2, 3, by class K ∞ functions (a continuous function β is of class K ∞ if it is strictly increasing with β(0) = 0 and lim t→∞ β(t) = ∞). Furthermore, in inequalities (6) and (7), we replace the functions c 4 | · | and c 5 | · | 2 by a continuous function H.
We prove Results (i), (ii) and (iii) separately as follows.

Thus (9) leads to |y(t)| ≤ M, for all
This implies that y ∞ < ∞ so that ω = ∞, which completes the proof of the claim.
Claim 1 proves that each solution x(t) of the system (2) is continuable on [t 0 , ∞), which completes the proof of Result (i).
(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→∞ 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→∞ 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) ≤ W (t), for all t ≥ t 0 . Hence the fact that lim t→∞ W (t) = 0 leads to lim t→∞ z(t) = 0. Thus inequality (9) implies that lim t→∞ y(t) = 0 and Result (ii) is seen to be true.
Proof of Result (iii): Assume that h(·) is the zero function and that V (·, ·) is independent of its second component. Then we have x(·) = y(·). We have by inequality (10) that This makes V (·) a strict Lyapunov function [1, Theorem 3.2] which implies that the origin x = 0 is uniformly stable and is globally asymptotically stable.

Example and simulations
This section uses Theorem 3.1 to study the stability of a nonlinear slowly varying system without any restriction on the initial conditions or the magnitude of the functionu. Consider the nonlinear systeṁ where μ > 0, t ≥ t 0 , u ∈ C 1 (R, R) and x(t) = x 1 (t) x 2 (t) takes values in R 2 . The system (21)-(22) has the form of (2) where n = 1, m = 2 and , for all α = Let u ∈ Γ and let x(t) be a solution of the system (21)- (22) with maximal interval of the form [t 0 , ω). Consider the Lyapunov function V ∈ C 1 (R 2 , R + ) that is defined as Inequality (9) is satisfied with β 1 (·) = β 2 (·) = 1 2 (·) 2 . Moreover, one has This implies that inequality (11) is satisfied with H being the identity function.

Conclusion
We have provided sufficient conditions that ensure the stability of the slowly varying systemẋ(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.