Some new results on the global uniform asymptotic stability of time-varying dynamical systems

Some new results on the global uniform asymptotic stability of time-varying dynamical systems Abstract This article presents some general results on the global uniform asymptotic stability (GUAS) of time-varying nonlinear perturbed systems without the boundedness hypothesis with respect to time. The main contribution is the development of a new Lyapunov function to obtain a GUAS of some perturbed systems. Therefore, we generalized some works which are already made in the literature. Furthermore, some illustrative examples are presented. 1. Introduction This work studies the problem of ensuring global uniform asymptotic stability (GUAS) of time-varying nonlinear perturbed systems of the form x˙=f(t,x)+g(t,x), (1.1) where $$f,g:\mathbb{R}_+\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$$ are piecewise continuous in $$t,$$ locally Lipschitz in $$x,$$ such that $$f(t,0)=g(t,0)=0$$ for all $$t\geq0.$$ This system is seen as a perturbation of the nominal system x˙=f(t,x). (1.2) The perturbation term $$g(t,x)$$ could result from modelling errors, aging or uncertainties and disturbances, which exist in any realistic problem. Suppose that the nominal system (1.2), has a GUAS equilibrium point at the origin with $$V(t,x)$$ as an associate Lyapunov function, then if we calculate the derivative of $$V$$ along the trajectories of (1.1), one can conclude negative definiteness of $$\dot{V}$$ by imposing some restrictions on $$g.$$ The usual techniques in the literature for the stability analysis of (1.1) is based on the stability of the nominal system with $$V(t,x)$$ as a Lyapunov function candidate for the whole system provided that the size of perturbation is known (Hahn, 1967; Lin et al., 1996; Grammel, 2001; Panteley & Loria, 2001; Khalil, 2002; Chaillet & Loria, 2007; Dlala & Hammami, 2007; Benabdallah et al., 2009). However, we cannot usually conclude the behaviour of the solutions of the perturbed system (1.1), by using $$V(t,x)$$ as a Lyapunov function candidate. This fact can be viewed from the following example. Example 1.1 x˙=−a(t)x+δ(t)x|x|1+|x|12, (1.3) where $$x\in\mathbb{R},$$$$a$$ is a bounded continuous function such that, $$0<c_1\leq a(t)\leq c_2,$$ for all $$t\geq0,$$ and $$\delta$$ is a positive continuous unbounded integrable function. The nominal system $$\dot{x}=-a(t)x,$$ is GUAS with a Lyapunov function $$V(t,x)=x^2.$$ Nevertheless, if we use $$V(t,x)$$ as a Lyapunov function for the perturbed system (1.3), we cannot conclude the behaviour of the solutions of system (1.3) on $$\mathbb{R}.$$ Indeed, the derivative of $$V(t,x)$$ along the trajectories of (1.3) is given by V˙(t,x)=2(δ(t)|x|1+|x|12−a(t))x2≥2(δ(t)−a(t))x2, for all $$x\in S:=\{x\in\mathbb{R}/\vert x \vert\geq 1+\vert x \vert^{1\over 2}\}.$$ Since, $$\delta(t)-a(t)$$ is a continuous unbounded function, then there exists a bounded interval $$I,$$ such that δ(t)−a(t)≥1,for all t∈I. It follows that $$\dot{V}(t,x)\geq 2x^2,$$ for all $$t\in I$$ and $$x\in S,$$ although system (1.3) is GUAS. For the proof refer to Remark 2.3. Strict Lyapunov functions are used for this purpose (Mazenc et al., 1999; Mazenc & Nesic, 2007; Malisoff & Mazenc, 2008; Mazenc et al., 2011; Weng & Mao, 2013; Utkin, 2015). Roughly speaking, strict Lyapunov functions are characterized by having negative definite time derivatives along all trajectories of the system. Even when a system is known to be GUAS, one often still needs an explicit strict Lyapunov function, e.g., to build feedbacks that provide input-to-state stability to actuator errors. Converse Lyapunov function theory guarantees the existence of strict Lyapunov functions for many globally asymptotically stable nonlinear systems. However, the Lyapunov functions provided by converse theory are often abstract and not explicit. This yields us to a search for an appropriate Lyapunov function for system (1.1). The challenge is then how to prove the GUAS of such systems by using Lyapunov function theory. This is in fact our objective. Thus, we have addressed the problem in two different ways. First, we start to study a special case of the perturbed system (1.1), which is the time-varying nonlinear cascaded system of the form {x˙1=f1(t,x1)+g(t,x)x2x˙2=f2(t,x2), (1.4) where $$x_1\in\mathbb{R}^{n},$$$$x_2\in\mathbb{R}^{m},$$$$x:=\hbox{col}(x_1,x_2),$$ and the functions $$f_1(t,x_1),$$$$f_2(t,x_2)$$ and $$g(t,x)$$ are continuous, locally Lipschitz in $$x,$$ uniformly in $$t,$$ and $$f_1(t,x_1)$$ is continuously differentiable in both arguments. In fact, the term $$g(t,x)x_2,$$ can be considered as the perturbation of the nominal system. For instance, in Panteley & Loria (1998), the authors established sufficient conditions for GUAS of (1.4) based on a similar linear growth condition as in Jankovic et al. (1996) and an integrability assumption on the input $$x_2,$$ while in Panteley & Loria (2001) they assume that the interconnection term $$g(t,x)$$ satisfies the following condition ‖g(t,x)‖≤G(‖x‖), (1.5) where $$G(.)$$ is a nondecreasing function, and prove that the integrability of the solutions of x˙2=f2(t,x2) (1.6) is sufficient to obtain the global uniform asymptotic stablity of (1.4). So, we have extended the results given in Panteley & Loria (2001) and Seibert & Suarez (1990) concerning the boundedness of solutions of (1.4) without assuming the boundedness hypothesis with respect to time. Thus, we have shown by using the Lyapunov function associated to the nominal system, that system (1.4) is globally uniformly asymptotically stabile. Second, we construct a new strict Lyapunov function for (1.1) which can be applied to (1.4). For instance, the authors in Benabdallah et al. (2007) have studied this problem using the idea given by Jankovic et al. (1996) for the cascade nonlinear system and they prove that, under some restrictions on the dynamic of the system, (1.1) is GUAS with W(t,x)=V(t,x)+ψ(t,x), as a Lyapunov function, where $$\psi$$ is chosen such that $$\dot{W}$$ is definite negative. However, in all previously mentioned works, one of the main hypotheses is that the dynamics of the system are bounded in time. From an engineering point of view, this is a strong assumption, since in some design problems like tracking control and feedback stabilization, the stability analysis concerns systems whose dynamics are in general unbounded with respect to time (Karafyllis & Tsinias, 2003; Karafyllis & Tsinias, 2009). In this article, we are interested, in the beginning, to give sufficient conditions to achieve that a GUAS nonlinear time-varying system x˙1=f1(t,x1) (1.7) remains GUAS when it is perturbed by the output of another GUAS system (1.6), i.e., we establish sufficient conditions to ensure the global uniform asymptotic stablity for system (1.4), without the boundedness hypothesis with respect to time. Next, a new strict Lyapunov function is established that achieves the global uniform asymptotic stablity of (1.1), which can be applied to cascaded system (1.4). The rest of this article is organized as follows. In section two, we present our main results concerning the GUAS of (1.1) and in particularly of (1.4). Furthermore, we present illustrative examples showing the importance of this study. 1.1. Notations and definitions In this article, the solution of a differential equation x˙=f(t,x) (1.8) with initial condition $$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^n,$$ is denoted by $$\phi(.,t,x).$$ For $$r\geq0$$ we define $$\mathcal{B}_r=\{x\in\mathbb{R}^n/\Vert x\Vert\leq r\},$$$$\mathcal{B}^c_r=\{x\in\mathbb{R}^n/\Vert x\Vert> r\},$$ and $$\mathcal{B}^*_r=\mathcal{B}_r\backslash\{0\}$$, where $$\Vert .\Vert$$ is the usual euclidian norm on $$\mathbb{R}^n.$$ We provide below some standard definitions. A continuous function $$\alpha:[0,+\infty[\rightarrow [0,+\infty[$$ is said to belong to class $$\mathcal{K}$$ if it is strictly increasing and $$\alpha(0)=0.$$ It is said to belong to class $$\mathcal{K}_\infty,$$ if $$\alpha(r)\rightarrow +\infty$$ as $$ r\rightarrow +\infty.$$ A continuous function $$\beta:[0,+\infty[\times[0,+\infty[\rightarrow [0,+\infty[$$ is said to belong to class $$\mathcal{KL}$$ if, for each fixed $$s,$$ the mapping $$\beta(r,s)$$ belongs to class $$\mathcal{K}$$ with respect to $$r$$ and, for each fixed $$r,$$ the mapping $$\beta(r,s)$$ is decreasing with respect to $$s$$ and $$\beta(r,s)\rightarrow 0$$ as $$s\rightarrow +\infty.$$ System (1.8) is said to be globally uniformly bounded, if there exist a class $$\mathcal{K}_\infty$$ function $$\alpha,$$ and a positive constant $$a$$ such that $$\Vert \phi(s,t,x) \Vert \leq a+\alpha(\Vert x \Vert),\;\;\; \hbox{for all}\;\; s\geq t\geq0.$$ System (1.8) is said to be globally uniformly stable (GUS), if there exists a class $$\mathcal{K}_\infty$$ function $$\alpha,$$ such that $$\Vert \phi(s,t,x) \Vert \leq \alpha(\Vert x \Vert),\;\;\; \hbox{for all}\;\; s\geq t\geq0.$$ System (1.8) is said to be uniformly asymptotically stable (UAS), if there exists a class $$\mathcal{KL}$$ function $$\beta,$$ and a positive constant $$c,$$ independent of $$t,$$ such that $$\Vert \phi(s,t,x) \Vert \leq \beta(\Vert x \Vert,s-t),\;\;\; \hbox{for all}\;\; s\geq t\geq0,\; \hbox{and}\;\ \Vert x\Vert \leq c.$$ System (1.8) is said to be globally uniformly asymptotically stable (GUAS), if there exists a class $$\mathcal{KL}$$ function $$\beta,$$ such that $$\Vert \phi(s,t,x) \Vert \leq \beta(\Vert x \Vert,s-t),\;\;\; \hbox{for all}\;\; s\geq t\geq0.$$ This is equivalent to saying that (1.8) is GUS and for all $$\varepsilon>0,$$$$c>0,$$ there exists $$T(\varepsilon,c)>0$$ such that for all $$t\geq0,$$$$\Vert \phi(s,t,x) \Vert \leq \varepsilon,\;\;\; \hbox{for all}\;\; s\geq t+T(\varepsilon,c),\;\; \Vert x\Vert\leq c.$$ 2. Main results 2.1. Global uniform asymptotic stability of cascaded systems First, consider system (1.4) with the following assumptions. $$(\mathcal{H}_1)$$ There exist an integrable continuous function $$\delta:\mathbb{R}_+\rightarrow \mathbb{R}_+$$ and class $$\mathcal{K}_\infty$$ functions $$\gamma$$ and $$\theta,$$ such that, for all $$t\geq0$$ and $$(x_1,x_2)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$$, we have ‖g(t,x)‖≤δ(t)θ(‖x2‖)γ(‖x1‖), where ∫0+∞δ(t)dt<M and $$M$$ is a positive constant. $$(\mathcal{H}_2)$$ There exist a continuous differentiable function $$V(t,x_1),$$ class $$\mathcal{K}_\infty$$ functions $$\gamma_i,$$$$i=1,2,3$$ and a constant $$\lambda>0,$$ such that for all $$t\geq0$$ and all $$x_1\in\mathbb{R}^n,$$ we have γ1(‖x1‖)≤V(t,x1)≤γ2(‖x1‖),∂V∂t(t,x1)+∂V∂x1(t,x1)f1(t,x1)≤−λV(t,x1),‖∂V∂x1(t,x1)‖≤γ3(‖x1‖). $$(\mathcal{H}_3)$$ There exists a $$\mathcal{KL}$$-function $$\beta$$ such that the solutions of (1.6) satisfy ‖ϕ2(s,t,x2)‖≤β(‖x2‖,s−t),for alls≥tandx2∈Rm. $$(\mathcal{H}_4)$$ There exists a constant $$c>0,$$ such that ∫c+∞dsγ3(γ1−1(s))γ(γ1−1(s))=∞. We are now ready to present an auxiliary but fundamental result. Lemma 2.1 If assumptions $$(\mathcal{H}_1),$$$$(\mathcal{H}_2)$$ and $$(\mathcal{H}_3)$$ are satisfied and the solutions of (1.4) are globally uniformly bounded then (1.4) is GUAS. Proof. Since the solutions of (1.4) are globally uniformly bounded, then, there exist a $$\mathcal{K}_\infty$$ function $$\alpha$$ and a positive constant $$a$$ such that given any initial state $$x,$$ the solution of (1.4) satisfies ‖ϕ(s,t,x)‖≤a+α(‖x‖),for alls≥t. Let $$\phi_1(.,t,x)$$ and $$\phi_2(.,t,x_2),$$ respectively, be the solutions of x˙1=f1(t,x1)+g(t,x)x2 (2.1) and (1.6). The derivative of $$V$$ along the trajectories of system (2.1) is given by dds(V(s,ϕ1(s,t,x))) =∂V∂t(s,ϕ1(s,t,x))+∂V∂x1(s,ϕ1(s,t,x))f(s,ϕ1(s,t,x)) +∂V∂x1(s,ϕ1(s,t,x))g(s,ϕ(s,t,x))ϕ2(s,t,x2) ≤−λV(s,ϕ1(s,t,x))+γ3(‖ϕ1(s,t,x)‖)γ(‖ϕ1(s,t,x)‖) ×θ(‖ϕ2(s,t,x2)‖)‖ϕ2(s,t,x2)‖δ(s) ≤−λV(s,ϕ1(s,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,s−t)δ(s). Let v(s,ϕ1(s,t,x))=V(s,ϕ1(s,t,x))eλ(s−t). Then, dds(v(s,ϕ1(s,t,x)))≤γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,s−t)δ(s)eλ(s−t). Let $$t_0\geq t.$$ Integrating between $$t_0$$ and $$s,$$ one obtains for all $$s\geq t_0,$$ v(s,ϕ1(s,t,x)) ≤v(t0,ϕ1(t0,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0)) ×∫t0sδ(u)β(‖x2‖,u−t)eλ(u−t)du. (2.2) Moreover, v(s,ϕ1(s,t,x)) ≤v(t0,ϕ1(t0,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0)) ×β(‖x2‖,0)∫t0sδ(u)eλ(u−t)du. It follows that v(s,ϕ1(s,t,x)) ≤v(t0,ϕ1(t0,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,0)Meλ(s−t). Hence, for all $$s\geq t_0,$$ we have V(s,ϕ1(s,t,x)) ≤V(t0,ϕ1(t0,t,x))e−λ(s−t0)+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,0)M. (2.3) Since $$t\geq t_0,$$ then, (2.3) implies that for all $$s\geq t,$$ V(s,ϕ1(s,t,x)) ≤V(t,x1)e−λ(s−t)+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,0)M. Thus, for all $$s\geq t,$$ we have V(s,ϕ1(s,t,x))≤γ2(‖x1‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,0)M. Since, $$\Vert x_1\Vert \leq \Vert x\Vert$$ and $$\Vert x_2\Vert \leq \Vert x\Vert,$$ then, for all $$s\geq t,$$ we have V(s,ϕ1(s,t,x))≤γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M (2.4) We deduce that, ‖ϕ1(s,t,x)‖ ≤γ1−1(γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M). As the map r↦γ1−1(γ2(r)+γ3(a+α(r))γ(a+α(r))θ(β(r,0))β(r,0)M), is a $$\mathcal{K}_\infty$$ function, then, the system (2.1) is GUS. On the other hand, we have $$\beta(\Vert x_2\Vert,s)\rightarrow 0$$ as $$s\rightarrow\infty.$$ Then, for each $$\varepsilon>0,$$ let $$T(\Vert x_2\Vert,\varepsilon)>0$$ such that γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,T(‖x2‖,ε))M≤ε2. Also, we can deduce from (2.2) that for all $$s\geq t+T(\Vert x_2\Vert,\varepsilon),$$ we have v(s,ϕ1(s,t,x)) ≤v(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))∫t+T(‖x2‖,ε)sδ(u)β(‖x2‖,T(‖x2‖,ε))eλ(u−t)du ≤v(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,T(‖x2‖,ε))∫t+T(‖x2‖,ε)sδ(u)eλ(u−t)du. Which in turn implies that for all $$s\geq t+T(\Vert x_2\Vert,\varepsilon),$$ we have V(s,ϕ1(s,t,x)) ≤V(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x))e−λ(s−t−T(‖x2‖,ε)) +γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,T(‖x2‖,ε))M. Since $$t+T(\Vert x_2\Vert,\varepsilon)>t,$$ then, using (2.4), we obtain V(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x)) ≤γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0)) ×β(‖x‖,0)M. Therefore, V(s,ϕ1(s,t,x)) ≤(γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M) ×e−λ(s−t−T(‖x2‖,ε))+ε2. It follows that $$V(s,\phi_1(s,t,x))\leq\varepsilon,$$ for all $$s\geq t+\tilde{T}(\Vert x\Vert,\varepsilon)$$ with T~(‖x‖,ε) =T(‖x2‖,ε) +1λlog⁡(2(γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M)ε). Finally, defining $$\tilde{\varepsilon}=\gamma_1^{-1}(\varepsilon),$$ we conclude that $$\Vert \phi_1(s,t,x)\Vert \leq \tilde{\varepsilon},$$ for all $$s\geq t+\tilde{T}(\Vert x\Vert,\varepsilon).$$ Thus, the system (2.1) is GUAS. And by observing that system (1.6) is GUAS, we deduce that system (1.4) is GUAS. □ From Lemma 2.1, it is sufficient to show that the solutions of the system (2.1) are globally uniformly bounded to obtain the GUAS of (1.4). To achieve our goal, we present the following proposition given in Benabdallah et al. (2007). Proposition 2.1 If there exists a continuous differentiable function $$V:\mathbb{R}_+\times\mathbb{R}^n\rightarrow\mathbb{R}^n,$$ and class $$\mathcal{K}_\infty$$ functions $$\alpha_i,$$$$i=1,2,3,4,$$ such that for all $$t\geq0$$ and all $$x\in\mathbb{R}^n$$ α1(‖x‖)≤V(t,x)≤α2(‖x‖),∂V∂t(t,x)+∂V∂x(t,x)f(t,x)≤−α3(‖x‖),‖∂V∂x(t,x)‖≤α4(‖x‖), there exist an integrable continuous function $$\rho:\mathbb{R}_+\rightarrow\mathbb{R}_+,$$ and a class $$\mathcal{K}$$ function $$\alpha,$$ such that ‖g(t,x)‖≤ρ(t)α(‖x‖) for all $$t\geq0$$ and all $$x\in\mathbb{R}^n$$ and there exist a constant $$c>0,$$ such that ∫c+∞dsα4(α1−1(s))α(α1−1(s))=∞, then the solution of (1.8) is globally uniformly bounded. Now, one has the following theorem. Theorem 2.1 Under assumptions $$(\mathcal{H}_1),$$$$(\mathcal{H}_2),$$$$(\mathcal{H}_3)$$ and $$(\mathcal{H}_4),$$ system (1.4) is GUAS. Proof. Since the component $$\phi_2(.,t,x)$$ is globally uniformly bounded for all $$s\geq t,$$ then by using Lemma 2.1, it suffices to prove that the component $$\phi_1(.,t,x)$$ of (1.4) is globally uniformly bounded. Since, ‖g(t,x)x2‖≤δ(t)θ(‖x2‖)γ(‖x1‖), then, conditions of Proposition 2.1 are satisfied with $$\rho=\delta,$$$$\alpha_1=\gamma_1,$$$$\alpha_2=\gamma_2,$$$$\alpha_3=-\lambda\gamma_1,$$$$\alpha_4=\gamma_3,$$ and α(‖x‖)=θ(β(‖x2‖,0))γ(‖x1‖)β(‖x2‖,0)∈K∞. Therefore, due to Proposition 2.1, the component $$\phi_1(.,t,x)$$ of (1.4) is globally uniformly bounded. Thus, the system (1.4) is GUAS. □ Remark 2.1 We note that in Panteley & Loria (2001), an integrability condition on the state $$x_2$$ was used to establish the GUAS of the cascade system. Here, the integrability condition is imposed on the function $$\delta$$ that bounds $$g(t,x)$$ and is not imposed on the state of the system. This point can be seen in the following example. Example 2.1 Consider the following system {x˙1=−x1+δ(t)x1log⁡(|x1|+1)x2x˙2=−x23, (2.5) where $$x_1\in\mathbb{R},$$ the function $$\delta$$ is a positive continuous unbounded integrable function. Let V(t,x1)=x12 as a Lyapunov function for the nominal system $$\dot{x}_1=-x_1.$$ One can see that assumption $$(\mathcal{H}_2)$$ is satisfied with $$\gamma_1(r)=r^2,$$$$\gamma_2(r)=2r^2$$ and $$\gamma_3(r)=r.$$ Thus, system $$\dot{x}_1=-x_1$$ is GUAS. Let g(t,x1,x2)=δ(t)x1log⁡(|x1|+1)x2, which satisfies assumption $$(\mathcal{H}_1)$$ with $$\gamma(r )=r\log(r+1)$$ and $$\theta(r)=r.$$ Also, we have ∫1+∞drγ3(γ1−1(r))γ(γ1−1(r))=∫1+∞dsrln⁡(r+1)=∞. It follows that, assumption $$(\mathcal{H}_4)$$ is satisfied. On the one hand, the solution of $$\dot{x}_2=-x^3_2,$$$$\phi_2(s,t,x_2)=(2s+{1\over x^2_2})^{-{1\over 2}}$$ satisfies assumption $$(\mathcal{H}_3)$$ with $$\beta(r,s)=\displaystyle {r\over \sqrt{2sr^2+1}}.$$ Therefore, conditions of Theorem 2.1 are satisfied. Thus, the system (2.5) is GUAS. On the other hand, the solution $$\phi_2(s,t,x_2)=(2s+{1\over x^2_2})^{-{1\over 2}}$$ does not satisfy the assumption $$\mathcal{A}_6$$ in Panteley & Loria (2001) i.e., there exists $$\alpha\in\mathcal{K}$$ such that, the trajectory of (1.6), satisfies ∫t+∞‖ϕ2(s,t,x2)‖ds≤α(‖x2‖). Indeed, ∫t+∞‖ϕ2(s,t,x2)‖ds=∞. It means that, we cannot show the GUAS of (2.5) using Panteley & Loria (2001). Now, the question which can be addressed is: is there a Lyapunov function that achieves the GUAS of (1.4), without the boundedness hypothesis with respect to time? This question leads us to study the perturbed system (1.1). 2.2. Construction of a Lyapunov function for perturbed systems Now, let us return to the system (1.1). In this section, we are going to construct a Lyapunov function that achieves the GUAS of (1.1) without the boundedness hypothesis with respect to time. In their article, Jankovic et al. (1996), have constructed a Lyapunov function for autonomous cascaded nonlinear system having the form {x˙=f(x)+h(x,ξ)ξ˙=a(ξ), where $$\dot{x}=f(x)$$ is assumed to be GAS with a known Lyapunov function $$V,$$ and $$\dot{\xi}=a(\xi)$$ is GAS and locally exponentially stable (LES) with Lyapunov function $$U.$$ Under a linear growth assumption on the function $$h$$ and some conditions taken on $$V$$ and $$\displaystyle{\partial V \over \partial x},$$ the considered Lyapunov function has the form W(x,ξ)=V(x)+ψ(x,ξ)+U(ξ), where the cross term $$\psi$$ must guarantee that $$V$$ is nonincreasing along the solutions of the cascaded system. The authors in Benabdallah et al. (2007) use the same idea as in Jankovic et al. (1996) for the perturbed systems of the form (1.1) and they have considered W(t,x)=V(t,x)+ψ(t,x) to ensure the GUAS of (1.1) under the hypothesis is that the dynamics of the system is bounded in time and other restrictions (Benabdallah et al. (2007), Theorem 2). Here, we are going to use almost the same idea as in Benabdallah et al. (2007) and Jankovic et al. (1996) to construct a new Lyapunov function for (1.1) that ensures GUAS of the equilibrium point, without the boundedness hypothesis with respect to time. Let us consider a Lyapunov function for (1.1) of the form W(t,x)={V(t,x)exp⁡(φ(t,x))ifx≠00otherwise. (2.6) Our goal is to seek a suitable function $$\varphi$$ which can compensate the perturbation term. Therefore, if we consider the derivative of $$W$$ along the trajectories of the system (1.1), we get W˙(t,x) =dds(W(s,ϕ(s,t,x)))/s=t =[∂V∂t(t,x)+∂V∂x(t,x)f(t,x)]exp⁡(φ(t,x)) +∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x)). The first term of the right-hand side constitutes the derivative of $$W(t,x)$$ along the trajectories of the nominal system. The second term is the effect of the perturbation, while the third term is the derivative of $$\exp\left(\varphi(t,x)\right)$$ multiplied by $$V(t,x).$$ In order to guarantee that $$\dot{W}(t,x)$$ is a negative definite function, we shall choose φ(t,x)=∫t+∞1V(s,ϕ(s,t,x))∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x))ds. (2.7) Thus one can verify that ∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x))=0,for all(t,x)∈[0,+∞[×Rn∖{0}. This implies with this choice that W˙(t,x)=[∂V∂t(t,x)+∂V∂x(t,x)f(t,x)]exp⁡(φ(t,x)). This equality will now be used for the analysis of (1.1). To this end, we must impose some conditions in view to prove that the new Lyapunov function is continuous positive definite radially unbounded and decreasing along the trajectories of solutions of (1.1). Hence, let us consider the following assumptions. $$(\mathcal{H}_5)$$ There exist a continuous differentiable function $$V(t,x),$$ class $$\mathcal{K}_\infty$$ functions $$\xi_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x\in\mathbb{R}^n,$$ we have ξ1(‖x‖)≤V(t,x)≤ξ2(‖x‖),∂V∂t(t,x)+∂V∂x(t,x)f(t,x)≤−ξ3(‖x‖),‖∂V∂x(t,x)‖≤ξ4(‖x‖). $$(\mathcal{H}_6)$$ There exist an integrable and continuous function $$\rho:\mathbb{R}_+\rightarrow \mathbb{R}_+$$ and a class $$\mathcal{K}_\infty,$$ function $$\vartheta$$ such that for all $$t\geq0$$ and $$x\in\mathbb{R}^n,$$ we have −ρ(t)V(t,x)≤∂V∂x(t,x)g(t,x)≤ρ(t)ϑ(‖x‖)V(t,x), where ∫0+∞ρ(t)dt<Mρ, with $$M_\rho$$ is a positive constant. $$(\mathcal{H}_7)$$ There exists a constant $$c>0$$ such that ∫c+∞dsξ4(ξ1−1(s))ϑ(ξ1−1(s))=∞. Then, we have the following Theorem. Theorem 2.2 If assumptions $$(\mathcal{H}_5),$$$$(\mathcal{H}_6)$$ and $$(\mathcal{H}_7)$$ are satisfied then, system (1.1) is globally uniformly asymptotically stabile. Proof. Let consider the function $$W(t,x)$$ defined in (2.6). Since, $$\vartheta$$ is a class $$\mathcal{K}_\infty$$ function, there exists a positive constant $$r$$ such that ϑ(‖x‖)>1,for allx∈Brc. It follows that |∂V∂x(t,x)g(t,x)|≤ρ(t)ϑ(‖x‖)V(t,x),for allx∈Brc. Therefore, by assumptions $$(\mathcal{H}_5)$$ and $$(\mathcal{H}_7),$$ we deduce as in Benabdallah et al. (2007), Proposition 1), that there exist a class $$\mathcal{K}_\infty$$ function and a positive constant $$a,$$ such that for all $$s\geq t\geq0,$$ we have ‖ϕ(s,t,x)‖≤a+α(‖x‖),for allx∈Brc. (2.8) Now, if $$x\in\mathcal{B}^*_r$$ and we calculate the derivative of $$V$$ along the trajectories of the system (1.1), one can reach the following result: ‖ϕ(s,t,x)‖≤ξ1−1(ξ2(‖x‖)eMρ),for allx∈Br∗. (2.9) Let, for all $$(t,x) \in [0,+\infty[\times\mathbb{R}^n\backslash\{0\},$$ and all $$s\geq t \geq 0,$$ denote ψ(s,t,x)=1V(s,ϕ(s,t,x))∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x)). Since, −ρ(s)V(s,ϕ(s,t,x))≤∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x))≤ρ(s)ϑ(‖ϕ(s,t,x)‖)V(s,ϕ(s,t,x)), we have the following two cases: Case 1: $$x\in\mathcal{B}^c_r.$$ |ψ(s,t,x)| ≤max(ϑ(‖ϕ(s,t,x)‖),1)ρ(s). Using (2.8), we obtain |ψ(s,t,x)| ≤max(ϑ(a+α(‖x‖)),1)ρ(s)∈L1([0,+∞[). This implies that the integral defined in (2.7) exists. Then, ξ1(‖x‖)e−Mρ≤W(t,x)≤ξ2(‖x‖)exp⁡(ϑ(a+α(‖x‖))Mρ) for all t∈[0,+∞[. Case 2: $$x\in\mathcal{B}^*_r.$$ |ψ(s,t,x)| ≤max(ϑ(‖ϕ(s,t,x)‖),1)ρ(s). Using (2.9), we obtain |ψ(s,t,x)| ≤max(ϑ(ξ1−1(ξ2(‖x‖)eMρ)),1)ρ(s)∈L1([0,+∞[). Which implies also that the integral defined in (2.7) exists. Then, ξ1(‖x‖)e−Mρ≤W(t,x)≤ξ2(‖x‖)exp⁡(ϑ(ξ1−1(ξ2(‖x‖)eMρ))Mρ) for all t∈[0,+∞[. Hence, $$W$$ is positive definite and radially unbounded. The continuity of $$\varphi(t,x)$$ can be shown by observing that, for all $$s\geq t\geq0,$$ the function (t,x)⟼ψ(s,t,x) is continuous on $$[0,+\infty[\times\mathbb{R}^n\backslash\{0\},$$ and by using the fact that for each compact set $$K\in \mathbb{R}^n\backslash\{0\},$$ and for all $$(t,x)\in [0,+\infty[\times K,$$$$s\geq t\geq0$$ the following holds |ψ(s,t,x)|≤max(MK,1)ρ(s)∈L1([0,+∞[), where $$M_K$$ is a positive constant which depends only on $$K.$$ Now, to prove that $$W$$ is continuous, it suffices to show that $$W$$ is continuous on the following set S={(t,x)∈[0,+∞[×Rn/x=0}. We have for all $$x\in\mathcal{B}^*_r,$$ W(t,x)≤ξ2(‖x‖)exp⁡(ϑ(ξ1−1(ξ2(‖x‖)eMρ))Mρ), hence lim(t,‖x‖)→(t,0)W(t,x)=0, which implies that $$W$$ is continuous on $$\mathcal{S}.$$ Now, to prove that the derivative of $$\varphi$$ along the trajectories of (1.1) exists, it suffices to use that, for all $$s\geq t\geq0$$ φ(s,ϕ(s,t,x))=∫s+∞1V(u,ϕ(u,s,ϕ(s,t,x)))∂V∂x(u,ϕ(u,s,ϕ(s,t,x)))g(u,ϕ(u,s,ϕ(s,t,x)))du. Since the following two solutions of (1.1): u⟼ϕ(u,t,x) and ϕ(u,s,ϕ(s,t,x)) are equal at the time $$u=s,$$ ϕ(u,t,x)=ϕ(u,s,ϕ(s,t,x)) for alls≥t≥0. Thus, φ(s,ϕ(s,t,x))=∫s+∞1V(u,ϕ(u,t,x))∂V∂x(u,ϕ(u,t,x))g(u,ϕ(u,t,x))du. This implies that the derivative of $$\varphi(t,x)$$ along the trajectories of (1.1) exists and it is given by φ˙(t,x) =dds(φ(s,ϕ(s,t,x)))/s=t =−1V(t,x)∂V∂x(t,x)g(t,x). Hence, the derivative of $$W(t,x)$$ along the trajectories of system (1.1) exists and is given by W˙(t,x) =dds(W(s,ϕ(s,t,x)))/s=t =V˙(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x)) =[∂V∂t(t,x)+∂V∂x(t,x)f(t,x)]exp⁡(φ(t,x)) +∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x)) ≤−ξ3(‖x‖)exp⁡(φ(t,x))+∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))−∂V∂x(t,x)g(t,x)exp⁡(φ(t,x)) ≤−ξ3(‖x‖)e−Mρ, for all t≥0 and x∈Rn∖{0}. Therefore, $$W$$ is decreasing along the trajectories of solutions of (1.1). Thus, system (1.1) is GUAS. □ Remark 2.2 The above theorem covers a larger class of perturbed systems than those in Benabdallah et al. (2007), Theorem $$2).$$ Indeed, in this reference (Benabdallah et al. 2007), the authors prove the GUAS of (1.1), with the $$L_\infty$$ condition under the following restrictions: $$(\mathcal{R}_1)$$ The integrable function $$\rho$$ is bounded by a positive constant $$\bar{\rho}.$$ $$(\mathcal{R}_2)$$ There exists a nondecreasing function $$G$$ such that, for all $$t\geq0$$ and $$x\in\mathbb{R}^n,$$ ‖f(t,x)‖≤G(‖x‖) and the function $$G$$ satisfies another condition. $$(\mathcal{R}_3)$$ There exists a class $$\mathcal{K}$$ function $$\xi_5,$$ such that |∫t+∞∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x))ds|≤ξ5(‖x‖). Remark 2.3 Now, let us return to the differential equation (1.3) given in Example 1.1. It is clear that assumptions $$(\mathcal{H}_5)$$ and $$(\mathcal{H}_7)$$ are satisfied with $$\xi_{i}(r)=d_ir^2,$$$$d_i>0,$$$$i=1,2;$$$$\xi_{3}(r)=c_1r^2,$$$$\xi_4(r)=2r$$ and $$\vartheta(r)=r.$$ Also, we have −ρ(t)V(t,x)≤∂V∂x(t,x)g(t,x) =2δ(t)x2|x|1+|x|12 ≤2δ(t)|x|V(t,x). It follows that, assumption $$(\mathcal{H}_{6})$$ is satisfied with $$\rho(t)=2\displaystyle\delta(t)\in L^1([0,+\infty[).$$ Then, W(t,x)={x2exp⁡(∫t+∞2δ(s)|ϕ(s,t,x)|ds),ifx≠00otherwise. guarantees the GUAS of (1.3). Next, we give a mechanical example which represents a nonlinear mass-spring-damper system, see Slotine & Li (1991). Example 2.2 Consider the following system q¨+c(t)q˙+k0q=0 (2.10) which represents a nonlinear mass-spring-damper system. Where $$c(t)$$ is a time-varying damping coefficient, and $$k_o$$ is a spring constant. The variable $$q\in\mathbb{R}$$ represents the position of the mass with respect to its rest position. We use the notation $$\dot{q}$$ to denote the derivative of $$q$$ with respect to time (i.e., the velocity of the mass) and $$\ddot{q}$$ to represent the second derivative (acceleration). Such a model is natural to use for celestial mechanics, because it is difficult to influence the motion of the planets. In many examples, it is useful to model the effects of external disturbances or controlled forces on the system. One way to capture this is to replace equation (2.10) by q¨+c(t)q˙+k0q=u (2.11) where $$u$$ represents the effect of external influences. The model (2.11) is called a forced or controlled differential equation. It implies that the rate of change of the state can be influenced by the input $$u.$$ Adding the input makes the model richer and allows new questions to be posed. For example, we can examine what influence external disturbances have on the trajectories of a system. Or, in the case when the input variable is something that can be modulated in a controlled way, we can analyse whether it is possible to steer the system from one point in the state space to another through proper choice of the input. Let $$\dot{q}=x.$$ Then system (2.11) can be rewritten as {x˙=−c(t)x−k0q+u(t,q˙,q)q˙=x (2.12) The system has the form of (1.1) with f(t,x,q)=[f1(t,x,q)f2(t,x,q)]=[−c(t)x−k0q0] and g(t,x,q)=[g1(t,x,q)g2(t,x,q)]=[u(t,x,q)0] Let V(t,x,q)=12(x+αq)2+12b(t)q2, as a Lyapunov function for the nominal system, with $$0<\alpha<\min(1,k_0^{1 \over 2}),$$$$b(t)=k_0-\alpha^2+\alpha c(t),$$$$\alpha<c(t)<\beta$$ and $$\dot{c}(t)\leq\gamma<2k_0,$$ and let u(t,x,q)={12ρ(t)(x+αq)3+(x+αq)b(t)q2(x2+q2)12if(x,q)≠(0,0)0otherwise. Where $$\rho$$ is a positive continuous integrable unbounded function, i.e, $$\rho$$ is $$L^1([0,+\infty[)$$ but not $$L^\infty([0,+\infty[).$$ For instance, one can take $$\rho$$ as follows: ρ(t)={0ift∈[0,2−18]n4t+(n−n5)ift∈[n−1n3,n], n≥2−n4t+(n+n5)ift∈[n,n+1n3], n≥20ift∈[n+1n3,(n+1)−1(n+1)3], n≥2. One can see that assumptions $$(\mathcal{H}_5),$$$$(\mathcal{H}_6)$$ and $$(\mathcal{H}_7),$$ are satisfied with ξ1(r)=max((1+α),(α+k0+αβ))r2,ξ2(r)=min(12(1−α),12(−α+k0+α2))r2,ξ3(r)=min(2α,αk0−αγ2)r2,ξ4(r)=(max(1+α(1+k0)+α2(1+β),(1+α)(α+αβ+k0)))12r and ϑ(r)=max(1+α,2α)r. Then, for all $$(t,x,q)\in \mathcal{S}=\{(t,x,q)\backslash (x,q)\neq0\},$$ one has φ(t,x,q) =∫t+∞max(1+α,2α)(ϕ12(s,t,x,q)+ϕ12(s,t,x,q))12ds. It follows that system (2.11) is GUAS with W(t,x,q)={V(t,x,q)exp⁡(φ(t,x,q)),if(t,x,q)∈S0otherwise. as a Lyapunov function candidate. Also, an interesting application of this new Lyapunov function can be seen for system (1.4). Then, let us consider the following assumptions. $$(\mathcal{H}_8)$$ There exist a continuous differentiable function $$V_1(t,x_1),$$ class $$\mathcal{K}_\infty$$ functions $$\xi_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x_1\in\mathbb{R}^n,$$ we have ξ1(‖x1‖)≤V1(t,x1)≤ξ2(‖x1‖),∂V1∂t(t,x1)+∂V1∂x1(t,x)f1(t,x1)≤−ξ3(‖x1‖),‖∂V1∂x1(t,x1)‖≤ξ4(‖x1‖). $$(\mathcal{H}_9)$$ There exist a continuous differentiable function $$V_2(t,x_2),$$ class $$\mathcal{K}_\infty$$ functions $$\tilde{\xi}_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x_2\in\mathbb{R}^m,$$ we have ξ~1(‖x2‖)≤V2(t,x2)≤ξ~2(‖x2‖),∂V2∂t(t,x2)+∂V2∂x2(t,x2)f2(t,x2)≤−ξ~3(‖x2‖),‖∂V2∂x2(t,x2)‖≤ξ~4(‖x2‖). Assumptions $$(\mathcal{H}_8)$$ and $$(\mathcal{H}_9)$$ assert the GUAS of the subsystems (1.7) and (1.6). $$(\mathcal{H}_{10})$$ There exist an integrable and continuous function $$\delta:\mathbb{R}_+\rightarrow \mathbb{R}_+$$ and a class $$\mathcal{K}_\infty$$ function $$\vartheta,$$ such that for all $$t\geq0$$ and $$x_1\in\mathbb{R}^n,$$ we have −δ(t)V1(t,x1)≤∂V1∂x1(t,x1)g(t,x)x2≤δ(t)ϑ(‖x1‖)V1(t,x1), where ∫0+∞δ(t)dt<Mδ, with $$M_\delta$$ is a positive constant. Then, one has the following corollary. Corollary 2.1 If assumptions $$(\mathcal{H}_7),$$$$(\mathcal{H}_8),$$$$(\mathcal{H}_9)$$ and $$(\mathcal{H}_{10})$$ are satisfied, then, system (1.4) is GUAS with W~(t,x)={W(t,x1)+V2(t,x2)ifx1≠00otherwise, (2.13) where W(t,x1)=V1(t,x1)exp⁡(∫t+∞1V1(s,ϕ1(s,t,x))∂V1∂x1(τ,ϕ1(s,t,x))g(s,ϕ(s,t,x))ϕ2(s,t,x2)ds), as a Lyapunov function candidate. Proof. Under assumptions $$(\mathcal{H}_7),$$$$(\mathcal{H}_8)$$ and $$(\mathcal{H}_{10})$$ and using Theorem 2.2, there exist class $$\mathcal{K}_\infty$$ functions $$\sigma_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x_1\in\mathbb{R}^n,$$ we have σ1(‖x1‖)≤W(t,x1)≤σ2(‖x1‖),W˙(t,x1)=dds(W(s,ϕ(s,t,x)))/s=t≤−σ3(‖x1‖),‖∂W∂x1(t,x1)‖≤σ4(‖x1‖). Using assumption $$(\mathcal{H}_9),$$$$\tilde{W}(t,x)$$ satisfies the following inequalities for all $$x\in\mathbb{R}^{n}\times \mathbb{R}^{m},$$ σ1(‖x1‖)+ξ~1(‖x2‖)≤W~(t,x)≤σ2(‖x1‖)+ξ~2(‖x2‖),W~˙(t,x)=dds(W~(s,ϕ(s,t,x)))/s=t≤−σ3(‖x1‖)−ξ~2(‖x3‖),‖∂W~∂x(t,x)‖≤σ4(‖x1‖)+ξ~4(‖x2‖). Hence, (1.4) is GUAS. □ Remark 2.4 The difference between the assumptions Theorem 2.1 and those of corollary 2.1 for the case (1.4) can be summarized in two essential points. On the one hand, we have assumed in the hypothesis $$(\mathcal{H}_2)$$ of Theorem 2.1 that the derivative of the Lyapunov function $$V$$ associated to the nominal system (1.7) satisfies $$\dot{V} (t, x_1) \leq-\lambda V(t,x_1),$$$$\lambda>0.$$ Using the fact that $$V$$ satisfies γ1(‖x1‖)≤V(t,x1), then, V˙(t,x1)≤−λγ1(‖x1‖). Which can be considered as a special case of hypothesis $$(\mathcal{H}_8)$$ of Corollary 2.1. It suffices to take $$\xi=\lambda \gamma_1\in \mathcal{K}_\infty.$$ On the other hand, assumption $$(\mathcal{H}_1)$$ of Theorem 2.1 gives a wide class of systems which contain a term of perturbation. These points can be seen in the following example. Let us consider the following two-dimensional cascaded system. Example 2.3 {x˙1=−x11+|x1|+ρ(t)x2x1|x1|1+x22x2x˙2=−x2(1+e−t) (2.14) where $$(x_1,x_2)\in\mathbb{R}^2$$ and $$\rho$$ is a positive continuous integrable unbounded function. This system has the form of (1.4) with f1(t,x1)=−x11+|x1|, f2(t,x2)=−x2(1+e−t) and g(t,x)=ρ(t)x2x1|x1|1+x22. Let $$V_1(t,x_1)=x_{1}^2$$ and $$V_2(t,x_2)=x_{2}^2,$$ respectively, be the Lyapunov functions for x˙1=f1(t,x1) (2.15) and x˙2=f2(t,x2). (2.16) Their derivative along the trajectories, respectively, of (2.15) and (2.16) are given by V˙1(t,x1) =−x121+|x1| and V˙2(t,x2) =−x22(1+e−t), It is clear that assumptions $$(\mathcal{H}_7),$$$$(\mathcal{H}_8)$$ and $$(\mathcal{H}_9),$$ are satisfied with $$\xi_{i}(r)=c_ir^2,$$$$\tilde{\xi}_{i}(r)=\tilde{c}_ir^2,$$$$c_i, \tilde{c}_i>0,$$$$i=1,2;$$$$\xi_{3}(r)={r^2\over 1+r},$$$$\tilde{\xi}_{3}(r)=r^2,$$$$\xi_4(r)=\tilde{\xi}_4(r)=2r$$ and $$\vartheta(r)=r.$$ Which implies that the systems (2.15) and (2.16), are GUAS. Also, we have 0≤∂V1∂x1(t,x1)g(t,x)x2 =2ρ(t)x12|x1|1+x22x22 ≤2ρ(t)|x1|V1(t,x1). It follows that, assumption $$(\mathcal{H}_{10})$$ is satisfied with $$\delta(t)=2\rho(t)\in L^1([0,+\infty[).$$ Then, system (2.14) is GUAS with W~(t,x)={W(t,x1)+V2(t,x2)ifx1≠00otherwise, where W(t,x1)=V1(t,x1)exp⁡(∫t+∞δ(s)|ϕ1(s,t,x)|ϕ22(s,t,x2)1+ϕ2(s,t,x2)2ds), as a Lyapunov function candidate. However, one can see that here the assumption $$(\mathcal{H}_2)$$ of Theorem 2.1 is not satisfied. In fact, for all $$x_1\in\mathbb{R},$$ we have $$0<\displaystyle{1\over 1+\vert x_1\vert}\leq1.$$ Then, there is no $$\lambda>0$$ such that $$\dot{V}_1(t,x_1)\leq -\lambda V(t,x_1).$$ Thus, we cannot show the GUAS of (2.14) by using Theorem 2.1. Remark 2.5 Lyapunov technique remains a very important tool for studying the stability of dynamic systems such as robotic systems (Chen et al., 2013a; Chen, 2014) and choatic systems (Chen et al., 2013b). Thus, the built lyapunov function can be a good tool to show stability and stabilization of these systems. 3. Conclusion In this article, we dealt with the analysis problem of nonlinear time-varying perturbed systems without the boundedness hypothesis with respect to time. Indeed, we established the GUAS for general classes of perturbed systems by using a new Lyapunov function. In addition, an interesting application of this new Lyapunov function is given for cascaded time-varying systems. As example, we applied the present results to a nonlinear mass-spring-damper system with a disturbance. Acknowledgements The authors wish to thank the reviewers for their valuable and careful comments. References Benabdallah A. , Dlala M. & Hammami M. A. ( 2007 ) A new Lyapunov function for stability of time-varying nonlinear perturbed systems . Syst. Control Lett . 56 , 179 – 187 . Google Scholar Crossref Search ADS Benabdallah A. , Ellouze I. & Hammami M. A. ( 2009 ) Practical stability of nonlinear time varying cascade systems , J. Dyn. Control Syst. , 15 , 45 – 62 . Google Scholar Crossref Search ADS Chaillet A. & Loria A. ( 2007 ) Necessary and sufficient conditions for uniform semi-global practical asymptotic stability: application to cascaded systems , Automatica 56 , 179 – 187 . Chen H. ( 2014 ) Robust stabilization for a class of dynamic feedback uncertain nonholonomic mobile robots with input saturation , Int. J. Control, Autom. Syst . 12 , 1216 – 1224 . Google Scholar Crossref Search ADS Chen H. , Zhang J. , Chen B. & Baoujun L. ( 2013 ) Global practical stabilization for non-holonomic mobile robots with uncalibrated visual parameters by using a switching controller , IMA J Math. Control Info . 30 , 543 – 557 . Google Scholar Crossref Search ADS Chen H. , Chen W. , Zhang B. , Chen B. & Haitao C. ( 2013 ) Robust synchronization of incommensurate fractional-order chaotic systems via second-order sliding mode technique , J. Appl. Math. , http://dx.doi.org/10.1155/2013/321253. Dlala M. & Hammami M. A. ( 2007 ) Uniform exponential practical stability of impulsive perturbed systems , J. Dyn. Control Syst . 13 , 373 – 386 . Google Scholar Crossref Search ADS Grammel G. ( 2001 ) Exponential stability via the averaged system . Dyn. Control Syst . 7 , 327 – 338 . Google Scholar Crossref Search ADS Hahn W. ( 1967 ) Stability of Motion , Berlin : Springer . Jankovic M. , Sepulchre R. & Kokotovic P. V. ( 1996 ) Constructive Lyapunov stabilisation of nonlinear cascade systems , IEEE Trans. Autom. Control 41 , 1723 – 1735 . Google Scholar Crossref Search ADS Karafyllis I. & Tsinias J. ( 2003 ) Non-uniform in time stabilization for linear systems and traking control for non-holonomic systems in chained form , Int. J. Control 76 , 2003 : 1536 – 1546 . Google Scholar Crossref Search ADS Karafyllis I. & Tsinias J. ( 2009 ) Control Lyapunov functions and stabilization by means of continuous time-varying feedback , ESAIM-COCV 15 , 599 – 625 . Google Scholar Crossref Search ADS Khalil H. ( 2002 ) Nonlinear Systems , 3rd edn . Englewood Cliffs, NJ : Prentice-Hall . Lin Y. , Sontag E. & Wang Y. ( 1996 ) A smooth converse Lyapunov theorem for robust stability , SIAM J. Control Optim . 34 , 124 – 160 . Google Scholar Crossref Search ADS Malisoff M. & Mazenc F. ( 2008 ) Constructions of Strict Lyapunov functions for discrete time and hybrid time-varying systems , Nonlinear Anal.: Hybrid Syst . 2 , 394 – 407 . Google Scholar Crossref Search ADS Mazenc F. , Sepulchre R. & Jankovic M. ( 1999 ) Lyapunov functions for stable cascades and applications to stabilisation , IEEE Transcations on Automatic Control 44 , 1795 – 1800 . Google Scholar Crossref Search ADS Mazenc F. & Nesic D. ( 2007 ) Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem , Math. Control Signals Syst . 19 , 151 – 182 . Google Scholar Crossref Search ADS Mazenc F. , Malisoff M. & de Queiroz M. ( 2011 ) Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains , Nonlinear Anal . 74 , 1132 – 1145 . Google Scholar Crossref Search ADS Panteley E. & Loria A. ( 1998 ) On global uniform asymptotic stability of nonlinear time-varying non autonomous systems in cascade , Syst. Control Lett . 33 , 131 – 138 . Google Scholar Crossref Search ADS Panteley E. & Loria A. ( 2001 ) Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems , Automatica 37 , 453 – 460 . Google Scholar Crossref Search ADS Seibert P. & Suarez R. ( 1990 ) Global stabilisation of nonlinear cascade systems , Syst. Control Lett . 14 , 347 – 352 . Google Scholar Crossref Search ADS Slotine J. & Li W. ( 1991 ) Applied nonlinear control , Prentice-Hall , Englewood Cliffs, NJ . Utkin V. ( 2015 ) Mechanical energy-based Lyapunov function design for twisting and super-twisting sliding mode control , IMA J. Math. Control Inf . 32 , 675 – 688 . Weng F. & Mao W. ( 2013 ) Robust stability and stabilization of uncertain discrete singular time-delay systems based on PNP Lyapunov functional , IMA J. Math. Control Inf. , 30 , 301 – 314 . Google Scholar Crossref Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

Some new results on the global uniform asymptotic stability of time-varying dynamical systems

Loading next page...
 
/lp/ou_press/some-new-results-on-the-global-uniform-asymptotic-stability-of-time-Qohclvf3O0
Publisher
Oxford University Press
Copyright
© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dnx006
Publisher site
See Article on Publisher Site

Abstract

Abstract This article presents some general results on the global uniform asymptotic stability (GUAS) of time-varying nonlinear perturbed systems without the boundedness hypothesis with respect to time. The main contribution is the development of a new Lyapunov function to obtain a GUAS of some perturbed systems. Therefore, we generalized some works which are already made in the literature. Furthermore, some illustrative examples are presented. 1. Introduction This work studies the problem of ensuring global uniform asymptotic stability (GUAS) of time-varying nonlinear perturbed systems of the form x˙=f(t,x)+g(t,x), (1.1) where $$f,g:\mathbb{R}_+\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$$ are piecewise continuous in $$t,$$ locally Lipschitz in $$x,$$ such that $$f(t,0)=g(t,0)=0$$ for all $$t\geq0.$$ This system is seen as a perturbation of the nominal system x˙=f(t,x). (1.2) The perturbation term $$g(t,x)$$ could result from modelling errors, aging or uncertainties and disturbances, which exist in any realistic problem. Suppose that the nominal system (1.2), has a GUAS equilibrium point at the origin with $$V(t,x)$$ as an associate Lyapunov function, then if we calculate the derivative of $$V$$ along the trajectories of (1.1), one can conclude negative definiteness of $$\dot{V}$$ by imposing some restrictions on $$g.$$ The usual techniques in the literature for the stability analysis of (1.1) is based on the stability of the nominal system with $$V(t,x)$$ as a Lyapunov function candidate for the whole system provided that the size of perturbation is known (Hahn, 1967; Lin et al., 1996; Grammel, 2001; Panteley & Loria, 2001; Khalil, 2002; Chaillet & Loria, 2007; Dlala & Hammami, 2007; Benabdallah et al., 2009). However, we cannot usually conclude the behaviour of the solutions of the perturbed system (1.1), by using $$V(t,x)$$ as a Lyapunov function candidate. This fact can be viewed from the following example. Example 1.1 x˙=−a(t)x+δ(t)x|x|1+|x|12, (1.3) where $$x\in\mathbb{R},$$$$a$$ is a bounded continuous function such that, $$0<c_1\leq a(t)\leq c_2,$$ for all $$t\geq0,$$ and $$\delta$$ is a positive continuous unbounded integrable function. The nominal system $$\dot{x}=-a(t)x,$$ is GUAS with a Lyapunov function $$V(t,x)=x^2.$$ Nevertheless, if we use $$V(t,x)$$ as a Lyapunov function for the perturbed system (1.3), we cannot conclude the behaviour of the solutions of system (1.3) on $$\mathbb{R}.$$ Indeed, the derivative of $$V(t,x)$$ along the trajectories of (1.3) is given by V˙(t,x)=2(δ(t)|x|1+|x|12−a(t))x2≥2(δ(t)−a(t))x2, for all $$x\in S:=\{x\in\mathbb{R}/\vert x \vert\geq 1+\vert x \vert^{1\over 2}\}.$$ Since, $$\delta(t)-a(t)$$ is a continuous unbounded function, then there exists a bounded interval $$I,$$ such that δ(t)−a(t)≥1,for all t∈I. It follows that $$\dot{V}(t,x)\geq 2x^2,$$ for all $$t\in I$$ and $$x\in S,$$ although system (1.3) is GUAS. For the proof refer to Remark 2.3. Strict Lyapunov functions are used for this purpose (Mazenc et al., 1999; Mazenc & Nesic, 2007; Malisoff & Mazenc, 2008; Mazenc et al., 2011; Weng & Mao, 2013; Utkin, 2015). Roughly speaking, strict Lyapunov functions are characterized by having negative definite time derivatives along all trajectories of the system. Even when a system is known to be GUAS, one often still needs an explicit strict Lyapunov function, e.g., to build feedbacks that provide input-to-state stability to actuator errors. Converse Lyapunov function theory guarantees the existence of strict Lyapunov functions for many globally asymptotically stable nonlinear systems. However, the Lyapunov functions provided by converse theory are often abstract and not explicit. This yields us to a search for an appropriate Lyapunov function for system (1.1). The challenge is then how to prove the GUAS of such systems by using Lyapunov function theory. This is in fact our objective. Thus, we have addressed the problem in two different ways. First, we start to study a special case of the perturbed system (1.1), which is the time-varying nonlinear cascaded system of the form {x˙1=f1(t,x1)+g(t,x)x2x˙2=f2(t,x2), (1.4) where $$x_1\in\mathbb{R}^{n},$$$$x_2\in\mathbb{R}^{m},$$$$x:=\hbox{col}(x_1,x_2),$$ and the functions $$f_1(t,x_1),$$$$f_2(t,x_2)$$ and $$g(t,x)$$ are continuous, locally Lipschitz in $$x,$$ uniformly in $$t,$$ and $$f_1(t,x_1)$$ is continuously differentiable in both arguments. In fact, the term $$g(t,x)x_2,$$ can be considered as the perturbation of the nominal system. For instance, in Panteley & Loria (1998), the authors established sufficient conditions for GUAS of (1.4) based on a similar linear growth condition as in Jankovic et al. (1996) and an integrability assumption on the input $$x_2,$$ while in Panteley & Loria (2001) they assume that the interconnection term $$g(t,x)$$ satisfies the following condition ‖g(t,x)‖≤G(‖x‖), (1.5) where $$G(.)$$ is a nondecreasing function, and prove that the integrability of the solutions of x˙2=f2(t,x2) (1.6) is sufficient to obtain the global uniform asymptotic stablity of (1.4). So, we have extended the results given in Panteley & Loria (2001) and Seibert & Suarez (1990) concerning the boundedness of solutions of (1.4) without assuming the boundedness hypothesis with respect to time. Thus, we have shown by using the Lyapunov function associated to the nominal system, that system (1.4) is globally uniformly asymptotically stabile. Second, we construct a new strict Lyapunov function for (1.1) which can be applied to (1.4). For instance, the authors in Benabdallah et al. (2007) have studied this problem using the idea given by Jankovic et al. (1996) for the cascade nonlinear system and they prove that, under some restrictions on the dynamic of the system, (1.1) is GUAS with W(t,x)=V(t,x)+ψ(t,x), as a Lyapunov function, where $$\psi$$ is chosen such that $$\dot{W}$$ is definite negative. However, in all previously mentioned works, one of the main hypotheses is that the dynamics of the system are bounded in time. From an engineering point of view, this is a strong assumption, since in some design problems like tracking control and feedback stabilization, the stability analysis concerns systems whose dynamics are in general unbounded with respect to time (Karafyllis & Tsinias, 2003; Karafyllis & Tsinias, 2009). In this article, we are interested, in the beginning, to give sufficient conditions to achieve that a GUAS nonlinear time-varying system x˙1=f1(t,x1) (1.7) remains GUAS when it is perturbed by the output of another GUAS system (1.6), i.e., we establish sufficient conditions to ensure the global uniform asymptotic stablity for system (1.4), without the boundedness hypothesis with respect to time. Next, a new strict Lyapunov function is established that achieves the global uniform asymptotic stablity of (1.1), which can be applied to cascaded system (1.4). The rest of this article is organized as follows. In section two, we present our main results concerning the GUAS of (1.1) and in particularly of (1.4). Furthermore, we present illustrative examples showing the importance of this study. 1.1. Notations and definitions In this article, the solution of a differential equation x˙=f(t,x) (1.8) with initial condition $$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^n,$$ is denoted by $$\phi(.,t,x).$$ For $$r\geq0$$ we define $$\mathcal{B}_r=\{x\in\mathbb{R}^n/\Vert x\Vert\leq r\},$$$$\mathcal{B}^c_r=\{x\in\mathbb{R}^n/\Vert x\Vert> r\},$$ and $$\mathcal{B}^*_r=\mathcal{B}_r\backslash\{0\}$$, where $$\Vert .\Vert$$ is the usual euclidian norm on $$\mathbb{R}^n.$$ We provide below some standard definitions. A continuous function $$\alpha:[0,+\infty[\rightarrow [0,+\infty[$$ is said to belong to class $$\mathcal{K}$$ if it is strictly increasing and $$\alpha(0)=0.$$ It is said to belong to class $$\mathcal{K}_\infty,$$ if $$\alpha(r)\rightarrow +\infty$$ as $$ r\rightarrow +\infty.$$ A continuous function $$\beta:[0,+\infty[\times[0,+\infty[\rightarrow [0,+\infty[$$ is said to belong to class $$\mathcal{KL}$$ if, for each fixed $$s,$$ the mapping $$\beta(r,s)$$ belongs to class $$\mathcal{K}$$ with respect to $$r$$ and, for each fixed $$r,$$ the mapping $$\beta(r,s)$$ is decreasing with respect to $$s$$ and $$\beta(r,s)\rightarrow 0$$ as $$s\rightarrow +\infty.$$ System (1.8) is said to be globally uniformly bounded, if there exist a class $$\mathcal{K}_\infty$$ function $$\alpha,$$ and a positive constant $$a$$ such that $$\Vert \phi(s,t,x) \Vert \leq a+\alpha(\Vert x \Vert),\;\;\; \hbox{for all}\;\; s\geq t\geq0.$$ System (1.8) is said to be globally uniformly stable (GUS), if there exists a class $$\mathcal{K}_\infty$$ function $$\alpha,$$ such that $$\Vert \phi(s,t,x) \Vert \leq \alpha(\Vert x \Vert),\;\;\; \hbox{for all}\;\; s\geq t\geq0.$$ System (1.8) is said to be uniformly asymptotically stable (UAS), if there exists a class $$\mathcal{KL}$$ function $$\beta,$$ and a positive constant $$c,$$ independent of $$t,$$ such that $$\Vert \phi(s,t,x) \Vert \leq \beta(\Vert x \Vert,s-t),\;\;\; \hbox{for all}\;\; s\geq t\geq0,\; \hbox{and}\;\ \Vert x\Vert \leq c.$$ System (1.8) is said to be globally uniformly asymptotically stable (GUAS), if there exists a class $$\mathcal{KL}$$ function $$\beta,$$ such that $$\Vert \phi(s,t,x) \Vert \leq \beta(\Vert x \Vert,s-t),\;\;\; \hbox{for all}\;\; s\geq t\geq0.$$ This is equivalent to saying that (1.8) is GUS and for all $$\varepsilon>0,$$$$c>0,$$ there exists $$T(\varepsilon,c)>0$$ such that for all $$t\geq0,$$$$\Vert \phi(s,t,x) \Vert \leq \varepsilon,\;\;\; \hbox{for all}\;\; s\geq t+T(\varepsilon,c),\;\; \Vert x\Vert\leq c.$$ 2. Main results 2.1. Global uniform asymptotic stability of cascaded systems First, consider system (1.4) with the following assumptions. $$(\mathcal{H}_1)$$ There exist an integrable continuous function $$\delta:\mathbb{R}_+\rightarrow \mathbb{R}_+$$ and class $$\mathcal{K}_\infty$$ functions $$\gamma$$ and $$\theta,$$ such that, for all $$t\geq0$$ and $$(x_1,x_2)\in\mathbb{R}^{n}\times\mathbb{R}^{m}$$, we have ‖g(t,x)‖≤δ(t)θ(‖x2‖)γ(‖x1‖), where ∫0+∞δ(t)dt<M and $$M$$ is a positive constant. $$(\mathcal{H}_2)$$ There exist a continuous differentiable function $$V(t,x_1),$$ class $$\mathcal{K}_\infty$$ functions $$\gamma_i,$$$$i=1,2,3$$ and a constant $$\lambda>0,$$ such that for all $$t\geq0$$ and all $$x_1\in\mathbb{R}^n,$$ we have γ1(‖x1‖)≤V(t,x1)≤γ2(‖x1‖),∂V∂t(t,x1)+∂V∂x1(t,x1)f1(t,x1)≤−λV(t,x1),‖∂V∂x1(t,x1)‖≤γ3(‖x1‖). $$(\mathcal{H}_3)$$ There exists a $$\mathcal{KL}$$-function $$\beta$$ such that the solutions of (1.6) satisfy ‖ϕ2(s,t,x2)‖≤β(‖x2‖,s−t),for alls≥tandx2∈Rm. $$(\mathcal{H}_4)$$ There exists a constant $$c>0,$$ such that ∫c+∞dsγ3(γ1−1(s))γ(γ1−1(s))=∞. We are now ready to present an auxiliary but fundamental result. Lemma 2.1 If assumptions $$(\mathcal{H}_1),$$$$(\mathcal{H}_2)$$ and $$(\mathcal{H}_3)$$ are satisfied and the solutions of (1.4) are globally uniformly bounded then (1.4) is GUAS. Proof. Since the solutions of (1.4) are globally uniformly bounded, then, there exist a $$\mathcal{K}_\infty$$ function $$\alpha$$ and a positive constant $$a$$ such that given any initial state $$x,$$ the solution of (1.4) satisfies ‖ϕ(s,t,x)‖≤a+α(‖x‖),for alls≥t. Let $$\phi_1(.,t,x)$$ and $$\phi_2(.,t,x_2),$$ respectively, be the solutions of x˙1=f1(t,x1)+g(t,x)x2 (2.1) and (1.6). The derivative of $$V$$ along the trajectories of system (2.1) is given by dds(V(s,ϕ1(s,t,x))) =∂V∂t(s,ϕ1(s,t,x))+∂V∂x1(s,ϕ1(s,t,x))f(s,ϕ1(s,t,x)) +∂V∂x1(s,ϕ1(s,t,x))g(s,ϕ(s,t,x))ϕ2(s,t,x2) ≤−λV(s,ϕ1(s,t,x))+γ3(‖ϕ1(s,t,x)‖)γ(‖ϕ1(s,t,x)‖) ×θ(‖ϕ2(s,t,x2)‖)‖ϕ2(s,t,x2)‖δ(s) ≤−λV(s,ϕ1(s,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,s−t)δ(s). Let v(s,ϕ1(s,t,x))=V(s,ϕ1(s,t,x))eλ(s−t). Then, dds(v(s,ϕ1(s,t,x)))≤γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,s−t)δ(s)eλ(s−t). Let $$t_0\geq t.$$ Integrating between $$t_0$$ and $$s,$$ one obtains for all $$s\geq t_0,$$ v(s,ϕ1(s,t,x)) ≤v(t0,ϕ1(t0,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0)) ×∫t0sδ(u)β(‖x2‖,u−t)eλ(u−t)du. (2.2) Moreover, v(s,ϕ1(s,t,x)) ≤v(t0,ϕ1(t0,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0)) ×β(‖x2‖,0)∫t0sδ(u)eλ(u−t)du. It follows that v(s,ϕ1(s,t,x)) ≤v(t0,ϕ1(t0,t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,0)Meλ(s−t). Hence, for all $$s\geq t_0,$$ we have V(s,ϕ1(s,t,x)) ≤V(t0,ϕ1(t0,t,x))e−λ(s−t0)+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,0)M. (2.3) Since $$t\geq t_0,$$ then, (2.3) implies that for all $$s\geq t,$$ V(s,ϕ1(s,t,x)) ≤V(t,x1)e−λ(s−t)+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,0)M. Thus, for all $$s\geq t,$$ we have V(s,ϕ1(s,t,x))≤γ2(‖x1‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,0)M. Since, $$\Vert x_1\Vert \leq \Vert x\Vert$$ and $$\Vert x_2\Vert \leq \Vert x\Vert,$$ then, for all $$s\geq t,$$ we have V(s,ϕ1(s,t,x))≤γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M (2.4) We deduce that, ‖ϕ1(s,t,x)‖ ≤γ1−1(γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M). As the map r↦γ1−1(γ2(r)+γ3(a+α(r))γ(a+α(r))θ(β(r,0))β(r,0)M), is a $$\mathcal{K}_\infty$$ function, then, the system (2.1) is GUS. On the other hand, we have $$\beta(\Vert x_2\Vert,s)\rightarrow 0$$ as $$s\rightarrow\infty.$$ Then, for each $$\varepsilon>0,$$ let $$T(\Vert x_2\Vert,\varepsilon)>0$$ such that γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,T(‖x2‖,ε))M≤ε2. Also, we can deduce from (2.2) that for all $$s\geq t+T(\Vert x_2\Vert,\varepsilon),$$ we have v(s,ϕ1(s,t,x)) ≤v(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))∫t+T(‖x2‖,ε)sδ(u)β(‖x2‖,T(‖x2‖,ε))eλ(u−t)du ≤v(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x))+γ3(a+α(‖x‖))γ(a+α(‖x‖)) ×θ(β(‖x2‖,0))β(‖x2‖,T(‖x2‖,ε))∫t+T(‖x2‖,ε)sδ(u)eλ(u−t)du. Which in turn implies that for all $$s\geq t+T(\Vert x_2\Vert,\varepsilon),$$ we have V(s,ϕ1(s,t,x)) ≤V(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x))e−λ(s−t−T(‖x2‖,ε)) +γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x2‖,0))β(‖x2‖,T(‖x2‖,ε))M. Since $$t+T(\Vert x_2\Vert,\varepsilon)>t,$$ then, using (2.4), we obtain V(t+T(‖x2‖,ε),ϕ1(t+T(‖x2‖,ε),t,x)) ≤γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0)) ×β(‖x‖,0)M. Therefore, V(s,ϕ1(s,t,x)) ≤(γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M) ×e−λ(s−t−T(‖x2‖,ε))+ε2. It follows that $$V(s,\phi_1(s,t,x))\leq\varepsilon,$$ for all $$s\geq t+\tilde{T}(\Vert x\Vert,\varepsilon)$$ with T~(‖x‖,ε) =T(‖x2‖,ε) +1λlog⁡(2(γ2(‖x‖)+γ3(a+α(‖x‖))γ(a+α(‖x‖))θ(β(‖x‖,0))β(‖x‖,0)M)ε). Finally, defining $$\tilde{\varepsilon}=\gamma_1^{-1}(\varepsilon),$$ we conclude that $$\Vert \phi_1(s,t,x)\Vert \leq \tilde{\varepsilon},$$ for all $$s\geq t+\tilde{T}(\Vert x\Vert,\varepsilon).$$ Thus, the system (2.1) is GUAS. And by observing that system (1.6) is GUAS, we deduce that system (1.4) is GUAS. □ From Lemma 2.1, it is sufficient to show that the solutions of the system (2.1) are globally uniformly bounded to obtain the GUAS of (1.4). To achieve our goal, we present the following proposition given in Benabdallah et al. (2007). Proposition 2.1 If there exists a continuous differentiable function $$V:\mathbb{R}_+\times\mathbb{R}^n\rightarrow\mathbb{R}^n,$$ and class $$\mathcal{K}_\infty$$ functions $$\alpha_i,$$$$i=1,2,3,4,$$ such that for all $$t\geq0$$ and all $$x\in\mathbb{R}^n$$ α1(‖x‖)≤V(t,x)≤α2(‖x‖),∂V∂t(t,x)+∂V∂x(t,x)f(t,x)≤−α3(‖x‖),‖∂V∂x(t,x)‖≤α4(‖x‖), there exist an integrable continuous function $$\rho:\mathbb{R}_+\rightarrow\mathbb{R}_+,$$ and a class $$\mathcal{K}$$ function $$\alpha,$$ such that ‖g(t,x)‖≤ρ(t)α(‖x‖) for all $$t\geq0$$ and all $$x\in\mathbb{R}^n$$ and there exist a constant $$c>0,$$ such that ∫c+∞dsα4(α1−1(s))α(α1−1(s))=∞, then the solution of (1.8) is globally uniformly bounded. Now, one has the following theorem. Theorem 2.1 Under assumptions $$(\mathcal{H}_1),$$$$(\mathcal{H}_2),$$$$(\mathcal{H}_3)$$ and $$(\mathcal{H}_4),$$ system (1.4) is GUAS. Proof. Since the component $$\phi_2(.,t,x)$$ is globally uniformly bounded for all $$s\geq t,$$ then by using Lemma 2.1, it suffices to prove that the component $$\phi_1(.,t,x)$$ of (1.4) is globally uniformly bounded. Since, ‖g(t,x)x2‖≤δ(t)θ(‖x2‖)γ(‖x1‖), then, conditions of Proposition 2.1 are satisfied with $$\rho=\delta,$$$$\alpha_1=\gamma_1,$$$$\alpha_2=\gamma_2,$$$$\alpha_3=-\lambda\gamma_1,$$$$\alpha_4=\gamma_3,$$ and α(‖x‖)=θ(β(‖x2‖,0))γ(‖x1‖)β(‖x2‖,0)∈K∞. Therefore, due to Proposition 2.1, the component $$\phi_1(.,t,x)$$ of (1.4) is globally uniformly bounded. Thus, the system (1.4) is GUAS. □ Remark 2.1 We note that in Panteley & Loria (2001), an integrability condition on the state $$x_2$$ was used to establish the GUAS of the cascade system. Here, the integrability condition is imposed on the function $$\delta$$ that bounds $$g(t,x)$$ and is not imposed on the state of the system. This point can be seen in the following example. Example 2.1 Consider the following system {x˙1=−x1+δ(t)x1log⁡(|x1|+1)x2x˙2=−x23, (2.5) where $$x_1\in\mathbb{R},$$ the function $$\delta$$ is a positive continuous unbounded integrable function. Let V(t,x1)=x12 as a Lyapunov function for the nominal system $$\dot{x}_1=-x_1.$$ One can see that assumption $$(\mathcal{H}_2)$$ is satisfied with $$\gamma_1(r)=r^2,$$$$\gamma_2(r)=2r^2$$ and $$\gamma_3(r)=r.$$ Thus, system $$\dot{x}_1=-x_1$$ is GUAS. Let g(t,x1,x2)=δ(t)x1log⁡(|x1|+1)x2, which satisfies assumption $$(\mathcal{H}_1)$$ with $$\gamma(r )=r\log(r+1)$$ and $$\theta(r)=r.$$ Also, we have ∫1+∞drγ3(γ1−1(r))γ(γ1−1(r))=∫1+∞dsrln⁡(r+1)=∞. It follows that, assumption $$(\mathcal{H}_4)$$ is satisfied. On the one hand, the solution of $$\dot{x}_2=-x^3_2,$$$$\phi_2(s,t,x_2)=(2s+{1\over x^2_2})^{-{1\over 2}}$$ satisfies assumption $$(\mathcal{H}_3)$$ with $$\beta(r,s)=\displaystyle {r\over \sqrt{2sr^2+1}}.$$ Therefore, conditions of Theorem 2.1 are satisfied. Thus, the system (2.5) is GUAS. On the other hand, the solution $$\phi_2(s,t,x_2)=(2s+{1\over x^2_2})^{-{1\over 2}}$$ does not satisfy the assumption $$\mathcal{A}_6$$ in Panteley & Loria (2001) i.e., there exists $$\alpha\in\mathcal{K}$$ such that, the trajectory of (1.6), satisfies ∫t+∞‖ϕ2(s,t,x2)‖ds≤α(‖x2‖). Indeed, ∫t+∞‖ϕ2(s,t,x2)‖ds=∞. It means that, we cannot show the GUAS of (2.5) using Panteley & Loria (2001). Now, the question which can be addressed is: is there a Lyapunov function that achieves the GUAS of (1.4), without the boundedness hypothesis with respect to time? This question leads us to study the perturbed system (1.1). 2.2. Construction of a Lyapunov function for perturbed systems Now, let us return to the system (1.1). In this section, we are going to construct a Lyapunov function that achieves the GUAS of (1.1) without the boundedness hypothesis with respect to time. In their article, Jankovic et al. (1996), have constructed a Lyapunov function for autonomous cascaded nonlinear system having the form {x˙=f(x)+h(x,ξ)ξ˙=a(ξ), where $$\dot{x}=f(x)$$ is assumed to be GAS with a known Lyapunov function $$V,$$ and $$\dot{\xi}=a(\xi)$$ is GAS and locally exponentially stable (LES) with Lyapunov function $$U.$$ Under a linear growth assumption on the function $$h$$ and some conditions taken on $$V$$ and $$\displaystyle{\partial V \over \partial x},$$ the considered Lyapunov function has the form W(x,ξ)=V(x)+ψ(x,ξ)+U(ξ), where the cross term $$\psi$$ must guarantee that $$V$$ is nonincreasing along the solutions of the cascaded system. The authors in Benabdallah et al. (2007) use the same idea as in Jankovic et al. (1996) for the perturbed systems of the form (1.1) and they have considered W(t,x)=V(t,x)+ψ(t,x) to ensure the GUAS of (1.1) under the hypothesis is that the dynamics of the system is bounded in time and other restrictions (Benabdallah et al. (2007), Theorem 2). Here, we are going to use almost the same idea as in Benabdallah et al. (2007) and Jankovic et al. (1996) to construct a new Lyapunov function for (1.1) that ensures GUAS of the equilibrium point, without the boundedness hypothesis with respect to time. Let us consider a Lyapunov function for (1.1) of the form W(t,x)={V(t,x)exp⁡(φ(t,x))ifx≠00otherwise. (2.6) Our goal is to seek a suitable function $$\varphi$$ which can compensate the perturbation term. Therefore, if we consider the derivative of $$W$$ along the trajectories of the system (1.1), we get W˙(t,x) =dds(W(s,ϕ(s,t,x)))/s=t =[∂V∂t(t,x)+∂V∂x(t,x)f(t,x)]exp⁡(φ(t,x)) +∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x)). The first term of the right-hand side constitutes the derivative of $$W(t,x)$$ along the trajectories of the nominal system. The second term is the effect of the perturbation, while the third term is the derivative of $$\exp\left(\varphi(t,x)\right)$$ multiplied by $$V(t,x).$$ In order to guarantee that $$\dot{W}(t,x)$$ is a negative definite function, we shall choose φ(t,x)=∫t+∞1V(s,ϕ(s,t,x))∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x))ds. (2.7) Thus one can verify that ∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x))=0,for all(t,x)∈[0,+∞[×Rn∖{0}. This implies with this choice that W˙(t,x)=[∂V∂t(t,x)+∂V∂x(t,x)f(t,x)]exp⁡(φ(t,x)). This equality will now be used for the analysis of (1.1). To this end, we must impose some conditions in view to prove that the new Lyapunov function is continuous positive definite radially unbounded and decreasing along the trajectories of solutions of (1.1). Hence, let us consider the following assumptions. $$(\mathcal{H}_5)$$ There exist a continuous differentiable function $$V(t,x),$$ class $$\mathcal{K}_\infty$$ functions $$\xi_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x\in\mathbb{R}^n,$$ we have ξ1(‖x‖)≤V(t,x)≤ξ2(‖x‖),∂V∂t(t,x)+∂V∂x(t,x)f(t,x)≤−ξ3(‖x‖),‖∂V∂x(t,x)‖≤ξ4(‖x‖). $$(\mathcal{H}_6)$$ There exist an integrable and continuous function $$\rho:\mathbb{R}_+\rightarrow \mathbb{R}_+$$ and a class $$\mathcal{K}_\infty,$$ function $$\vartheta$$ such that for all $$t\geq0$$ and $$x\in\mathbb{R}^n,$$ we have −ρ(t)V(t,x)≤∂V∂x(t,x)g(t,x)≤ρ(t)ϑ(‖x‖)V(t,x), where ∫0+∞ρ(t)dt<Mρ, with $$M_\rho$$ is a positive constant. $$(\mathcal{H}_7)$$ There exists a constant $$c>0$$ such that ∫c+∞dsξ4(ξ1−1(s))ϑ(ξ1−1(s))=∞. Then, we have the following Theorem. Theorem 2.2 If assumptions $$(\mathcal{H}_5),$$$$(\mathcal{H}_6)$$ and $$(\mathcal{H}_7)$$ are satisfied then, system (1.1) is globally uniformly asymptotically stabile. Proof. Let consider the function $$W(t,x)$$ defined in (2.6). Since, $$\vartheta$$ is a class $$\mathcal{K}_\infty$$ function, there exists a positive constant $$r$$ such that ϑ(‖x‖)>1,for allx∈Brc. It follows that |∂V∂x(t,x)g(t,x)|≤ρ(t)ϑ(‖x‖)V(t,x),for allx∈Brc. Therefore, by assumptions $$(\mathcal{H}_5)$$ and $$(\mathcal{H}_7),$$ we deduce as in Benabdallah et al. (2007), Proposition 1), that there exist a class $$\mathcal{K}_\infty$$ function and a positive constant $$a,$$ such that for all $$s\geq t\geq0,$$ we have ‖ϕ(s,t,x)‖≤a+α(‖x‖),for allx∈Brc. (2.8) Now, if $$x\in\mathcal{B}^*_r$$ and we calculate the derivative of $$V$$ along the trajectories of the system (1.1), one can reach the following result: ‖ϕ(s,t,x)‖≤ξ1−1(ξ2(‖x‖)eMρ),for allx∈Br∗. (2.9) Let, for all $$(t,x) \in [0,+\infty[\times\mathbb{R}^n\backslash\{0\},$$ and all $$s\geq t \geq 0,$$ denote ψ(s,t,x)=1V(s,ϕ(s,t,x))∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x)). Since, −ρ(s)V(s,ϕ(s,t,x))≤∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x))≤ρ(s)ϑ(‖ϕ(s,t,x)‖)V(s,ϕ(s,t,x)), we have the following two cases: Case 1: $$x\in\mathcal{B}^c_r.$$ |ψ(s,t,x)| ≤max(ϑ(‖ϕ(s,t,x)‖),1)ρ(s). Using (2.8), we obtain |ψ(s,t,x)| ≤max(ϑ(a+α(‖x‖)),1)ρ(s)∈L1([0,+∞[). This implies that the integral defined in (2.7) exists. Then, ξ1(‖x‖)e−Mρ≤W(t,x)≤ξ2(‖x‖)exp⁡(ϑ(a+α(‖x‖))Mρ) for all t∈[0,+∞[. Case 2: $$x\in\mathcal{B}^*_r.$$ |ψ(s,t,x)| ≤max(ϑ(‖ϕ(s,t,x)‖),1)ρ(s). Using (2.9), we obtain |ψ(s,t,x)| ≤max(ϑ(ξ1−1(ξ2(‖x‖)eMρ)),1)ρ(s)∈L1([0,+∞[). Which implies also that the integral defined in (2.7) exists. Then, ξ1(‖x‖)e−Mρ≤W(t,x)≤ξ2(‖x‖)exp⁡(ϑ(ξ1−1(ξ2(‖x‖)eMρ))Mρ) for all t∈[0,+∞[. Hence, $$W$$ is positive definite and radially unbounded. The continuity of $$\varphi(t,x)$$ can be shown by observing that, for all $$s\geq t\geq0,$$ the function (t,x)⟼ψ(s,t,x) is continuous on $$[0,+\infty[\times\mathbb{R}^n\backslash\{0\},$$ and by using the fact that for each compact set $$K\in \mathbb{R}^n\backslash\{0\},$$ and for all $$(t,x)\in [0,+\infty[\times K,$$$$s\geq t\geq0$$ the following holds |ψ(s,t,x)|≤max(MK,1)ρ(s)∈L1([0,+∞[), where $$M_K$$ is a positive constant which depends only on $$K.$$ Now, to prove that $$W$$ is continuous, it suffices to show that $$W$$ is continuous on the following set S={(t,x)∈[0,+∞[×Rn/x=0}. We have for all $$x\in\mathcal{B}^*_r,$$ W(t,x)≤ξ2(‖x‖)exp⁡(ϑ(ξ1−1(ξ2(‖x‖)eMρ))Mρ), hence lim(t,‖x‖)→(t,0)W(t,x)=0, which implies that $$W$$ is continuous on $$\mathcal{S}.$$ Now, to prove that the derivative of $$\varphi$$ along the trajectories of (1.1) exists, it suffices to use that, for all $$s\geq t\geq0$$ φ(s,ϕ(s,t,x))=∫s+∞1V(u,ϕ(u,s,ϕ(s,t,x)))∂V∂x(u,ϕ(u,s,ϕ(s,t,x)))g(u,ϕ(u,s,ϕ(s,t,x)))du. Since the following two solutions of (1.1): u⟼ϕ(u,t,x) and ϕ(u,s,ϕ(s,t,x)) are equal at the time $$u=s,$$ ϕ(u,t,x)=ϕ(u,s,ϕ(s,t,x)) for alls≥t≥0. Thus, φ(s,ϕ(s,t,x))=∫s+∞1V(u,ϕ(u,t,x))∂V∂x(u,ϕ(u,t,x))g(u,ϕ(u,t,x))du. This implies that the derivative of $$\varphi(t,x)$$ along the trajectories of (1.1) exists and it is given by φ˙(t,x) =dds(φ(s,ϕ(s,t,x)))/s=t =−1V(t,x)∂V∂x(t,x)g(t,x). Hence, the derivative of $$W(t,x)$$ along the trajectories of system (1.1) exists and is given by W˙(t,x) =dds(W(s,ϕ(s,t,x)))/s=t =V˙(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x)) =[∂V∂t(t,x)+∂V∂x(t,x)f(t,x)]exp⁡(φ(t,x)) +∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))+φ˙(t,x)V(t,x)exp⁡(φ(t,x)) ≤−ξ3(‖x‖)exp⁡(φ(t,x))+∂V∂x(t,x)g(t,x)exp⁡(φ(t,x))−∂V∂x(t,x)g(t,x)exp⁡(φ(t,x)) ≤−ξ3(‖x‖)e−Mρ, for all t≥0 and x∈Rn∖{0}. Therefore, $$W$$ is decreasing along the trajectories of solutions of (1.1). Thus, system (1.1) is GUAS. □ Remark 2.2 The above theorem covers a larger class of perturbed systems than those in Benabdallah et al. (2007), Theorem $$2).$$ Indeed, in this reference (Benabdallah et al. 2007), the authors prove the GUAS of (1.1), with the $$L_\infty$$ condition under the following restrictions: $$(\mathcal{R}_1)$$ The integrable function $$\rho$$ is bounded by a positive constant $$\bar{\rho}.$$ $$(\mathcal{R}_2)$$ There exists a nondecreasing function $$G$$ such that, for all $$t\geq0$$ and $$x\in\mathbb{R}^n,$$ ‖f(t,x)‖≤G(‖x‖) and the function $$G$$ satisfies another condition. $$(\mathcal{R}_3)$$ There exists a class $$\mathcal{K}$$ function $$\xi_5,$$ such that |∫t+∞∂V∂x(s,ϕ(s,t,x))g(s,ϕ(s,t,x))ds|≤ξ5(‖x‖). Remark 2.3 Now, let us return to the differential equation (1.3) given in Example 1.1. It is clear that assumptions $$(\mathcal{H}_5)$$ and $$(\mathcal{H}_7)$$ are satisfied with $$\xi_{i}(r)=d_ir^2,$$$$d_i>0,$$$$i=1,2;$$$$\xi_{3}(r)=c_1r^2,$$$$\xi_4(r)=2r$$ and $$\vartheta(r)=r.$$ Also, we have −ρ(t)V(t,x)≤∂V∂x(t,x)g(t,x) =2δ(t)x2|x|1+|x|12 ≤2δ(t)|x|V(t,x). It follows that, assumption $$(\mathcal{H}_{6})$$ is satisfied with $$\rho(t)=2\displaystyle\delta(t)\in L^1([0,+\infty[).$$ Then, W(t,x)={x2exp⁡(∫t+∞2δ(s)|ϕ(s,t,x)|ds),ifx≠00otherwise. guarantees the GUAS of (1.3). Next, we give a mechanical example which represents a nonlinear mass-spring-damper system, see Slotine & Li (1991). Example 2.2 Consider the following system q¨+c(t)q˙+k0q=0 (2.10) which represents a nonlinear mass-spring-damper system. Where $$c(t)$$ is a time-varying damping coefficient, and $$k_o$$ is a spring constant. The variable $$q\in\mathbb{R}$$ represents the position of the mass with respect to its rest position. We use the notation $$\dot{q}$$ to denote the derivative of $$q$$ with respect to time (i.e., the velocity of the mass) and $$\ddot{q}$$ to represent the second derivative (acceleration). Such a model is natural to use for celestial mechanics, because it is difficult to influence the motion of the planets. In many examples, it is useful to model the effects of external disturbances or controlled forces on the system. One way to capture this is to replace equation (2.10) by q¨+c(t)q˙+k0q=u (2.11) where $$u$$ represents the effect of external influences. The model (2.11) is called a forced or controlled differential equation. It implies that the rate of change of the state can be influenced by the input $$u.$$ Adding the input makes the model richer and allows new questions to be posed. For example, we can examine what influence external disturbances have on the trajectories of a system. Or, in the case when the input variable is something that can be modulated in a controlled way, we can analyse whether it is possible to steer the system from one point in the state space to another through proper choice of the input. Let $$\dot{q}=x.$$ Then system (2.11) can be rewritten as {x˙=−c(t)x−k0q+u(t,q˙,q)q˙=x (2.12) The system has the form of (1.1) with f(t,x,q)=[f1(t,x,q)f2(t,x,q)]=[−c(t)x−k0q0] and g(t,x,q)=[g1(t,x,q)g2(t,x,q)]=[u(t,x,q)0] Let V(t,x,q)=12(x+αq)2+12b(t)q2, as a Lyapunov function for the nominal system, with $$0<\alpha<\min(1,k_0^{1 \over 2}),$$$$b(t)=k_0-\alpha^2+\alpha c(t),$$$$\alpha<c(t)<\beta$$ and $$\dot{c}(t)\leq\gamma<2k_0,$$ and let u(t,x,q)={12ρ(t)(x+αq)3+(x+αq)b(t)q2(x2+q2)12if(x,q)≠(0,0)0otherwise. Where $$\rho$$ is a positive continuous integrable unbounded function, i.e, $$\rho$$ is $$L^1([0,+\infty[)$$ but not $$L^\infty([0,+\infty[).$$ For instance, one can take $$\rho$$ as follows: ρ(t)={0ift∈[0,2−18]n4t+(n−n5)ift∈[n−1n3,n], n≥2−n4t+(n+n5)ift∈[n,n+1n3], n≥20ift∈[n+1n3,(n+1)−1(n+1)3], n≥2. One can see that assumptions $$(\mathcal{H}_5),$$$$(\mathcal{H}_6)$$ and $$(\mathcal{H}_7),$$ are satisfied with ξ1(r)=max((1+α),(α+k0+αβ))r2,ξ2(r)=min(12(1−α),12(−α+k0+α2))r2,ξ3(r)=min(2α,αk0−αγ2)r2,ξ4(r)=(max(1+α(1+k0)+α2(1+β),(1+α)(α+αβ+k0)))12r and ϑ(r)=max(1+α,2α)r. Then, for all $$(t,x,q)\in \mathcal{S}=\{(t,x,q)\backslash (x,q)\neq0\},$$ one has φ(t,x,q) =∫t+∞max(1+α,2α)(ϕ12(s,t,x,q)+ϕ12(s,t,x,q))12ds. It follows that system (2.11) is GUAS with W(t,x,q)={V(t,x,q)exp⁡(φ(t,x,q)),if(t,x,q)∈S0otherwise. as a Lyapunov function candidate. Also, an interesting application of this new Lyapunov function can be seen for system (1.4). Then, let us consider the following assumptions. $$(\mathcal{H}_8)$$ There exist a continuous differentiable function $$V_1(t,x_1),$$ class $$\mathcal{K}_\infty$$ functions $$\xi_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x_1\in\mathbb{R}^n,$$ we have ξ1(‖x1‖)≤V1(t,x1)≤ξ2(‖x1‖),∂V1∂t(t,x1)+∂V1∂x1(t,x)f1(t,x1)≤−ξ3(‖x1‖),‖∂V1∂x1(t,x1)‖≤ξ4(‖x1‖). $$(\mathcal{H}_9)$$ There exist a continuous differentiable function $$V_2(t,x_2),$$ class $$\mathcal{K}_\infty$$ functions $$\tilde{\xi}_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x_2\in\mathbb{R}^m,$$ we have ξ~1(‖x2‖)≤V2(t,x2)≤ξ~2(‖x2‖),∂V2∂t(t,x2)+∂V2∂x2(t,x2)f2(t,x2)≤−ξ~3(‖x2‖),‖∂V2∂x2(t,x2)‖≤ξ~4(‖x2‖). Assumptions $$(\mathcal{H}_8)$$ and $$(\mathcal{H}_9)$$ assert the GUAS of the subsystems (1.7) and (1.6). $$(\mathcal{H}_{10})$$ There exist an integrable and continuous function $$\delta:\mathbb{R}_+\rightarrow \mathbb{R}_+$$ and a class $$\mathcal{K}_\infty$$ function $$\vartheta,$$ such that for all $$t\geq0$$ and $$x_1\in\mathbb{R}^n,$$ we have −δ(t)V1(t,x1)≤∂V1∂x1(t,x1)g(t,x)x2≤δ(t)ϑ(‖x1‖)V1(t,x1), where ∫0+∞δ(t)dt<Mδ, with $$M_\delta$$ is a positive constant. Then, one has the following corollary. Corollary 2.1 If assumptions $$(\mathcal{H}_7),$$$$(\mathcal{H}_8),$$$$(\mathcal{H}_9)$$ and $$(\mathcal{H}_{10})$$ are satisfied, then, system (1.4) is GUAS with W~(t,x)={W(t,x1)+V2(t,x2)ifx1≠00otherwise, (2.13) where W(t,x1)=V1(t,x1)exp⁡(∫t+∞1V1(s,ϕ1(s,t,x))∂V1∂x1(τ,ϕ1(s,t,x))g(s,ϕ(s,t,x))ϕ2(s,t,x2)ds), as a Lyapunov function candidate. Proof. Under assumptions $$(\mathcal{H}_7),$$$$(\mathcal{H}_8)$$ and $$(\mathcal{H}_{10})$$ and using Theorem 2.2, there exist class $$\mathcal{K}_\infty$$ functions $$\sigma_i,$$$$i=1,2,3,4,$$ such that, for all $$t\geq0$$ and all $$x_1\in\mathbb{R}^n,$$ we have σ1(‖x1‖)≤W(t,x1)≤σ2(‖x1‖),W˙(t,x1)=dds(W(s,ϕ(s,t,x)))/s=t≤−σ3(‖x1‖),‖∂W∂x1(t,x1)‖≤σ4(‖x1‖). Using assumption $$(\mathcal{H}_9),$$$$\tilde{W}(t,x)$$ satisfies the following inequalities for all $$x\in\mathbb{R}^{n}\times \mathbb{R}^{m},$$ σ1(‖x1‖)+ξ~1(‖x2‖)≤W~(t,x)≤σ2(‖x1‖)+ξ~2(‖x2‖),W~˙(t,x)=dds(W~(s,ϕ(s,t,x)))/s=t≤−σ3(‖x1‖)−ξ~2(‖x3‖),‖∂W~∂x(t,x)‖≤σ4(‖x1‖)+ξ~4(‖x2‖). Hence, (1.4) is GUAS. □ Remark 2.4 The difference between the assumptions Theorem 2.1 and those of corollary 2.1 for the case (1.4) can be summarized in two essential points. On the one hand, we have assumed in the hypothesis $$(\mathcal{H}_2)$$ of Theorem 2.1 that the derivative of the Lyapunov function $$V$$ associated to the nominal system (1.7) satisfies $$\dot{V} (t, x_1) \leq-\lambda V(t,x_1),$$$$\lambda>0.$$ Using the fact that $$V$$ satisfies γ1(‖x1‖)≤V(t,x1), then, V˙(t,x1)≤−λγ1(‖x1‖). Which can be considered as a special case of hypothesis $$(\mathcal{H}_8)$$ of Corollary 2.1. It suffices to take $$\xi=\lambda \gamma_1\in \mathcal{K}_\infty.$$ On the other hand, assumption $$(\mathcal{H}_1)$$ of Theorem 2.1 gives a wide class of systems which contain a term of perturbation. These points can be seen in the following example. Let us consider the following two-dimensional cascaded system. Example 2.3 {x˙1=−x11+|x1|+ρ(t)x2x1|x1|1+x22x2x˙2=−x2(1+e−t) (2.14) where $$(x_1,x_2)\in\mathbb{R}^2$$ and $$\rho$$ is a positive continuous integrable unbounded function. This system has the form of (1.4) with f1(t,x1)=−x11+|x1|, f2(t,x2)=−x2(1+e−t) and g(t,x)=ρ(t)x2x1|x1|1+x22. Let $$V_1(t,x_1)=x_{1}^2$$ and $$V_2(t,x_2)=x_{2}^2,$$ respectively, be the Lyapunov functions for x˙1=f1(t,x1) (2.15) and x˙2=f2(t,x2). (2.16) Their derivative along the trajectories, respectively, of (2.15) and (2.16) are given by V˙1(t,x1) =−x121+|x1| and V˙2(t,x2) =−x22(1+e−t), It is clear that assumptions $$(\mathcal{H}_7),$$$$(\mathcal{H}_8)$$ and $$(\mathcal{H}_9),$$ are satisfied with $$\xi_{i}(r)=c_ir^2,$$$$\tilde{\xi}_{i}(r)=\tilde{c}_ir^2,$$$$c_i, \tilde{c}_i>0,$$$$i=1,2;$$$$\xi_{3}(r)={r^2\over 1+r},$$$$\tilde{\xi}_{3}(r)=r^2,$$$$\xi_4(r)=\tilde{\xi}_4(r)=2r$$ and $$\vartheta(r)=r.$$ Which implies that the systems (2.15) and (2.16), are GUAS. Also, we have 0≤∂V1∂x1(t,x1)g(t,x)x2 =2ρ(t)x12|x1|1+x22x22 ≤2ρ(t)|x1|V1(t,x1). It follows that, assumption $$(\mathcal{H}_{10})$$ is satisfied with $$\delta(t)=2\rho(t)\in L^1([0,+\infty[).$$ Then, system (2.14) is GUAS with W~(t,x)={W(t,x1)+V2(t,x2)ifx1≠00otherwise, where W(t,x1)=V1(t,x1)exp⁡(∫t+∞δ(s)|ϕ1(s,t,x)|ϕ22(s,t,x2)1+ϕ2(s,t,x2)2ds), as a Lyapunov function candidate. However, one can see that here the assumption $$(\mathcal{H}_2)$$ of Theorem 2.1 is not satisfied. In fact, for all $$x_1\in\mathbb{R},$$ we have $$0<\displaystyle{1\over 1+\vert x_1\vert}\leq1.$$ Then, there is no $$\lambda>0$$ such that $$\dot{V}_1(t,x_1)\leq -\lambda V(t,x_1).$$ Thus, we cannot show the GUAS of (2.14) by using Theorem 2.1. Remark 2.5 Lyapunov technique remains a very important tool for studying the stability of dynamic systems such as robotic systems (Chen et al., 2013a; Chen, 2014) and choatic systems (Chen et al., 2013b). Thus, the built lyapunov function can be a good tool to show stability and stabilization of these systems. 3. Conclusion In this article, we dealt with the analysis problem of nonlinear time-varying perturbed systems without the boundedness hypothesis with respect to time. Indeed, we established the GUAS for general classes of perturbed systems by using a new Lyapunov function. In addition, an interesting application of this new Lyapunov function is given for cascaded time-varying systems. As example, we applied the present results to a nonlinear mass-spring-damper system with a disturbance. Acknowledgements The authors wish to thank the reviewers for their valuable and careful comments. References Benabdallah A. , Dlala M. & Hammami M. A. ( 2007 ) A new Lyapunov function for stability of time-varying nonlinear perturbed systems . Syst. Control Lett . 56 , 179 – 187 . Google Scholar Crossref Search ADS Benabdallah A. , Ellouze I. & Hammami M. A. ( 2009 ) Practical stability of nonlinear time varying cascade systems , J. Dyn. Control Syst. , 15 , 45 – 62 . Google Scholar Crossref Search ADS Chaillet A. & Loria A. ( 2007 ) Necessary and sufficient conditions for uniform semi-global practical asymptotic stability: application to cascaded systems , Automatica 56 , 179 – 187 . Chen H. ( 2014 ) Robust stabilization for a class of dynamic feedback uncertain nonholonomic mobile robots with input saturation , Int. J. Control, Autom. Syst . 12 , 1216 – 1224 . Google Scholar Crossref Search ADS Chen H. , Zhang J. , Chen B. & Baoujun L. ( 2013 ) Global practical stabilization for non-holonomic mobile robots with uncalibrated visual parameters by using a switching controller , IMA J Math. Control Info . 30 , 543 – 557 . Google Scholar Crossref Search ADS Chen H. , Chen W. , Zhang B. , Chen B. & Haitao C. ( 2013 ) Robust synchronization of incommensurate fractional-order chaotic systems via second-order sliding mode technique , J. Appl. Math. , http://dx.doi.org/10.1155/2013/321253. Dlala M. & Hammami M. A. ( 2007 ) Uniform exponential practical stability of impulsive perturbed systems , J. Dyn. Control Syst . 13 , 373 – 386 . Google Scholar Crossref Search ADS Grammel G. ( 2001 ) Exponential stability via the averaged system . Dyn. Control Syst . 7 , 327 – 338 . Google Scholar Crossref Search ADS Hahn W. ( 1967 ) Stability of Motion , Berlin : Springer . Jankovic M. , Sepulchre R. & Kokotovic P. V. ( 1996 ) Constructive Lyapunov stabilisation of nonlinear cascade systems , IEEE Trans. Autom. Control 41 , 1723 – 1735 . Google Scholar Crossref Search ADS Karafyllis I. & Tsinias J. ( 2003 ) Non-uniform in time stabilization for linear systems and traking control for non-holonomic systems in chained form , Int. J. Control 76 , 2003 : 1536 – 1546 . Google Scholar Crossref Search ADS Karafyllis I. & Tsinias J. ( 2009 ) Control Lyapunov functions and stabilization by means of continuous time-varying feedback , ESAIM-COCV 15 , 599 – 625 . Google Scholar Crossref Search ADS Khalil H. ( 2002 ) Nonlinear Systems , 3rd edn . Englewood Cliffs, NJ : Prentice-Hall . Lin Y. , Sontag E. & Wang Y. ( 1996 ) A smooth converse Lyapunov theorem for robust stability , SIAM J. Control Optim . 34 , 124 – 160 . Google Scholar Crossref Search ADS Malisoff M. & Mazenc F. ( 2008 ) Constructions of Strict Lyapunov functions for discrete time and hybrid time-varying systems , Nonlinear Anal.: Hybrid Syst . 2 , 394 – 407 . Google Scholar Crossref Search ADS Mazenc F. , Sepulchre R. & Jankovic M. ( 1999 ) Lyapunov functions for stable cascades and applications to stabilisation , IEEE Transcations on Automatic Control 44 , 1795 – 1800 . Google Scholar Crossref Search ADS Mazenc F. & Nesic D. ( 2007 ) Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem , Math. Control Signals Syst . 19 , 151 – 182 . Google Scholar Crossref Search ADS Mazenc F. , Malisoff M. & de Queiroz M. ( 2011 ) Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains , Nonlinear Anal . 74 , 1132 – 1145 . Google Scholar Crossref Search ADS Panteley E. & Loria A. ( 1998 ) On global uniform asymptotic stability of nonlinear time-varying non autonomous systems in cascade , Syst. Control Lett . 33 , 131 – 138 . Google Scholar Crossref Search ADS Panteley E. & Loria A. ( 2001 ) Growth rate conditions for uniform asymptotic stability of cascaded time-varying systems , Automatica 37 , 453 – 460 . Google Scholar Crossref Search ADS Seibert P. & Suarez R. ( 1990 ) Global stabilisation of nonlinear cascade systems , Syst. Control Lett . 14 , 347 – 352 . Google Scholar Crossref Search ADS Slotine J. & Li W. ( 1991 ) Applied nonlinear control , Prentice-Hall , Englewood Cliffs, NJ . Utkin V. ( 2015 ) Mechanical energy-based Lyapunov function design for twisting and super-twisting sliding mode control , IMA J. Math. Control Inf . 32 , 675 – 688 . Weng F. & Mao W. ( 2013 ) Robust stability and stabilization of uncertain discrete singular time-delay systems based on PNP Lyapunov functional , IMA J. Math. Control Inf. , 30 , 301 – 314 . Google Scholar Crossref Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Journal

IMA Journal of Mathematical Control and InformationOxford University Press

Published: Sep 21, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off