STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS DEPENDING ON A PARAMETER

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1 Bull. Korean Math. Soc ), No. 4, pp pissn: / eissn: STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS DEPENDING ON A PARAMETER Abdellatif Ben Makhlouf, Mohamed Ali Hammami, and Khaled Sioud Abstract. In this paper, we present a practical Mittag Leffler stability for fractional-order nonlinear systems depending on a parameter. A sufficient condition on practical Mittag Leffler stability is given by using a Lyapunov function. In addition, we study the problem of stability and stabilization for some classes of fractional-order systems. 1. Introduction The fractional order dynamical systems have attracted remarkable attention in the last decade. Many dynamic systems are better characterized by a dynamic model of fractional order, usually based on the concept of differentiation or integration of fractional order. It is usually known that several physical systems are characterized by fractional-order state equations [16]), such as, fractional Langevin equation [3]), fractional Lotka-Volterra equation [1]) in biological systems, fractional-order oscillator equation [29]) in damping vibration, in anomalous diffusion and so on. Particularly, stability analysis is one of the most fundamental issues for systems. In this few years, there are many results about the stability and stabilization of fractional order systems [4, 5, 6, 7, 11, 12, 13, 17, 18, 21, 23, 25, 27, 3]). The problem of stability and stabilization for nonlinear integer-order dynamic systems is many studied in literature [2, 9, 15, 2, 22, 28, 31]). One type of stability studied, deals with the so called practical stability, this notion was studied in [1, 14]). In the present paper, we introduce the notion of practical stability of nonlinear fractional-order systems depending on a parameter, called ε practical Mittag Leffler stability. This stability ensures the Mittag Leffler stability of a ball containing the origin of the state space, the radius of the ball can be made arbitrarily small. Our goal is to investigate the practical Mittag Leffler stability of nonlinear fractional-order systems depending on Received July 3, 216; Accepted November 29, Mathematics Subject Classification. Primary 37C75, 93Dxx, 49J15. Key words and phrases. practical stability, Lyapunov functions, fractional differential equations, Caputo derivative. 139 c 217 Korean Mathematical Society

2 131 A. BEN MAKHLOUF, M. A. HAMMAMI, AND K. SIOUD a parameter by using the Lyapunov techniques. Precisely we give sufficient conditions that ensures the practical Mittag Leffler stability of such systems. The paper is organised as follows. In Section 2, some necessary definitions and lemmas are presented. In Section 3 a sufficient condition on practical Mittag Leffler stability of nonlinear fractional differential equations is given. In addition, some other classes of perturbed fractional systems are studied in point of view stability and a continuous feedback controller is proposed to stabilize a large class of nonlinear fractional dynamical systems with uncertainties. 2. Preliminaries In this section, some definitions, lemmas and theorems related to the fractional calculus are given. In the literature, there are many definitions for fractional derivative [17, 26]). In this paper we adopt the definition of Caputo fractional derivative. Definition. The Riemann-Liouville fractional integral of order α > is defined as, It α xt) = 1 Γα) Γα) = + t t τ) α 1 xτ)dτ, e t t α 1 dt, where Γ is the Gamma function generalizing factorial for non-integer arguments. Definition. The Caputo fractional derivative is defined as, C D α t,txt) = 1 Γm α) t t τ) m α 1 dm dτmxτ)dτ, m 1 < α < m). When < α < 1, then the Caputo fractional derivative of order α of f reduces to 2.1) C Dt α 1 t,txt) = t τ) α d Γ1 α) t dτ xτ)dτ. On the other hand, there exists a frequently used function in the solution of fractional order systems named the Mittag Leffler function. Indeed, the proposed function is a generalization of the exponential function. In this context, the following definitions and lemmas are presented. Definition. The Mittag-Leffler function with two parameters is defined as E α,β z) = + k= z k Γkα+β), where α >, β >, z C. When β = 1, one has E α z) = E α,1 z), furthermore, E 1,1 z) = e z.

3 STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS 1311 Lemma 2.1 [24]). For < α < 1, we have E α t) is nonincreasing in t. We consider the nonhomogeneous linear fractional differential equation with Caputo fractional derivative 2.2) C D α t,txt) = λx+ht), t t xt ) = x, The solution of 2.2) is given by 2.3) xt) = x E α λt t ) α )+ t t s) α 1 E α,α λt s) α )hs)ds. Lemma 2.2 [8]). If one sets ht) = d in 2.2) with a constant d, then the solution of 2.3) reduces to 2.4) xt) = x E α λt t ) α )+dt t ) α E α,α+1 λt t ) α ). Lemma 2.3 [8]). Let < α < 1 and argλ) > πα 2. Then, one has t α E α,α+1 λt α ) = 1 λ 1 Γ1 α)λ 2 t α +O 1 λ 3 t 2α ) as t. Lemma 2.4 [8]). Suppose that C D α t,t mt) λmt)+d, mt ) = m, t t, where λ, d R. Then, one has mt) mt )E α λt t ) α) +dt t ) α E α,α+1 λt t ) q), t t. Moreover if λ <, then mt) mt )E α λt t ) α) +M d, t t, where M = sup s α E α,α+1 λs α )). s Remark 2.5. Authors in[8]) studied the boundedness of solutions for fractional differential equations by using the previous lemma and the Lyapunov function. Lemma 2.6 [13]). Let x be a vector of functions. Then, for any time instant t t, the following relationship holds 1 2 C D α t,tx T t)pxt)) x T t)p C D α t,txt), α,1), t t where P R n n is a constant, square, symmetric and positive definite matrix. Lemma 2.7. For all p 1 and a,b, we have a+b) p 2 p 1 a p +b p ) and a+b) p a p +b p ).

4 1312 A. BEN MAKHLOUF, M. A. HAMMAMI, AND K. SIOUD 3. Stability of fractional differential equations depending on a parameter Consider a parameterized family of fractional differential equations with a Caputo derivative for < α < 1 having the following form: C D α t,t xt) = ft,x,ε), t t xt ) = x, where t R +, ε R +, xt) Rn, f,,ε) : R + R n R n is continuous and locally Lipschitz in x. The goal of the paper is to study the ε -practical Mittag Leffler stability of certain classes of form of 3.1). Definition. The system 3.1) is said to be ε -uniformly practically Mittag Leffler stable if for all < ε ε there exist positive scalars Kε), λε) and ρε) and such that the trajectory of 3.1) passing through any initial state x ε t ) at any initial time t evaluated at time t satisfies: [ 3.1) x ε t) Kε)m x ε t ) ) E α λε)t t ) α)] b +ρε), t t, with b >, ρε) as ε + and there exist K, λ 1, λ 2 > such that λ 1 λε) λ 2, < Kε) K for all ε ],ε ], m) =, mx) and m is locally Lipschitz. Uniformly Mittag Leffler stable if 3.1) is satisfied with ρ =. Proposition 3.1. Let p 1 and ε >. Assume that for all < ε ε, there exists a continuously differentiable function V ε : R + R n R, a continuous function µ : R + R + and positive constants scalar a 1 ε), a 2 ε), a 3 ε), r 1 ε) and r 2 ε) such that 1) 3.2) a 1 ε) x p V ε t,x) a 2 ε) x p +r 1 ε), t, x R n, 2) 3.3) with C D α t,t V εt,x ε t)) a 3 ε) x ε t) p +µt)r 2 ε), t t, ε ],ε ], a3ε) t a 2ε) a2ε) λ and < a 1ε) K, with λ, K >. t s) α 1 E α,α λt s) α) µs)ds is a bounded function. cε) as ε + where cε) = r 1 ε) a 2ε)+Ma 3 ε)) a 1 ε)a 2 ε) +r 2 ε) M a 1 ε),

5 STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS 1313 with, M = M 1 +M 2, where M 1 = sup s α E α,α+1 λs α )) s and M 2 = sup t t s) α 1 E α,α λt s) α) µs)ds. Then, the system 3.1) is ε -uniformly practically Mittag Leffler stable. Proof. Let us consider t. It follows from 3.2) and 3.3) that 3.4) C D α t,t V εt,x ε t)) a 3ε) a 2 ε) V εt,x ε t))+ρt)lε) λv ε t,x ε t))+ρt)lε), t t, where ρt) = 1+µt) ) and lε) = r 2 ε)+ r1ε)a3ε) a 2ε). There exists a nonnegative function ht) satisfying: 3.5) It follows from 2.3) that 3.6) We have then, 3.7) Hence, 3.8) C D α t,t V εt,x ε t)) = λv ε t,x ε t))+ρt)lε) ht). V ε t,x ε t)) = E α λt t ) α) V ε t,x ε t )) + t s) α 1 E α,α λt s) α) ) ρs)lε) hs) ds. t t t s) α 1 E α,α λt s) α) hs)ds, V ε t,x ε t)) E α λt t ) α) V ε t,x ε t )) +lε) t s) α 1 E α,α λt s) α) ρs)ds. t V ε t,x ε t)) E α λt t ) α) V ε t,x ε t ))+Mlε), t t. By 3.2) we have: 3.9) x ε t) p 1 a 1 ε) E α λt t ) α) a 2 ε) x ε t ) p +r 1 ε) ) + Mlε) a 1 ε), t t. It follows from Lemma 2.1 E α λs α ) 1, s, so 3.1) x ε t) p a 2ε) a 1 ε) E α λt t ) α) x ε t ) p +cε), t t.

6 1314 A. BEN MAKHLOUF, M. A. HAMMAMI, AND K. SIOUD Hence, from Lemma 2.7 we obtain [ a2 ε) 3.11) x ε t) a 1 ε) E α λt t ) α) x ε t ) p] 1 p +rε), t t with rε) = cε) 1 p. Hence, the system 3.1) is ε -uniformly practically Mittag Leffler stable Stability for a class of perturbed systems In this section, we consider a perturbed system: 3.12) C D α t,txt) = Axt)+gt,xt),ε), xt ) = x, where, < α < 1, xt) R n, A R n n is a constant matrix and gt,xt),ε) R n. Consider the following assumption: H) The perturbation term gt,xt),ε) satisfies, for all t, ε > and x R n 3.13) gt,x,ε) δ 1 ε)νt)+δ 2 ε) x, such that δ 1 ε), δ 2 ε) > and δ 1 ε), δ 2 ε) as ε + and ν is a nonnegative continuous function. Then, we have the following theorem: Theorem 3.2. Suppose H) holds, the system 3.12) is ε 1 -uniformly practically Mittag Leffler stable for some ε 1 > if there exists a symmetric and positive definite matrix P, η > and λ ],η[ such that 3.14) A T P +PA+ηI <, and t t s) α 1 η λ) E α,α λ max P) t s)α) ν 2 s)ds is a bounded function. Proof. It follows from 3.13) that: 3.15) Let < λ 1 < λ, we have 2x T Pgt,x,ε) 2 x P gt,x,ε) 2 x P δ 1 ε)νt)+δ 2 ε) x ). 3.16) 2 x P δ 1 ε)νt) λ 1 x 2 + P 2 δ 1 ε) 2 ν 2 t) λ 1. Substituting 3.16) into 3.15) yields 2x T Pgt,x,ε) λ 1 +2δ 2 ε) P ) x 2 + P 2 δ 1 ε) 2 ν 2 t) λ 1. Since δ 2 ε) as ε + then there exists ε 1 > such that for all ε ],ε 1 ], 2δ 2 ε) P +λ 1 < λ. Then, ε ],ε 1 ], we have 3.17) 2x T Pgt,x,ε) λ x 2 + P 2 δ 1 ε) 2 ν 2 t) λ 1.

7 STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS 1315 Let ε ],ε 1 ]. Choose a Lyapunov function Vt,x) = x T Px. It follows from Lemma 6 that: 3.18) C D α t,tvt,x ε t)) 2x ε t) T P C D α t,tx ε t) By 3.17) and 3.18) we have [ Ax ε +gt,x ε,ε) ] T Pxε +x T εp [ Ax ε +gt,x ε,ε) ] x T ε A T P +PA ) x ε t)+2x T ε Pgt,x ε,ε). C D α t,t Vt,x εt)) x ε t) T A T P +PA ) x ε t)+λ x ε t) 2 + P 2 δ 1 ε) 2 ν 2 t) λ 1. Hence by 3.14), C D α t,t Vt,x εt)) η 2 x ε t) 2 + P 2 δ 1 ε) 2 ν 2 t) λ 1, with η 2 = η λ. Then, all hypothesis of Proposition 1 are satisfied. Therefore, the system 3.12) is ε 1 -uniformly practically Mittag Leffler stable. Example 3.3. Consider the following fractional system: { C D,t α E) x 1 = 2x 1 +x 2 +εe t x 1 +ε t 2 C D,t α x 2 = x 1 x 2 +εe t x 2 +ε 2 2t 1+t 2 where, < α < 1 and xt) = x 1 t),x 2 t) ) R 2. This system has the same from of 3.12) with ) 2 1 A = 1 1 and gt,x,ε) = εe t 2ε 2 t x 1,x 2 )+ 1+t 2, 1+t 2). The perturbed term g satisfies H) with δ 1 ε) = ε 2, δ 2 ε) = ε and νt) = 2. Select P = 2I, since ) A T 7 4 P +PA+I = <, 4 3 then, the assumptions of Theorem 3.2 are satisfied. Wehaveη = 1, wechooseλ = 3 4 andλ 1 = 1 2. So, 2ε P +λ 1 < λ, ε,ε 1 ] with < ε 1 < Hence, the system E) is ε 1-uniformly practically Mittag Leffler stable. Note that in this case, the state approaches the origin, for some sufficiently small ε, when t tends to infinity. ε 2

8 1316 A. BEN MAKHLOUF, M. A. HAMMAMI, AND K. SIOUD x 1 t) Figure 1. Time evolution of the state x 1 t) of system E) with ε =.1.1 x 2 t) Figure 2. Time evolution of the state x 2 t) of system E) with ε = Practical Mittag Leffler stability of a class of nonlinear fractional differential equations with uncertainties In this section we discuss the problem of stabilization for a class of nonlinear fractional-order systems with uncertainties. Consider the system 3.19) C D α t,txt) = Ax+B φx,u)+u ),

9 STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS 1317 where x R n, u R q, A and B are respectively n n), n q) constant matrices, φ : R n R q R q. Assume that the following assumptions are satisfied: H 1 ) There exist a symmetric and positive definite matrix P and η > such that the following inequality holds: 3.2) A T P +PA+ηI <. H 2 ) There exists a nonnegative continuous function ψ : R n R such that: 3.21) φx,u) ψx), x R n, u R q. Theorem 3.4. Suppose that assumptions H 1 ), H 2 ) hold, then the feedback law B T Pxψx) ) uε,x) = B T Px ψx)+ρε), where ρε) as ε +, ρε) >, ε >, uniformly practically Mittag Leffler stabilizes the system 3.19). Proof. Choose a Lyapunov function Vt,x) = x T Px. It follows from Lemma 2.6 that: 3.23) C D α t,t Vt,x εt)) 2x ε t) T P C D α t,t x εt) 2x T ε P[ Ax ε +B φx ε,u)+u )] Thus, by H 1 ) and H 2 ) we have x T ε A T P +PA ) x ε 2xT ε PBBT Px ε ψx ε ) 2 B T Px ε ψx ε )+ρε) +2x T ε PBφx ε,u). C D α t,tvt,x ε t)) η x ε 2 2xT εpbb T Px ε ψx ε ) 2 B T Px ε ψx ε )+ρε) +2 BT Px ε ψx ε ) 3.24) Using the following inequality then, 3.25) η x ε BT Px ε ψx ε )ρε) B T Px ε ψx ε )+ρε). B T Px ε ψx ε )ρε) B T Px ε ψx ε )+ρε) ρε), C D α t,tvt,x ε t)) η x ε t) 2 +2ρε). Hence, all hypothesis of Proposition 3.1 are satisfied. Therefore the uncertain closed-loop dynamical system 3.19) is ε -uniformly practically Mittag Leffler stable for some ε >.

10 1318 A. BEN MAKHLOUF, M. A. HAMMAMI, AND K. SIOUD x 1 t) time Figure 3. Time evolution of the state x 1 t) of system E ) with ε =.1 Example 3.5. Consider the following fractional system: E ) { C D α,tx 1 = 4x 1 +2x 2 +u+cosu)x 2 C D α,t x 2 = 2x 1 3x 2 where, < α < 1, xt) = x 1 t),x 2 t) ) R 2. This system has the same from of 3.19) with A = ), B = 1 ), φx,u) = cosu)x 2. Select P = 2I, since ) 15 8 A T P +PA+I = <, 8 11 then, the assumptions of Theorem 3.4 are satisfied. Hence, we obtain the ε - practical Mittag Leffler stability of the closed loop fractional-order system E ) for some ε > with 2x 1 x 2 2 u = 2 x 1 x 2 +ε Conclusion In this paper, we have introduced a notion of practical Mittag Leffler stability for fractional differential equations depending on a parameter. Sufficiently conditions are given by using Lyapunov theory. These results are applied to the analysis of some fractional systems.

11 STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS x 2 t) time Figure 4. Time evolution of the state x 2 t) of system E ) with ε =.1 References [1] B. Ben Hamed, Z. Haj Salem, and M. A. Hammami, Stability of nonlinear time-varying perturbed differential equations, Nonlinear Dynam ), no. 3, [2] A. Ben Makhlouf and M. A. Hammami, The convergence relation between ordinary and delay-integro-differential equations, Int. J. Dyn. Syst. Differ. Equ ), no. 3, [3] S. Burov and E. Barkai, Fractional Langevin equation: overdamped, underdamped, and critical behaviors, Phys. Rev. 3) 78 28), no. 3, 31112, 18 pp. [4] L. Chen, Y. He, Y. Chai, and R. Wu, New results on stability and stabilization of a class of nonlinear fractional-order systems, Nonlinear Dynam ), no. 4, [5] L. Chen, Y. He, R. Wu, and L. Yin, Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations, Int. J. Control Autom ), no. 3, [6], New result on finite-time stability of fractional-order nonlinear delayed systems, J. Comput. Nonlin. Dyn ), [7], Finite-time stability criteria for a class of fractional-order neural networks with delay, Neural Comput. Appl ), [8] S. K. Choi, B. Kang, and N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal. 214), Article ID 61547, 6 pages. [9] M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Automat. Control ), no. 5, [1] S. Das and P. Gupta, A mathematical model on fractional Lotka-Volterra equations, J. Theoret. Biol ), 1 6. [11] D. Ding, D. Qi, J. Peng, and Q. Wang, Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance, Nonlinear Dynam ), no. 1-2, [12] D. Ding, D. Qi, and Q. Wang, Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems, IET Control Theory Appl ), no. 5,

12 132 A. BEN MAKHLOUF, M. A. HAMMAMI, AND K. SIOUD [13] M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul ), no. 1-3, [14] B. Ghanmi, Stability of impulsive systems depending on a parameter, Mathematical Methods in the Applied Sciences, DOI: 1.12/mma ) [15] M. A. Hammami, On the Stability of Nonlinear Control Systems with Uncertainty, J. Dynam. Control Systems 7 21), no. 2, [16] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2. [17] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, New York, 215. [18] A. Leung, X. Li, Y. Chu, and X. Rao, Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling, Nonlinear Dynam ), no. 1-2, [19] Y. Li, Y. Q. Chen, and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 29), no. 8, [2] Y. Lin, D. Sontag, and Y. Wang, A smooth converse Lyapunov theorem for Robust Stability, SIAM J. Control Optim ), no. 1, [21] S. Liu, W. Jiang, X. Li, and X. Zhou, Lyapunov stability analysis of fractional nonlinear systems, Appl. Math. Lett ), [22] M. S. Mahmoud, Improved robust stability and feedback stabilization criteria for timedelay systems, IMA J. Math. Control Inform ), no. 4, [23] D. Matignon, Stability result on fractional differential equations with applications to control processing, Proceedings of IMACS-SMC, , [24] K. Miller and S. Samko, A note on the complete monotonicity of the generalized Mittag- Leffler function, Real Anal. Exchange ), no. 2, [25] O. Naifar, A. Ben Makhlouf, and M. A. Hammami, Comments on Lyapunov stability theorem about fractional system without and with delay, Commun. Nonlinear Sci. Numer. Simul ), no. 1-3, [26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego California, [27] D. Qian, C. Li, R. P. Agarwal, and P. Wong, Stability analysis of fractional differential system with Riemann-Liouville derivative, Math. Comput. Modelling 52 21), no. 5-6, [28] Z. Qu, Exponential stability of a class of nonlinear dynamical systems with uncertainties, Systems Control Lett ), [29] Y. Ryabov and A. Puzenko, Damped oscillation in view of the fractional oscillator equation, Phys. Rev ), [3] Y. Wei, Y. Chen, S. Liang, and Y. Wang, A novel algorithm on adaptive backstepping control of fractional order systems, Math. Comput. Modelling ), [31] H. Wu and K. Mizukami, Exponential stability of a class of nonlinear dynamical systems with uncertainties, Systems and Control Letters ), Abdellatif Ben Makhlouf Department of Mathematics University of Sfax Faculty of Sciences of Sfax, Tunisia address: benmakhloufabdellatif@gmail.com

13 STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS 1321 Mohamed Ali Hammami Department of Mathematics University of Sfax Faculty of Sciences of Sfax, Tunisia address: Khaled Sioud Department of Mathematics Taibah University Faculty of Sciences at Yanbu, Saudi Arabia address: