New fractional differential inequalities with their implications to the stability analysis of fractional order systems
aa r X i v : . [ m a t h . G M ] M a y New fractional differential inequalities with theirimplications to the stability analysis of fractionalorder systems
Bichitra Kumar LenkaDepartment of Mathematics and Statistics,Indian Institute of Science Education and Research Kolkata,Nadia-741246, West Bengal, India
Email : [email protected] 16, 2019
Abstract
It is well known that the Leibniz rule for the integer derivative of order one doesnot hold for the fractional derivative case when the fractional order lies between0 and 1. Thus it poses a great difficulty in the calculation of fractional derivativeof given functions as well as in the analysis of fractional order systems. In thiswork, we develop a few fractional differential inequalities which involve the Caputofractional derivative of the product of continuously differentiable functions. Weestablish some of their properties and propose a few propositions. We show thatthese inequalities play a very essential role in the Lyapunov stability analysis ofnonautonomous fractional order systems.
Fractional calculus plays a great role in many areas of science, engineering and interdis-ciplinary subjects, and has a broad range of applications [1, 2, 3, 4]. Fractional operators(fractional derivative and/or integral) are global operators [1, 2]. Unlike the usual localderivative operators (integer order derivative operators), which have easy properties (e.g.Leibniz rule, Chain rule and so on), the properties of fractional derivative operators arenot simple (e.g. Leibniz rule, Chain rule and so on). Indeed, fractional operators possessvery rich and complex properties [1, 2]. In [5, 6, 7], it has been shown that the Leibnizrule, Chain rule, etc., do not hold for fractional derivatives whenever the fractional orderlies between 0 and 1. In fact, these rules do not hold for the Caputo fractional derivativestoo. Thus, due to the global nature and the complicated properties of Caputo fractionalderivative operator, in practice, it is a very difficult task for the estimation of fractionalderivatives of the given functions. 1ractional differential inequalities (which involve fractional derivatives) exist in thetheory of fractional order systems. Indeed, they play a very essential role in the qual-itative theory of fractional order systems. It may be noted that stability is an impor-tant qualitative property of the solutions to fractional order systems. In the literature,there has been a growing interest in the investigation of the stability or asymptotic sta-bility of the solutions to the autonomous and nonautonomous fractional order systems[8, 9, 10, 11, 12, 13, 14].It may be noted that Lyapunov direct method [15], fractional Lyapunov direct method[8, 9, 10, 11, 12], distributed order Lyapunov direct method [14] and fractional compar-ison method [13] are powerful methods for the stability analysis of the fractional ordersystems. However, the application of these methods require some suitable candidate func-tions or time-varying Lyapunov functions (which involve both state variables (dependentvariables) and time (independent variable)) that are continuously differentiable, and thecalculation of its fractional derivatives. Indeed, the application of these methods requirefractional differential inequalities.Recently, in [16, 12, 20, 21, 17, 18, 19, 22], the authors have developed some interest-ing fractional differential inequalities which involve the Caputo fractional derivatives. Inthis paper, we present some new fractional differential inequalities which involve the Ca-puto fractional derivatives of the product of continuously differentiable functions. Then,we propose some equivalence results and also establish some of their interesting proper-ties. By presenting a few interesting examples, we demonstrate the usefulness of someappropriate fractional differential inequalities and their importance in the application offractional Lyapunov direct method.
Let us denote by N be the set of natural numbers, Z + the set of positive integers, R + theset of positive real numbers, R the set of real numbers, C the set of complex numbers, R ( z ) the real part of complex number z , R n the n -dimensional Euclidean space. Definition 2.1. [1] The Gamma function is defined asΓ( z ) = Z ∞ t z − e − t dt (1)where z ∈ C and R ( z ) > Definition 2.2. [1, 2] The Riemann-Liouville fractional integral of order α of a continuousfunction x : [ t , T ) → R , −∞ < t < T ≤ ∞ is defined as RL D − αt ,t x ( t ) = 1Γ( α ) Z tt ( t − τ ) α − x ( τ ) dτ (2)where α ∈ R + . Definition 2.3. [1, 2] The Caputo fractional derivative of order α of a n th continuously2ifferentiable function x : [ t , T ) → R , −∞ < t < T ≤ ∞ is defined as C D αt ,t x ( t ) = ( RL D − ( n − α ) t ,t (cid:16) d n x ( t ) dt n (cid:17) , if α ∈ ( n − , n ) d n x ( t ) dt n , if α = n (3)where α ∈ R + and n ∈ Z + . Result 3.1. [22]
Let φ : [ t , ∞ ) → R be a monotonically decreasing and continuouslydifferentiable function. Suppose x : [ t , ∞ ) → R is a non-negative and continuouslydifferentiable function. Then, the inequality C D αt ,t { φ ( t ) x ( t ) } ≤ φ ( t ) C D αt ,t x ( t ) , ∀ t ≥ t , ∀ α ∈ (0 , , (4) holds. Result 3.2. [22]
Let φ : [ t , ∞ ) → R be a monotonically increasing and continuouslydifferentiable function. Suppose x : [ t , ∞ ) → R is a non-negative and continuouslydifferentiable function. Then, the inequality C D αt ,t { φ ( t ) x ( t ) } ≥ φ ( t ) C D αt ,t x ( t ) , ∀ t ≥ t , ∀ α ∈ (0 , , (5) holds. Result 3.3.
Let φ : [ t , ∞ ) → R be a monotonically decreasing and continuously differ-entiable function. Suppose x : [ t , ∞ ) → R is a non-negative continuously differentiablefunction. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) φ n +1 ( t ) x β ( t ) (cid:9) ≤ φ n +1 ( t ) C D αt ,t x β ( t ) , (6) where n ∈ N and the constant β is a non-negative real number. Result 3.4.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a non-negative continuouslydifferentiable function. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) C D αt ,t x β ( t ) , (7) where the constant β is a non-negative real number.Proof. The proof follows from the Result 3.1.
Result 3.5.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a continuously differentiablefunction. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) C D αt ,t x β ( t ) , (8)3 here the constant β = 2 k , and k ∈ N .Proof. The proof directly follows from the Result 3.4.
Result 3.6.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a non-negative continuouslydifferentiable function. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the inequality C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) βx β − ( t ) C D αt ,t x ( t ) , (9) holds, if the inequality C D αt ,t (cid:8) x β ( t ) (cid:9) ≤ βx β − ( t ) C D αt ,t x ( t ) , (10) holds, where the real constant β ≥ .Proof. We can write C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) − φ ( t ) βx β − ( t ) C D αt ,t x ( t ) = C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) − φ ( t ) C D αt ,t x β ( t )+ φ ( t ) (cid:2) C D αt ,t (cid:8) x β ( t ) (cid:9) − βx β − ( t ) C D αt ,t x ( t ) (cid:3) . (11)Thus the result follows by using the Result 3.4 and the inequality (10) in the equation(11). Result 3.7.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a continuously differentiablefunction. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the inequality C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) βx β − ( t ) C D αt ,t x ( t ) , (12) holds, if the inequality C D αt ,t (cid:8) x β ( t ) (cid:9) ≤ βx β − ( t ) C D αt ,t x ( t ) , (13) holds, where the real constant β = 2 n , and n ∈ N .Proof. The proof is a consequence of the Result 3.6.
Result 3.8.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a positive continuouslydifferentiable function. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the inequality C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) βx β − ( t ) C D αt ,t x ( t ) , (14) holds, if the inequality C D αt ,t (cid:8) x β ( t ) (cid:9) ≤ βx β − ( t ) C D αt ,t x ( t ) , (15)4 olds, where the real constant β ≥ .Proof. The proof is similar to the proof of the Result 3.6.
Result 3.9.
Let φ : [ t , ∞ ) → R be a positive, monotonically decreasing and continuouslydifferentiable function. Suppose x : [ t , ∞ ) → R is a positive continuously differentiablefunction. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the inequality C D αt ,t (cid:8) x β ( t ) (cid:9) ≤ βx β − ( t ) C D αt ,t x ( t ) , (16) holds, if the inequality C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) βx β − ( t ) C D αt ,t x ( t ) , (17) holds, where the real constant β ≥ .Proof. Note that C D αt ,t (cid:8) x β ( t ) (cid:9) − βx β − ( t ) C D αt ,t x ( t ) = 1 φ ( t ) (cid:2) C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) − φ ( t ) βx β − ( t ) C D αt ,t x ( t ) (cid:3) + 1 φ ( t ) (cid:2) φ ( t ) C D αt ,t x β ( t ) − C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9)(cid:3) . (18)Let us denote by f ( t ) = 1 φ ( t ) (cid:2) C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) − φ ( t ) βx β − ( t ) C D αt ,t x ( t ) (cid:3) , (19)and g ( t ) = 1 φ ( t ) (cid:2) φ ( t ) C D αt ,t x β ( t ) − C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9)(cid:3) . (20)Clearly f ( t ) ≤
0, by inequality (17). Let ψ ( t ) = φ ( t ) . Since ψ ( t ) is positive, monotonicallyincreasing and continuously differentiable function, it follows from Result 3.2, that g ( t ) ≤
0. This completes the proof.
Lemma 3.1. [17]
Suppose x : [ t , ∞ ) → R is a non-negative continuously differentiablefunction. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) x β ( t ) (cid:9) ≤ βx β − ( t ) C D αt ,t x ( t ) , (21) where β ≥ . Lemma 3.2. [17]
Suppose x : [ t , ∞ ) → R is a continuously differentiable function.Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) x β ( t ) (cid:9) ≤ βx β − ( t ) C D αt ,t x ( t ) , (22)5 here β = pq ≥ , p = 2 n , and n, q ∈ N . Result 3.10.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a continuously differentiablefunction. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) φ ( t ) x β ( t ) (cid:9) ≤ φ ( t ) βx β − ( t ) C D αt ,t x ( t ) , (23) where β = pq ≥ , p = 2 k , and k, q ∈ N .Proof. The proof follows by using the Lemma 3.2. Also it can be easily observed thatthe inequality (23) holds from the Result 3.6 where the Lemma 3.2 is used.
Result 3.11.
Let φ : [ t , ∞ ) → R be a non-negative, monotonically decreasing and con-tinuously differentiable function. Suppose x : [ t , ∞ ) → R is a non-negative continuouslydifferentiable function. Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t (cid:8) φ p ( t ) x β ( t ) (cid:9) ≤ φ p ( t ) βx β − ( t ) C D αt ,t x ( t ) , (24) where the real positive constants p ≥ and β ≥ .Proof. It follows from the Result 3.4 that C D αt ,t (cid:8) φ p ( t ) x β ( t ) (cid:9) ≤ φ p ( t ) C D αt ,t x β ( t ) , (25)where the real positive constants p ≥ β ≥
1. Then, the application of the Result 3.6or the Lemma 3.1 into the inequality (25) yields the inequality (24).
Result 3.12.
Let φ i : [ t , ∞ ) → R are non-negative, monotonically decreasing and con-tinuously differentiable functions for i = 1 , , · · · , n . Suppose x i : [ t , ∞ ) → R arenon-negative continuously differentiable functions for i = 1 , , · · · , n . Then, ∀ t ≥ t , ∀ α ∈ (0 , , the following inequality holds C D αt ,t ( n X i =1 φ p i i ( t ) x β i i ( t ) ) ≤ n X i =1 φ p i i ( t ) β i x β i − i ( t ) C D αt ,t x i ( t ) , (26) where the real positive constants p i ≥ and β i ≥ for i = 1 , , · · · , n .Proof. The proof is similar to the proof of Result 3.11.
Result 3.13.
Let φ i : [ t , ∞ ) → R are non-negative, monotonically decreasing and con-tinuously differentiable functions for i = 1 , , · · · , n . Suppose x i : [ t , ∞ ) → R arecontinuously differentiable functions for i = 1 , , · · · , n . Then, ∀ t ≥ t , ∀ α ∈ (0 , , thefollowing inequality holds C D αt ,t ( n X i =1 φ p i i ( t ) x β i i ( t ) ) ≤ n X i =1 φ p i i ( t ) β i x β i − i ( t ) C D αt ,t x i ( t ) , (27) where the real positive constants p i ≥ and β i = u i v i ≥ , u i = 2 k i , k i , v i ∈ N for i = 1 , , · · · , n . roof. The proof is a consequence of the Result 3.12 where the Lemma 3.2 is used.
Result 3.14.
Let φ i : [ t , ∞ ) → R are non-negative, monotonically decreasing and con-tinuously differentiable functions for i = 1 , , · · · , n . Suppose x i : [ t , ∞ ) → R arecontinuously differentiable functions for i = 1 , , · · · , n . Then, ∀ t ≥ t , ∀ α ∈ (0 , , thefollowing inequality holds C D αt ,t ( n X i =1 c i φ p i i ( t ) x β i i ( t ) + n X i =1 d i x β i i ( t ) ) ≤ n X i =1 c i φ p i i ( t ) β i x β i − i ( t ) C D αt ,t x i ( t ) (28)+ n X i =1 d i β i x β i − i ( t ) C D αt ,t x i ( t ) , (29) where the real positive constants c i > , d i > , p i ≥ , and β i = u i v i ≥ , u i = 2 k i , k i , v i ∈ N for i = 1 , , · · · , n .Proof. The result follows by combining the Lemma 3.2 with the Result 3.13.
Result 3.15.
Let φ i : [ t , ∞ ) → R are non-negative, monotonically decreasing and con-tinuously differentiable functions for i = 1 , , · · · , n . Suppose x i : [ t , ∞ ) → R arecontinuously differentiable functions for i = 1 , , · · · , n . Then, ∀ t ≥ t , ∀ α ∈ (0 , , thefollowing inequality holds C D αt ,t ( n X i =1 c i φ p i i ( t ) x β i i ( t ) + n X i =1 d i x β i i ( t ) ) + C D αt ,t ( n X i =1 a i x γ i i ( t ) ) ≤ n X i =1 c i φ p i i ( t ) β i x β i − i ( t ) C D αt ,t x i ( t )+ n X i =1 d i β i x β i − i ( t ) C D αt ,t x i ( t )+ n X i =1 a i γ i x γ i − i ( t ) C D αt ,t x i ( t ) , (30) where the real constants a i ≥ , c i ≥ , d i ≥ , p i ≥ , β i = u i v i ≥ , u i = 2 k i , k i , v i ∈ N ,and γ i = m i n i ≥ , m i = 2 ℓ i , ℓ i , n i ∈ N for i = 1 , , · · · , n .Proof. The proof is a consequence of the Result 3.14.
In this section, we discuss the fractional order generalizations or versions of a few in-teresting examples [15]. We apply the fractional Lyapunov direct method and utilizeappropriate fractional differential inequalities, in order to analyse the stability of the zerosolutions to nonautonomous fractional order systems. We use the numerical predictor-corrector method [23] in order to carry out the numerical solutions to the presentedexamples. 7 xample 4.1.
Consider the nonautonomous linear fractional order system C D α ,t x ( t ) = − x ( t ) −
11 + t x ( t ) , x (0) = x , C D α ,t x ( t ) = x ( t ) − x ( t ) , x (0) = x , (31) where < α ≤ . Let us choose the function V ( t, x ) = x + x + t x . Note that x + x ≤ V ( t, x ) ≤ x + 2 x , ∀ x = ( x , x ) T ∈ R . Then, the application of the Result 3.15, allows us tocalculate the Caputo fractional derivative of V ( t, x ) along the solution x ( t ) to (31) asfollows C D α ,t V ( t, x ( t )) ≤ (cid:20) − x ( t ) −
21 + t x ( t ) x ( t ) (cid:21) + (cid:20) (2 + 21 + t ) x ( t ) x ( t ) − (2 + 21 + t ) x ( t ) (cid:21) = − x ( t ) + 2 x ( t ) x ( t ) − (2 + 21 + t ) x ( t ) ≤ − x ( t ) + 2 x ( t ) x ( t ) − x ( t )= − x T ( t ) (cid:18) − − (cid:19) x ( t ) , ∀ x = (cid:18) x x (cid:19) ∈ R . (32)Note that x T (cid:18) − − (cid:19) x is a positive definite quadratic function. Then, it followsfrom (32) that C D α ,t V ( t, x ( t )) ≤ −k x ( t ) k . (33)Let γ ( r ) = r , γ ( r ) = 2 r and γ ( r ) = r , where r = k x k . Then, all the assumptionsof Theorem 6.2 of [9] are satisfied. This observation confirms that V ( t, x ) is indeed atime-varying Lyapunov function. Hence, by Theorem 6.2 of [9], we conclude that thezero solution is asymptotically stable.Further, from (33), we deduce V ( t, x ( t )) ≤ E α (cid:18) − t α (cid:19) V (0 , x (0)) , ∀ t ≥ . (34)Thus, it follows that k x ( t ) k ≤ s E α (cid:18) − t α (cid:19) k x (0) k , ∀ t ≥ . (35)As a result, the zero solution to the system (31) is Mittag-Leffler stable. Thus, the zerosolution is asymptotically stable. The numerical solution shown in the Figure 1 indicatesthat the non-trivial solutions approach to the zero solution.8
20 40 60 80 100 t -10-50510 x , x x ( t ) x ( t ) Figure 1: Numerical solution to (31) for the value of fractional order α = 0 . x (0) = −
10 and x (0) = 10. Example 4.2.
Consider the nonautonomous nonlinear fractional differential equation C D α ,t x ( t ) = − x ( t ) − e t/ x ( t ) , x (0) = x , < α ≤ . (36)Let V ( t, x ) = x + e − t/ x be the function, which depends on time t and variable x .Then, by using the Result 3.15, we get the Caputo fractional derivative of V ( t, x ) alongthe solution x ( t ) to (4.2) as follows C D α ,t V ( t, x ( t )) ≤ (cid:2) − x ( t ) − e t/ x ( t ) (cid:3) + (cid:2) − e − t/ x ( t ) − x ( t ) (cid:3) = − (cid:0) e − t/ (cid:1) x ( t ) − (cid:0) e t/ (cid:1) x ( t ) ≤ − x ( t ) , ∀ x ∈ R , ∀ t ≥ . (37)Note that x ≤ V ( t, x ) ≤ x , ∀ x ∈ R , ∀ t ≥
0. Let γ ( r ) = r , γ ( r ) = 2 r and γ ( r ) =12 r , where r = | x | . Then, we see that all the assumptions of the Theorem 6.2 of [9]are satisfied. Hence, it follows from the Theorem 6.2 of [9] that the zero solution to theequation (36) is asymptotically stable. The numerical solution is shown in the Figure 2.9
10 20 30 40 t x x ( t ) Figure 2: Numerical solution to (36) for the value of fractional order α = 0 . x (0) = 0 . Example 4.3.
Consider the nonautonomous nonlinear fractional order system C D α ,t x ( t ) = − x ( t ) − x ( t ) + sin( t ) (cid:0) x ( t ) + x ( t ) (cid:1) , x (0) = x , C D α ,t x ( t ) = x ( t ) − x ( t ) + cos( t ) (cid:0) x ( t ) + x ( t ) (cid:1) , x (0) = x , (38) where < α ≤ . Let V ( t, x ) = x + x + φ ( t ) x + φ ( t ) x be the time-varying Lyapunov function, where φ ( t ) is a non-negative, monotonically decreasing, bounded and continuously differentiablefunction. Then, by using the Result 3.15 (or the Result 3.5 where the Lemma 1 of [16] isused), we get the Caputo fractional derivative of V ( t, x ) along the solution x ( t ) to (4.3)as follows C D α ,t V ( t, x ( t )) ≤ (cid:2) − x ( t ) − x ( t ) x ( t ) + x ( t ) sin( t ) (cid:0) x ( t ) + x ( t ) (cid:1)(cid:3) + (cid:2) x ( t ) x ( t ) − x ( t ) + x ( t ) cos( t ) (cid:0) x ( t ) + x ( t ) (cid:1)(cid:3) + φ ( t ) (cid:2) − x ( t ) − x ( t ) x ( t ) + x ( t ) sin( t ) (cid:0) x ( t ) + x ( t ) (cid:1)(cid:3) + φ ( t ) (cid:2) x ( t ) x ( t ) − x ( t ) + x ( t ) cos( t ) (cid:0) x ( t ) + x ( t ) (cid:1)(cid:3) = − (cid:2) (1 + φ ( t ))( x ( t ) + x ( t )) (cid:3) + (cid:2) (1 + φ ( t ))( x ( t ) + x ( t ))( x ( t ) sin( t ) + x ( t ) cos ( t )) (cid:3) ≤ − (cid:2) (1 + φ ( t ))( x ( t ) + x ( t )) (cid:3) + (cid:2) (1 + φ ( t ))( x ( t ) + x ( t )) (cid:3) = − (1 + φ ( t )) k x ( t ) k (1 − k x ( t ) k ) ≤ − (1 + φ ( t ))(1 − r ) k x ( t ) k , ∀k x ( t ) k ≤ r, ∀ r < , ≤ − (1 − r ) k x ( t ) k , ∀k x ( t ) k ≤ r, ∀ r < . (39)Hence, it follows from the Theorem 6.2 of [9] that the zero solution to the system (38) is10symptotically stable. t -0.4-0.200.20.4 x , x x ( t ) x ( t ) Figure 3: Numerical solution to (38), starting from the initial values x (0) = − . x (0) = 0 .
3, for the value of fractional order α = 0 . α = 0 .
85. It indicates that the zero solution is asymptoticallystable.
We have established several new fractional differential inequalities which involve the Ca-puto fractional derivative of continuously differentiable functions. By presenting a fewsimple illustrative examples, applying fractional direct method and using appropriatefractional differential inequalities, we have shown that the presented fractional differen-tial inequalities play a basic role in the analysis as well as the calculation of boundsof solutions to nonautonomous fractional order systems. These fractional differential in-equalities open up an opportunity and show a way to construct or discover different typesof potential candidate time-varying Lyapunov functions. They can also be useful for thecalculation of their fractional derivatives, at the same time, for the stability analysis offractional order systems.