2017 International Conference on Electrical Engineering and Automation Control (ICEEAC 2017) ISBN: 978-1-60595-447-9
Exponential Stability of Conformable Fractional Differential Equations Zhong-Qian WANGa and Ying SUNb School of Mathematics, University of Jinan, Jinan, China [email protected], [email protected]
Keywords: Fractional Differential Equation; Laplace Transform,; Conformable Fractional Calculus
Abstract. In this paper, by utilizing Laplace transforms and Lyapunove functions, we investigate the global ultimate boundedness and exponential stability for some impulsive conformable fractional differential equations. As an application, an example is given to demonstrate the main results.
Introduction Fractional differential equation is the theory of the differential and integral of any order, it is an extension of the integer order differential equation. Fractional calculus is nowadays one of the most intensively developing areas of mathematical analysis including several definitions of fractional operators like riemann-Liouville, Caputo and Grunwald-Letnikov, for details, see [1-4] and reference therein. Very recently, a new fractional derivative called the conformal be fractional derivative is given in [5, 6]. Some studies about theory and applications of the conformable fractional differential equations were conducted [7-10]. In this paper, we propose the application of Lyapunove's method to nonlinear fractional differential equations thanks to the definition of Laplace transforms of conformable fractional derivative, and give the global ultimate boundedness of such systems. The article is organized as follows. In Section 2 we recall the main concepts and tools necessary in the sequel. Our results are given in Section3. An example is given to demonstrate the main result in Section 4.
Preliminaries
Let RRn( n× n ) be the set of n( n× n) -dimensional real vectors (matrices). Let ||⋅ || denote the Euclidean norm in Rn , R =[0, ∞ ) , R=[,) t ∞ , and N ={1,2,3, …} . C RRR× n , denote the set of all + t0 0 ( t0 + ) piecewise continuous functions from RR× n to R . τ ()⋅ and τ ()⋅ denote the maximum and the t0 + M m minimum eigenvalue of the corresponding matrix, respectively. For convenience, we give some useful definitions and facts, which can be found in [5, 6]. Definition 2.1 Given a function f:[,) a∞ → R . Then the Conformable fractional derivative of order 0<α ≤ 1 is defined by
1−α a f(())() t+ε t − a − f t Tα f( t )= lim . (1) ε →0 ε
Ta f t Ta f a= lim Ta f t If α () exists on (a,b) then α () t→ a+ α () . Definition 2.2 Let a∈ R,α ∈ (0,1] , and f :[ a,∞) → R be real valued function. Then the fractional Laplace transform of order α starting from a of f is defined by
520
a Lα { f( t)}(s) = F a ()s α (2) ()t−a α ∞ −s = eα f()() t t− aα −1 dt ∫a
Lemma 2.1 Let a∈ R,α ∈ (0,1] and f:(,) a∞ → R be differentiable real valued function. Then
a a Lα{( T α f)( t)}(s) = s Fα ( s) − f( a) . (3)
a a Lemma 2.2 let f:(,) a∞ → R be a function such that Lα { f ( t )}( s )= Fα ( s) exists. Then
a 1/α Fα (s) =L { f ( a + (α t ) )}( s ), (4)
∞ where L{g ( t )}( s )= e−st g ( t ) dt is the usual Laplace transform. ∫0 After a simple calculation, we see that t 1 L0 ()()1 s=, s > 0. α s 1 1 1+ t t0 α L 0 t s = +α α ,s > 0. α ()() 1 s 1+ s α α ()t−t0 t λ 1 L0 e α = . α s − λ Consider the following systems:
Tα u( t) = Au( t) + g( t,, u( t)) u t u t+ u t − J t − u t − ∆()k =()()() k − k = k() k,, k (5) u t+ u ()0= 0
n× n where 0<α < 1, Tα u() t is the conformable fractional derivative of u() t of order α , A ∈ R is a n n n n constant matrix, g: Rt × R → R and JRRk : → , tk ( k∈ N ) satisfy t1< t 2 < < tk < and limtk = ∞ , 0 k→∞ + − with u( tk ) =lim u() tk + ε , u( tk ) = lim u() tk + ε . ε →0+ ε →0− n In this paper, we assume that for any given u0 ∈ R , system (5) owns at least one solution which is u t t u u t t u t− u t denoted by ( ,,0 0 ) (simply () ) and is left continuous at each k , that is ( k) = () k . For results on the existence and uniqueness of the solution of IFDEs, one can refer to [5-10].
Main Results
Theorem 3.1 Suppose there exist a function P t, u∈ C R × Rn , R and several constants () ( t0 + ) d≥0, a1 > 0, a 2 > 0,ωk > 0 and ρ > 0 such that (I) for all (t, u) ∈ R × Rn , t0
2 P 2 a1 || g ||≤( t , u) ≤ a2 || g || ; (6) 521 (II) For all k∈ N , and u∈ Rn ,
+ P (tk, u()() t k+ J k( t k, u t k)) ≤ γ kP ( t k,; u() t k ) (7)
n (III) for all k∈ N, t ∈ ∆k−1( t k − 1 ,] t k , and u∈ R ,
()(,)(,)Tα PP t u≤ −ρ t u + d (8)
(IV) for all i∈ N, letγ= sup{ γ i} < 1 , δk−1 =t k − t k −1 . Then system (5) is global ultimately bounded, and the solution u (t) converges to the compact set
P ={u ∈ Rn || u() t | ≤ G} , (9)
ρ d − δ α with G=(1 − e α ). ρ− ργ Proof: According to (8), we can choose a nonnegative function N( t) such that
(Tα PP )(,()) tut+ρ (,) tuNtdt + () = , ∈ ∆k , kN ∈ . (10)
Laplace transform of (10) yields d sPP( s )− ( t+ , u ( t + ))+ ρPN ( s )+ ( s ) = , α k−1 k − 1 α α s t t k−1 N k−1 where PPα()s = L α { ( t, u() t )} and α(s) = L α { N( t)} .
Writing Pα (s) in the form
d P t+, u t + N−() s + ( k() k −1 ) α s P ()s = . (11) s + ρ
The inverse Laplace transform of (11) gives
P (t, u( t))
α α −ρβk−1 −ρβk−1 P t+ u t+ eα N t e α = k−1, () k − 1 − ()* () (12) α βk d −ρ −1 α + 1−e , k ∈ N , t ∈ ∆k , ρ
where βk−1=t − t k − 1 , ∗ is the convolution operator. Using the no negativity, we obtain
β α −ρ k−1 PPt u t t+ u t+ e α (),() ≤ ()k−1 , () k − 1
α βk (13) d −ρ −1 +1 −eα , k ∈ N , t ∈ ∆ , ρ k
ρ Taking k=1 in (\ref{eq13}) and set = ω , we get α
522 −ωβα PP 0 (t, u() t) ≤ ( t0, u() t 0 ) e
d −ωβα (14) +1 −e0 , t ∈ ∆ ρ ( ) 1
Using (7) and (13), for t ∈ ∆2 we have
−ω βα + δ α PP ()1 0 (,t u() t≤ γ1 ( t0, u() t 0 ) e
dγ −ωδ α−ωβ αd −ωβ α (15) + 1 (1− e0 ) e1+ (1 − e 1 ) ρ ρ
Repeating the above procedure, for t ∈ ∆k , we have
P(t , u ( t )) k −ωδα − ωβ α i−1 k P ≤ ∏γ ie e( t0, u ( t 0 )) i=1 k −1 j α dγ k j α α (16) −ωδi+ k − j − −ωβk −ρδk− j + { γ i+ k − je e(1− e )} ∑ ∏ ρ j=1 i=1
d α αd α −ωδk−1 − ωβ k −ωβ k +γ k (1 − e ) e + (1 − e ). ρ ρ
Setting δ= sup{ δk } and considering γ= sup{ γ i } , βk < δ and the no negativity of some terms, from (16) we have
PP k− kωδ α (t, u() t) ≤ ( t0, u() t 0 )γ e k+1 α 1−γ d (17) + (1− e−ωδ ) 1− γ ρ
This together with Condition (i), we have
‖‖u( t)
k +1 a α α d 2‖‖2 −kωδ −ωδ (γ − 1) ≤ u0 γ k e+(1 − e ) , (18) a1 ρ( γ − 1)
t∈ ∆k +1, k ≥ 0
which implies the conclusion holds for γ < 1 .
Corollary 3.1 If d = 0 and the conditions (i)-(iv) of Theorem 3.2 hold, then the system (5) is globally exponetially stable with the exponential convergence rate
α 1 eωδ λ = ln( ), (19) 2δ γ
where δ= sup{} δk . k∈ N Proof. If d = 0 , then from (18) we have
a α ‖ ‖‖‖2 2 −kωδ u ()t≤ u0 γ k e , t ∈ ∆k+1 , k ≥ 0 . (20) a1
523 For t∈ ∆k+1, k ≥ 0 , we have
t− t0 ≤ tk+ 1 − t 0 ≤( k +1)δ . (21)
Using above two inequalities (20) and (21), we have
‖u( t)‖
a α 2‖ ‖2 −ωδ k ≤ u0 ()γ e a1
t− t0 a α α 2‖‖2 −ωδ −1 −ωδ 2δ ≤ u0 ()()γ eγ e a1
ωδ α a e −t − t ⋅λ 2‖‖2 ()0 = u0 e a1 γ
α 1 eωδ where λ = ln( ).Which ends the proof of Corollary 3.1. 2δ γ Lemma 3.2 Let y() t∈ Rn be a vector of differential function and B ∈ Rn× n a symmetric and positive definite matrix. Then
T yT tBB y t yT t T y t ()α ( ()()) = 2 ( )()α ( ), (22) for all 0<α < 1,t ≥ t0.
Proof. Since matrix B is symmetric, there exist an orthogonal matrix H ∈ Rn× n and a diagonal matrix Λ ∈ Rn× n such that
yT ()() tB y t TT =y ()()t H Λ H y t (23) T =()HTT y() t Λ () H y() t .
Let
z( t) = HT y( t),
from (23) we have
T yT ()()()() tB y t= z t Λ z t . (24)
Since matrix Λ is diagonal, we get
n T 2 (25) z()()() tΛ z t = ∑λii z ii t . i=1
Then
524 T z tT t α ( ()()Λ z ) n n T z2 t T z2 t = α(∑λii i ()) = ∑λii α i () (26) i=1 i=1 n
= 2∑λii zi ()() t Tα zi t ,∀ 0 <α < 1, ∀t ≥ t0. i=1
Noticing λii are the diagonal elements of matrix Λ , we rewrite
n T (27) ∑λii zi()()()() t Tα z i t= z t Λ Tα z t . i=1
So from (26) and (27) we can state that
T z tT t α ( ()()Λ z ) (28) T =2z()() t Λ Tα z t , ∀ 0 <α < 1, ∀t ≥ t0 .
Consequently, noticing z()() t= HT y t , we have
T T HT y t HT y t α ( ()) Λ ( ()) T =HT y t Λ T HT y t 2()() α ()() TT =2y() t H Λ H Tα y( t ),
∀0 <α < 1, ∀t ≥ t0.
Rearranging and using HT Λ H = B results
T yT tB y t yT tB T y t α ( ()()) = 2 ()()α (29) ∀0 <α <1, ∀t ≥ t0.
and the proof is complete. Lemma 3.3 ([11]). Let X ∈ Rn× n be a positive definite matrix and NR∈ n× n a symmetric matrix. n −1 TT −1 T Then for any u∈ R , τ m ( X NuXu) ≤ uNu ≤τ M ( X NuXu) , where τ m ()⋅ , τ M ()⋅ denote the minimum and maximum eigenvalue of the corresponding matrix respectively. Theorem3.2 Assume that there exist a symmetric positive definite matrix B ∈ Rn× n and several constants ξ > 0 , η ≥ 0 , γ k > 0 such that (a) for all (t, u) ∈ R × Rn , g( t , u ( t ))TTBB u≤ξ u u + η, t0
(30)
(b) for all k∈ N and u∈ Rn ,
T (()utk + Jtutk (,()) k k B (() utk+ Jtut k (,())) k k T (31) ≤ γ ku(t k )B u ( t k ),
(c) 0 <δk =t k − t k −1 < ∞,, k ∈ N
525 −1 T τ M (BBB (AA+ )) + 2ξ < 0 (32)
Then system (5) is global ultimately bounded, and the solution u( t ) converges to the compact set
P ={u( t ) ∈ Rn | || u ( t ) || ≤ G′ }
with
τ (BB−1 (AT + BA )) + 2ξ (1− exp ( M δ α )) 2η G′ = α , −1 T γ −1 τ M (BBB (AA+ ))+ 2ξ
where γ= sup{ γ k } , δ= sup{ δk }. k∈ N k∈ N Proof. Let WB(t, u) = uT u . It is easy to see that
τ m (BWB) ||u( t) ||≤( t , u) ≤ τ M ( ) ||u( t) || . (33)
Using Lemmas 3.2, 3.3 and Eq. (5), we have for any k∈ N, t ∈ ∆k +1,
T Tα W () t, u= 2 uB Tα u =2uT B [Au+ g ( t , u ( t ))] =uTTT [ ABB+ A ] u+ 2 g Bu ≤uTTT[ ABB + A ] u + 2ξ u Bu + 2 η ≤ τ BBBW−1 AT + A+ 2ξ t , u + 2η M ( ()) ()
According to Theorem 3.1, we have
k+1 B α α d ‖ ‖‖‖τ M () 2 −kωδ −ωδ (γ − 1) u ()t ≤ u0 γ k e+(1 − e ) , τ m ()B ωα( γ − 1)
for t ∈ ∆k +1 , k ≥ 0 , with
−1 −1 T ω= [ τ M (BBB (AA+ )) + 2ξ ]. α
Corollary 3.2 If η = 0 and the conditions (a)-(c) of Theorems 3.2 hold, then system (5) is globally exponentially stable with the exponential convergence rate
α 1 eωδ λ = ln( ), (34) 2δ γ
with
−1 −1 T ω= [ τ M (BBB (AA+ )) + 2ξ ]. α
Proof. By similar arguments with the proof of Theorem 3.2, it is clear from Corollary 3.1 we can get the conclusion.
526 Example Example 4.1 Consider system (5) with
T −3 2 g() t, u() t= () u1 ,sin u 2 , A = , 1− 3 (35) T Jk() t k, u() t k= () d1 k u 1,. d 2 k u 2
B E E Taking = 2 , where 2 denotes the 2-dimensional unit matrix. Obviously, and Condition (b) of d2 d 2 Theorem 3.2 is satisfied by taking γ k =max{()() 1 +1k , 1 + 2k } . By simple computation, we have
T g() t, u Eu≤ uT Eu,
with ξ = 1 , η = 0 , and
BBB−1 AAT τ M ( ( +)) + 2ξ T (36) = τ M () A+ A + 2ξ = −12 + 2 = − 10 < 0.
It follows from Corollary 3.2 that system (5) is globally exponentially stable.
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