Controllability of nonlinear higher order fractional ...

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Nonlinear Dyn (2013) 71:605–612 DOI 10.1007/s11071-012-0612-y

O R I G I N A L PA P E R

Controllability of nonlinear higher order fractional dynamical systems K. Balachandran · V. Govindaraj · L. Rodríguez-Germá · J.J. Trujillo

Received: 20 June 2012 / Accepted: 8 September 2012 / Published online: 28 September 2012 © Springer Science+Business Media B.V. 2012

Abstract The aim of this paper is to derive a set of sufficient conditions for controllability of nonlinear fractional dynamical system of order 1 < α < 2 in finite dimensional spaces. The results are obtained using the Schauder fixed point theorem. Examples are included to verify the result. Keywords Controllability · Fractional differential equations · Schauder fixed-point theorem

1 Introduction Differential equations of fractional order have recently proved to be valuable tools in the modeling of many

K. Balachandran · V. Govindaraj Department of Mathematics, Bharathiar University, Coimbatore 641 046, India K. Balachandran e-mail: [email protected] V. Govindaraj e-mail: [email protected] L. Rodríguez-Germá · J.J. Trujillo () Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain e-mail: [email protected] L. Rodríguez-Germá e-mail: [email protected]

phenomena in various fields of science and engineering. Although, controllability is one of the fundamental concept in mathematical control theory and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback or optimal control. Controllability of nonlinear systems in finite dimensional spaces has been studied extensively by means of fixed-point principles (see, for example [2, 15]). The use of fractionalorder derivatives and integrals in control theory leads to better results than integer-order approaches. Several authors discussed the controllability of fractional dynamical systems in finite dimensional spaces; see, for instance [1, 9, 18, 19, 24], and the references included in [7] and [23]. More recently, among several other authors Balachandran et al. [4, 6] established sufficient conditions for the controllability of nonlinear fractional dynamical systems using Schauder’s fixed-point theorem. The controllability problem for second-order dynamical systems in infinite dimensional spaces are studied in [3, 5, 16, 17, 20]. Controllability and observabillity of linear matrix second-order systems are discussed in [12] and the controllability of a matrix second order nonlinear dynamical systems using Banach fixed-point theorem is investigated in [22]. However, there is no work reported in the literature on the problem of controllability of nonlinear fractional dynamical system of order 1 < α < 2. In order to fill this gap in this paper, we study the controllability of nonlinear fractional dynamical system of order 1 < α < 2 in a finite dimensional space us-

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ing the Mittag–Leffler matrix function and Schauder fixed-point theorem. Examples are constructed to illustrate the theory.

Definition 2.3 ([10]) The Mittag–Leffler generalized matrix function Eα,β (A) =

k=0

2 Problem framework Consider the linear fractional dynamical system represented by a fractional differential equation of the form CD α x(t) + A2 x(t) = Bu(t), ⎫ ⎪ ⎬ (1) t ∈ [0, T ] := J, 1 < α < 2 ⎪ ⎭ x(0) = x0 , x (0) = y0 ,

CD α x(t) + A2 x(t) = Bu(t) + f (t, x(t), u(t))

x (0) = y0 ,

Ak Γ (αk + β)

for any α, β > 0, where A is an arbitrary square matrix. The solutions of (1) and (2) are given by [13] x(t) = Φ0 (t)x0 + Φ1 (t)y0 t + Φ(t − s)Bu(s) ds

(2)

x(t) = Φ0 (t)x0 + Φ1 (t)y0 t

+ Φ(t − s) Bu(s) + f s, x(s), u(s) ds, 0

where A and B are n × n and n × m matrices, x and u ∈ Rm and the nonlinear function f : J × Rn × Rm → Rn × Rm is continuous.

(4)

∈ Rn

Definition 2.1 ([14, 21]) The Caputo fractional derivative of order α ∈ C with n − 1 < α < n, n ∈ N, for a suitable function f is defined as t C α 1 D0+ f (t) = (t − s)n−α−1 f n (s)ds. Γ (n − α) 0 α For our brevity, the Caputo fractional derivative CD0+ is taken as CD α . The Laplace transform of Caputo fractional derivative is n−1

α f (t) (s) = s α F (s) − f (k) 0+ s α−1−k . L CD0+ k=0

In particular, if 1 < α < 2 then

α L CD0+ f (t) (s) = s α F (s) − f 0+ s α−1 − f 0+ s α−2 . Definition 2.2 The Mittag–Leffler function is a complex function which depands on two complex parameter α, β ∈ C is defined by Eα,β (z) =

∞ k=0

zk , Γ (αk + β)

Eα,1 (z) = Eα (z)

(3)

0

and the corresponding nonlinear system

x(0) = x0 ,

∞

where ∞ (−1)k A2k t αk Φ0 (t) = Eα −A2 t α = Γ (αk + 1) k=0

∞ (−1)k A2k t αk+1 2 α Φ1 (t) = tEα,2 −A t = Γ (αk + 2) k=0

∞ (−1)k A2k t α(k+1)−1 Φ(t) = t α−1 Eα,α −A2 t α = Γ (αk + α) k=0

are the Mittag–Leffler matrix functions. Definition 2.4 The system (1) is said to be controllable on J if for each vectors x0 , y0 , x1 ∈ Rn , there exists a control u(t) ∈ L2 (J, Rm ) such that the corresponding solution of (1) together with x(0) = x0 satisfies x(T ) = x1 . Lemma 2.1 ([8]) Let fi , for i = 1, 2, . . . , n, be 1 × p vector valued continuous functions defined on [t1 , t2 ]. Let F be the n × p matrix with fi as its ith row. Then f1 , f2 , . . . , fn are linearly independent on [t1 , t2 ] if and only if the n × n constant matrix

z ∈ C, α, β > 0

with β = 1.

M(t1 , t2 ) =

t2 t1

is nonsingular.

F (t)F ∗ (t) dt

Controllability of nonlinear higher order fractional dynamical systems

607

× x1 − Φ0 (T )x0 − Φ1 (T )y0 ds

Theorem 2.1 The following statements regarding the linear system (1) are equivalent (a) The linear system (1) is controllable on J . (b) The rows of Φ(t)B are linearly independent. (c) The controllability Gramian T W= Φ(T − s)BB ∗ Φ ∗ (T − s) ds (5) 0

is nonsingular. Proof First, we shall prove that (a) =⇒ (b). Suppose that the system (1) is controllable, but the rows of Φ(t)B are linearly dependent functions on J . Then there exists a nonzero constant n × 1 vector y such that y ∗ Φ(t)B = 0,

for every t ∈ J.

(6)

Let us choose x(0) = x0 = 0, x (0) = y0 = 0. Therefore, the solution of (1) becomes t x(t) = Φ(t − s)Bu(s) ds. 0

Since the system (1) is controllable on J , taking x(T ) = y. T x(T ) = y = Φ(T − s)Bu(s) ds y∗y =

0 T

y ∗ Φ(T − s)Bu(s) ds.

= Φ0 (T )x0 + Φ1 (T )y0

+ W W −1 x1 − Φ0 (T )x0 − Φ1 (T )y0 = x1 . Thus, (1) is controllable. The implications (b) =⇒ (c) and (c) =⇒ (b) follow directly from Lemma 2.1. Hence, the desired result.

3 Main result In this section, we study the controllability of the nonlinear system (2) with various conditions on the nonlinear function. Assume that C := C(J : Rn ) × C(J : Rm ) is a Banach space of all continuous Rn × Rm valued functions defined on the interval J with the norm (x, u) = x+u, where x = sup{|x(t)| : t ∈ J } and u = sup{|u(t)| : t ∈ J }. For our simplicity, let x¯ = x1 − Φ0 (T )x0 − Φ1 (T )y0 . Theorem 3.1 Let the continuous function f satisfy the condition lim

|(x,u)|→∞

|f (t, x, u)| = 0, |(x, u)|

(8)

uniformly in t ∈ J , and suppose that the linear system (1) is controllable. Then the nonlinear system (2) is controllable.

0

From (6), y ∗ y = 0, and hence y = 0. Hence, it contradicts our assumption that y is non-zero. Now we prove that (b) =⇒ (a). Suppose that the rows of Φ(t)B are linearly independent on J . Therefore, by Lemma 2.1, the n × n constant matrix T W= Φ(T − s)BB ∗ Φ ∗ (T − s) ds 0

Proof Define the operator P : C → C by P (z, v) = (x, u), where u(t) = B ∗ Φ ∗ (T − t)W −1 T × x¯ − Φ(T − s)f s, z(s), v(s) ds 0

is nonsingular. Now we define the control function as

u(t) = B ∗ Φ ∗ (T − t)W −1 x1 − Φ0 (T )x0 − Φ1 (T )y0 . (7) Substituting (7) in (3), we have x(T ) = Φ0 (T )x0 + Φ1 (T )y0 T + Φ(T − s)BB ∗ Φ ∗ (T − s)W −1 0

and x(t) = Φ0 (t)x0 + Φ1 (t)y0 t

+ Φ(t − s) Bu(s) + f s, z(s), v(s) ds. 0

Let us assume the following notations: a1 = supΦ(T − t);

a2 = sup Φ0 (t)x0 + Φ1 (t)y0 ;

608


b1 = 4a12 T B ∗ W −1 ;

such that P (z, v) = (z, v) = (x, u). Hence, we have

b2 = 4a1 T ; b = max{b1 , b2 }; ¯ c1 = 4a1 B ∗ W −1 |x|;

x(t) = Φ0 (t)x0 + Φ1 (t)y0 t

+ Φ(t − s) Bu(s) + f s, x(s), u(s) ds.

c2 = 4a2 ; c = max{c1 , c2 };

sup |f | = sup f s, z(s), v(s) : s ∈ J ; a = max{a1 T B, 1}. From the hypothesis of f , which satisfies the following conditions [11], for every pair of constants c, b, there exists a constant r > 0 such that if |(z, v)| ≤ r, then c + bf (t, z, v) ≤ r, for all t ∈ J. So let S(r) = {(z, v) ∈ C : (z, v) ≤ r} then if (z, v) ∈ S(r), we have u(t) ≤ B ∗ a1 W −1 |x| ¯ + a1 T sup |f | b1 1 c1 + sup |f | ≤ c + b sup |f | 4a 4a 4a r ≤ 4a

≤

x(t) ≤ a2 + a1 BT u + a1 T sup |f | c + 4 c ≤ + 2 r ≤ . 2 ≤

b 1 c + b sup |f | + sup |f | 4 4 b 1 sup |f | = c + b sup |f | 2 2

0

Thus, x(t) is the solution of the system (2) and it is verify that x(T ) = x1 . Hence, the system is controllable on J . If αi ∈ L1 (J ), i = 1, . . . , q, then αi is the L1 norm of αi (·), that is, αi = αi (s) ds, J

and let us assume

k1 = max Φ(t − s) : 0 ≤ s ≤ t ≤ T ;

k2 = max Φ(t − s)BT , 1 ; ai = 2k2 B ∗ Φ ∗ (T − t)W −1 Φ(T − s)αi ; bi = 2k1 αi ; ci = max{ai , bi }; ¯ d1 = 2k2 B ∗ Φ ∗ (T − t)W −1 |x|;

d2 = 2 Φ0 (t)x0 + Φ1 (t)y0 ; d = max{d1 , d2 }. Theorem 3.2 Let measurable functions ϕi : Rn × Rm → R+ and L1 - functions αi : J → R+ , i = 1, . . . , q, be such that q f (t, x, u) ≤ αi (t)ϕi (x, u).

(9)

i=1

Therefore, P maps S(r) into itself. Since f is continuous, this implies that the operator is continuous, and hence is completely continuous by the application of the Arzela–Ascoli theorem. Since S(r) is closed, bounded, and convex, the Schauder fixed-point theorem guarantees that P has a fixed point (z, v) ∈ S(r)

Then the controllability of (1) implies the controllability of (2) if q

ci sup ϕi (x, u) : (x, u) ≤ r lim sup r − r→∞

= +∞.

i=1

(10)

Proof Define the operator Q on C as follows: Q(x, u) = (z, v), where ∗

∗

v(t) = B Φ (T − t)W

−1

x¯ − 0

T

Φ (T − s)f s, x(s), u(s) ds ∗

(11)


609

and

t

z(t) = Φ0 (t)x0 + Φ1 (t)y0 +

Φ(t − s) Bv(s) + f s, x(s), u(s) ds.

(12)

0

Under our regularity assumptions on f , Q is continuous. Now the controllability result is to reduced to the existence of fixed point Q(x, u) = (x, u) with x(T ) = x1 . Let

ψi (r) = sup ϕi (x, u) : (x, u) ≤ r . Since (10) holds, there exists r0 > 0 such that r0 −

q

ci ψi (r0 ) ≥ d

i=1

or

q

ci ψi (r0 ) + d ≤ r0 .

i=1

Also,

Sr0 = (x, u) ∈ C : (x, u) ≤ r0 . If (x, u) ∈ Sr0 , from (11) and (12), we have q T ∗ ∗ −1 Φ(T − s) v ≤ B Φ (T − t)W x αi (s)ϕi x(s), u(s) ds ¯ + 0

≤ B ∗ Φ ∗ (T − t)W −1 x ¯ +

i=1

q Φ(T − s) αi (s)ψi (r0 ) ds

T

0

i=1

q q d1 1 1 r0 + ci ψi (r0 ) ≤ ci ψi (r0 ) ≤ d+ 2k2 2k2 2k2 2k2 i=1 i=1 t q Φ(t − s) Bv + z ≤ Φ0 (t)x0 + Φ1 (t)y0 + αi (s)ϕi x(s), u(s) ds

≤

0

q

i=1

t q d r0 1 d + k2 v + k1 αi (s)ψi (r0 ) ds ≤ + + ci ψi (r0 ) 2 2 2 2 0 i=1 i=1 q r0 r0 r0 1 ci ψi (r0 ) + ≤ + ≤ r0 . ≤ d+ 2 2 2 2 ≤

i=1

Hence, Q maps Sr0 into itself. Next, we need to show that Q(Sr ) is equicontinuous for all r > 0. To see that, we only have to note that, for every (x, u) ∈ Sr and s1 , s2 ∈ J, s1 < s2 , we have v(s1 ) − v(s2 ) ≤ B ∗ W −1 Φ ∗ (T − s1 ) − Φ ∗ (T − s2 ) q T Φ(T − s) × x ¯ + αi (s)ϕi x(s), u(s) ds 0

i=1

≤ B ∗ W −1 Φ ∗ (T − s1 ) − Φ ∗ (T − s2 ) q × x ¯ + Φ(T − s) αi ψi (r) i=1

(13)

610


z(s1 ) − z(s2 ) ≤ Φ0 (s1 ) − Φ0 (s2 )x0 + Φ1 (s1 ) − Φ1 (s2 )y0 s1 q Φ(s1 − s) − Φ(s2 − s) Bv + αi ψi (r) ds + 0

+

i=1

Φ(s2 − s) Bv +

s2

q

s1

αi ψi (r) ds

i=1

≤ Φ0 (s1 ) − Φ0 (s2 )x0 + Φ1 (s1 ) − Φ1 (s2 )y0 q αi ψi (r) + Φ(s1 − s) − Φ(s2 − s) BvT + i=1

+ (s2 − s1 )Φ(s2 − s) Bv +

q

αi ψi (r) .

(14)

i=1

Moreover, for all (x, u) ∈ Sr , q ∗ ∗ −1 Φ(T − s) v ≤ B Φ (T − t) W αi (s)ψi (r) ds x ¯ + J

i=1

≤ B ∗ Φ ∗ (T − t)W −1 x ¯ + Φ(T − s)

q

αi ψi (r) .

i=1

Thus, the right-hand sides of (13) and (14) do not depend on particular choices of (x, u). Hence, it is clear that Q(Sr ) is equicontinuous for all r > 0. By the Arzela–Ascoli theorem, C is a compact operator. Since Sr0 is nonempty, closed, bounded, and convex, by the Schauder fixed-point theorem guarantees the existence of solution. Also, x(T ) = x1 , shows that x(t) steers from x0 to x1 at time T . Hence, the system (2) is controllable on J . To apply the above theorem, one usually has to construct αi s and ϕi s such that (9) must be satisfied. These constructions are different for different situations. However, an obvious construction of αi s and ϕi s are easily achieved by letting q = 1, α1 = α = 1, and

ϕ1 (x, u) = ϕ(x, u) = sup f t, x(t), u(t) : t ∈ J . In this case, (10) holds if

1 lim inf(1/r) sup ϕ(x, u) : (x, u) ≤ r < . r→∞ c1

Corollary 3.1 Suppose that there exist α(t), β(t) ∈ L1 (J ), and monotonically nondecreasing functions ϕ, ψ such that f (t, x, u) ≤ α(t) ϕ x + ψ u + β(t), for all (t, x, u) ∈ J × Rn × Rm .

(16)

Let c = c2

= max 2k2 B ∗ Φ ∗ W −1 Φ(T − s)α, 2k2 α .

Then (2) is controllable if (1) is controllable and lim sup r − c ϕ(r) + ψ(r) = +∞. r→∞

In particular, this is true if

lim inf(1/r) ϕ(r) + ψ(r) < 1/c. r→∞

(15)

Proof In order to prove the corollary, it is enough to show that condition (10) holds for the following set-


611

⎛

tings:

f (t, x, u) = ⎝

α2 = α; ϕ2 = ϕ x + ψ u .

q = 2,

α = β,

ϕ1 = 1;

x(t) =

r→∞

+ ψ u : (x, u) ≤ r ≥ lim sup r − c1 − c ϕ(r) + ψ(r) = ∞. r→∞

The remaining proof of the corollary is obvious.

4 Example Example 4.1 Consider the fractional dynamical system D α x(t) + A2 x(t) = Bu(t) + f (t, x, u),

C

1 < α < 2, t ∈ J

(17)

x (0) = y0 ,

with √ √ 4 − 3√2 −3 + 3√2 , A= 4 − 4 2 −3 + 4 2

⎞ ⎠.

Here

However, this is trivial, since

lim sup r − sup c1 + c ϕ x

x(0) = x0 ,

x1 x12 +u21 +sin t x2 x22 +u22 +t

B=

1 0 0 1

x1 (t) x2 (t)

and using Mittag–Leffler matrix function for a given matrix A, we get L1 (s) L2 (s) Φ(T − s) = , L3 (s) L4 (s) where

L1 (s) = (T − s)α−1 4Eα,α −(T − s)α − 3Eα,α −2(T − s)α

L2 (s) = (T − s)α−1 3Eα,α −2(T − s)α − 3Eα,α −(T − s)α

L3 (s) = (T − s)α−1 4Eα,α −(T − s)α − 4Eα,α −2(T − s)α

L4 (s) = (T − s)α−1 4Eα,α −2(T − s)α − 3Eα,α −(T − s)α .

and

By the simple matrix calculation, one can see that the controllability matrix W=

T

0

=

0

T

Φ(T − s)BB ∗ Φ ∗ (T − s) ds

L21 (s) + L22 (s)

L1 (s)L3 (s) + L2 (s)L4 (s)

L1 (s)L3 (s) + L2 (s)L4 (s)

L21 (s) + L24 (s)

ds

is positive definite for any T > 0. Further, the nonlinear function f is bounded and continuous satisfies Theorem 3.1. Observe that the control defined by u(t) = B ∗ Φ ∗ (T − t)W −1 x¯ −

T

Φ(T − s)f s, x(s), u(s) ds

0

steers from x0 to x1 . Hence, the system (17) is controllable on [0, T ].

612


Example 4.2 Consider the fractional dynamical system D α x(t) + A2 x(t) = Bu(t) + f (t, x, u),

C

1 < α < 2, t ∈ J x(0) = x0 ,

(18)

x (0) = y0 ,

where A and B are defined above and f can be taken as 2 cos x1 f (t, x, u) = . u2 exp x2 Since the linear system is controllable and obviously f satisfies Theorem 3.2, hence the system (18) is controllable on J if (15) is holds.

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