Relative Controllability of Fractional Dynamical Systems with Multiple Delays in Control

Transcription

1 Chapter 4 Relative Controllability of Fractional Dynamical Systems with Multiple Delays in Control 4.1 Introduction A mathematical model for the dynamical systems with delayed controls denoting time varying multiple delays plays a highly significant role because the control of the physical plant is, in contemporary systems, performed by the electronic micro controllers. Time delays are admitted in the study of biological and physiological systems as well as electromagnetic systems composing of subsystems interconnected by hydraulic, mechanical linkages [7, 97. Controllability results for integer order nonlinear dynamical systems with several types of delays in control has been addressed in many monographs. Dauer and Gahl [49 studied the controllability for delay systems while Balachandran [12 obtained the controllability of nonlinear systems with time varying multiple delays in control by suitably adopting the technique of [46 and Darbo s fixed point theorem. Further Balachandran and Somasundaram [23 derived the sufficient conditions for the global relative controllability of the nonlinear system consisting of a bilinear mode with time-varying multiple delays in control by using Schauder s fixed point theorem and Klamka [67 investigated the controllability of nonlinear systems with different types of delays in control variables. However it should be emphasized that, to the best of our knowledge, the relative controllability of fractional dynamical system with multiple delays in control variables has 46

2 Chapter 4 47 not yet been established. In order to fill this gap, this chapter deals with the relative controllability for nonlinear fractional dynamical systems with time varying multiple delays in control variables. The common goal is to find the condition on the nonlinearity so that we choose the boundedness of the nonlinear function as per [52. The sufficient condition for the controllability results are established using Schauder s fixed point theorem. 4.2 Basic Assumptions In this chapter, consider the linear fractional dynamical system with multiple delays in control represented by the fractional differential equation of the form C D α x(t) = Ax(t) B i u(h i (t)), t J, (4.2.1) x() = x and the nonlinear fractional dynamical system with multiple delays in control represented by the fractional differential equation of the form C D α x(t) = Ax(t) x() = x, B i u(h i (t)) f(t, x(t), u(t)), t J, (4.2.2) where < α < 1, x R n, u R p and A is an n n matrix and B i are n p matrices, for i =, 1,, M and f : J R n R p R n is a continuous function. We impose the following assumptions: (H1) The functions h i : J R, i =, 1, 2,..., M, are twice continuously differentiable and strictly increasing in J. Moreover h i (t) t for t J, i =, 1, 2,..., M. (4.2.3) (H2) Introduce the time lead functions r i (t) : [h i (), h i (T ) [, T, i =, 1, 2,..., M, such that r i (h i (t)) = t for t J. Further h (t) = t and for t = T, the following inequalities hold h M (T ) h M1 (T )...h m1 (T ) = h m (T ) < h m 1 (T ) =...h 1 (T ) = h (T ) = T. (4.2.4)

3 Chapter 4 48 (H3) Let h > be given. For functions u : [ h, T R p and t J, we use the symbol u t to denote the function on [ h, defined by u t (s) = u(t s), for s [ h, ). Then the solution of the system (4.2.1) is expressed in the following form x(t) = E α (At α )x t (t s) α 1 E α,α (A(t s) α ) Using the time lead functions r i (t), the solution becomes x(t) = E α (A(t) α )x hi (t) h i () B i u(h i (s))ds. (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u(s)ds. (4.2.5) Now, using the inequalities (4.2.4), the above equation is expressed as x(t) = E α (At α )x (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u (s)ds h i () t (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u(s)ds hi (t) i=m1 h i () For brevity, we introduce the following notation: G(t) = h i () hi (t) i=m1 h i () 4.3 Controllability Results Linear System (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u (s)ds. (4.2.6) (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u (s)ds (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u (s)ds. (4.2.7) The following definitions of complete state of the system (4.2.1) at time t and relative controllability similar to integer order systems are assumed [7. Definition The set y(t) = {x(t), u t } is the complete state of the system (4.2.1) at time t.

4 Chapter 4 49 Definition System (4.2.1) is said to be globally relatively controllable on J if, for every complete state y() and every vector x 1 R n, there exists a control u(t) defined on J such that the corresponding trajectory of the system (4.2.1) satisfies x(t ) = x 1. Here the complete state y() and the vector x 1 R n are chosen arbitrarily and the denotes the matrix transpose. Theorem The linear control system (4.2.1) is relatively controllable on [, T if and only if the controllability Grammian matrix W (, T ) = (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) [E α,α (A(T r i (s)) α )B i ṙ i (s) ds (4.3.1) is positive definite, for some T >. Proof. Since W is positive definite, that is, it is non-singular, its inverse is well-defined. Define the control function by u(t) = [Bi E α,α (A (T r i (t)) α )ṙ i (t)w 1 [x 1 E α (At α )x G(T ), i =, 1,..., m, (4.3.2) where the complete state y() and the vector x 1 R n are chosen arbitrarily. Inserting t = T, (4.3.2) in (4.2.6) and using (4.2.7), we have x(t ) = E α (AT α )x G(T ) (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) [Bi E α,α (A (T r i (s)) α )ṙ i (s)w 1 [x 1 E α (AT α )x G(T )ds = x 1. Thus the control u(t) transfers the initial state y() to the desired vector x 1 R n at time T. Hence the system (4.2.1) is controllable. On the other hand, if it is not positive definite, there exists a nonzero y such that { y m (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) y W y =, } [E α,α (A(T r i (s)) α )B i ṙ i (s) y ds =,

5 Chapter 4 5 which implies y m (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) = on [, T. Let x = [E α (AT α ) 1 y. By assumption, there exists a control u such that it steers the complete initial state y() = {x, u (s)} to the origin in the interval [, T. It follows that x(t ) = E α (AT α )x G(T ) (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) Thus = y y [Bi E α,α (A (T r i (s)) α )ṙ i (s)w 1 [x 1 E α (AT α )x G(T )ds = y G(T ) (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) [Bi E α,α (A (T r i (s)) α )ṙ i (s)w 1 [x 1 E α (AT α )x G(T )ds =. y (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s)u(s)ds y G(T ). But the second and third terms are zero leading to the conclusion y y =. This is a contradiction to y. Thus W is positive definite. Hence the desired result Nonlinear Systems Now we assume the following assumptions to find the controllability results for the nonlinear systems (H4) Denote Q as the Banach space of continuous R n R p valued functions defined on the interval J with the uniform norm (z, v) = z v where z = sup{ z(t) : t J}. That is, Q = C n (J) C p (J), where C n (J) is the Banach space of continuous R n valued functions defined on the interval J with the sup norm.

6 Chapter 4 51 (H5) The continuous function f satisfies the condition that uniformly in t J. f(t, x, u) lim (x,u) (x, u) =, (H6) The continuous function f satisfies the condition that f(t, x, u) ρ j (t)φ j (x, u), where φ j : R n R p R are measurable functions and ρ j : J R are L 1 functions for j = 1, 2,..., q. For each (z, v) Q, consider the linear fractional dynamical system C D α x(t) = Ax(t) x() = x. B i u(h i (t)) f(t, z(t), v(t)), (4.3.3) Then the solution of the system (4.2.2) is expressed in the following form [42 x(t) = E α (A(t) α )x t (t s) α 1 E α,α (A(t s) α ) t Using the time lead functions r i (t), the solution becomes x(t) = E α (A(t) α )x t hi (t) h i () B i u(h i (s))ds (t s) α 1 E α,α (A(t s) α )f(s, z(s), v(s))ds. (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u(s)ds (t s) α 1 E α,α (A(t s) α )f(s, z(s), v(s))ds. (4.3.4) Now, using the inequalities (4.2.4), the above equation, for t = T, can be expressed as x(t ) = E α (A(T ) α )x (T r i (s)) α 1 E α,α (A(T r i (s)) α )B i ṙ i (s)u (s)ds h i ()

7 Chapter 4 52 i=m1 (T r i (s)) α 1 E α,α (A(T r i (s)) α )B i ṙ i (s)u(s)ds hi (T ) h i () (T r i (s)) α 1 E α,α (A(T r i (s)) α )B i ṙ i (s)u (s)ds (T s) α 1 E α,α (A(T s) α )f(s, z(s), v(s))ds. (4.3.5) For brevity, we introduce the following notation using (4.2.7) ψ(y(), x 1 ; z, v) = x 1 E α (AT α )x G(T ) (T s) α 1 E α,α (A(T s) α )f(s, z(s), v(s))ds. (4.3.6) Now we define the controllability Grammian matrix and the control function by W (, T ) = and (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) [E α,α (A(T r i (s)) α )B i ṙ i (s) ds (4.3.7) u(t) = [Bi E α,α (A (T r i (t)) α )ṙ i (t)w 1 ψ(y(), x 1 ; z, v), for i =, 1,, m. (4.3.8) Inserting (4.3.8) in (4.3.5) and by using (4.3.6), it is easy to verify that the control u(t) transfers the initial complete state y() to the desired vector x 1 R n at time T for each fixed (z, v) Q. Now, observing (4.3.6) and substituting (4.3.8) in (4.3.4), we get x(t) = E α (A(t) α )x (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u (s)ds h i () t (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s) Bi E α,α (A (T r i (s)) α )ṙ i (s)w 1 ψ(y(), x 1 ; z, v)ds hi (t) (t r i (s)) α 1 E α,α (A(t r i (s)) α )B i ṙ i (s)u (s)ds i=m1 t h i () (t s) α 1 E α,α (A(t s) α )f(s, z(s), v(s))ds. (4.3.9)

8 Chapter 4 53 Theorem Assume that the hypotheses (H1)-(H5) are satisfied and suppose that the linear fractional dynamical system (4.2.1) is globally relatively controllable. Then the nonlinear system (4.2.2) is globally relatively controllable on J. Proof. Define the operator Ψ : Q Q by Ψ(z, v) = (x, u), where u(t) = Bi E α,α (A (T r i (t)) α )ṙ i (t)w 1 ψ(y(), x 1 ; z, v) = Bi E α,α (A (T r i (t)) α )ṙ i (t)w [x 1 1 E α (AT α )x i=m1 h i () hi (T ) (T r i (s)) α 1 E α,α (A(T r i (s)) α )B i ṙ i (s)u (s)ds h i () (T r i (s)) α 1 E α,α (A(T r i (s)) α )B i ṙ i (s)u (s)ds (T s) α 1 E α,α (A(T s) α )f(s, z(s), v(s))ds. For our convenience, we introduce the following constants: ν = sup E α,α (A(T s) α ), a = max { bα 1 T α B i, 1 }, b = a i b i L i, c 2 = 4να 1 T α, d 2 = 4 [ β γµ, sup f =sup{ f(s, z(s), v(s)) :s J}, β =sup E α (At α )x, γ =sup u (s), µ = a i b i B i N i a i b i B i M i, i=m1 a i = sup E α,α (A(T r i (s)) α ), b i = sup ṙ i (s), i =, 1, 2,..., M, c i = 4a i b i Bi W 1 να 1 T α, c = max{ c i, c 2 }, i =, 1, 2,..., m, d i = 4a i b i Bi W 1 [ x 1 β µ, d = max{ d i, d 2 }, i =, 1, 2,..., m, L i = N i = M i = h i () hi (T ) h i () (T r i (s)) α 1 ds, i =, 1, 2,..., m, (T r i (s)) α 1 ds, i =, 1, 2,..., m, (T r i (s)) α 1 ds, i = m 1, m 2,..., M.

9 Chapter 4 54 Then [ u(t) a i b i Bi W 1 x 1 β γ γ i=m1 a i b i B i hi (T ) h i () a i b i B i (T r i (s)) α 1 ds h i () a i b i Bi W 1 να 1 T α sup f (T r i (s)) α 1 ds [ a i b i Bi W 1 x 1 β γµ a i b i Bi W 1 να 1 T α sup f [ di 4a c i sup f 4a 1 [d c sup f 4a and [ m x(t) β γµ a i b i B i L i α 1 T α 1 4a [d c sup f να 1 T α sup f d [d c sup f c sup f 4 d 2 c sup f. 2 By hypothesis, the function f satisfies the following conditions [48. For each pair of positive constants c and d, there exists a positive constant r such that, for p r, c f(t, p) d r, for all t J. (4.3.1) Also for given c and d, if r is a constant such that the inequality (4.3.1) is satisfied, then any r 1 such that r < r 1 will also satisfy (4.3.1). Now, take c and d as given above and choose r so that (4.3.1) is satisfied. Therefore if z r 2 and v r 2, then z(s) v(s) r, for all s J. It follows that d c sup f r. Therefore u(s) r for all s J, and hence u r, which gives x r. Thus we have proved that Ψ 4a 2 maps Q(r) into itself. Since f is continuous, it implies that the operator is continuous and hence is completely continuous by the application of Arzela-Ascoli s theorem. Since Q(r) is closed, bounded and convex, the Schauder fixed point theorem guarantees that Ψ has a fixed point (z, v) Q(r) such that Ψ(z, v) = (z, v) (x, u). Hence x(t) is the solution of the system (4.2.2) and it is easy to verify that x(t ) = x 1. Further the control function u(t) steers the system (4.2.2) from initial complete state y() to x 1 on J. Hence the system (4.2.2) is globally relatively controllable on J. 4a,

10 Chapter 4 55 Theorem Assume that the hypotheses (H1) - (H4), (H6) are satisfied and the equation (2.3.5) holds. Suppose that det W (, T ) >. Then the nonlinear system (4.2.2) is globally relatively controllable on J. Proof. Now let ψ j (r) = sup {φ j (x, u) : (x, u) r}. Since (H5) holds, there exists r > such that r c j ψ j (r ) d, which implies that Define the operator Ψ : Q Q by i=1 c j ψ j (r ) d r. Ψ(z, v) = (x, u). For our convenience, we introduce the following constants: λ = sup E α,α (A(T s) α ), ω = sup E α (At α )x, } ξ j = 4λα 1 T α ρ j, c j = max {ξ j, δ ij, a = max {b, 1}, δ ij = 4a i b i Bi W 1 λα 1 T α ρ j, γ = sup u (s), µ = a i b i B i N i a i b i B i M i, b = a i b i L i B i, i=m1 a i = sup E α,α (A(T r i (s)) α ), b i = sup ṙ i (s), i =, 1, 2,..., M, [ d i = 4 a i b i Bi W 1 [ x 1 ω µ, i =, 1, 2,..., m, d 2 = 4 [ ω γµ, d = max{ d i, d 2 }, i =, 1, 2,..., m, L i = N i = M i = h i () hi (T ) h i () (T r i (s)) α 1 ds, i =, 1, 2,..., m, (T r i (s)) α 1 ds, i =, 1, 2,..., m, (T r i (s)) α 1 ds, i = m 1, m 2,..., M.

11 Chapter 4 56 Then u(t) a i b i Bi W 1 [ x 1 ω γ γ i=m1 a i b i B i hi (T ) h i () (T r i (s)) α 1 ds a i b i Bi W 1 λα 1 T α ρ j ψ j (r ) [ a i b i Bi W 1 x 1 ω γµ d i 4a 1 4a 1 [ d 4a δ ij ψ j (r ) c j ψ j (r ) and [ m 1 x(t) ω γµ a i b i B i L i 4a [d d 4 1 [ d 4 1 [ d 2 c j ψ j (r ) 1 4 c j ψ j (r ). a i b i B i (T r i (s)) α 1 ds h i () a i b i Bi W 1 λα 1 T α ρ j ψ j (r ) ρ j ψ j (r ) λα 1 T α ξ j ψ j (r ) ρ j ψ j (r ) Therefore u(s) r 4a, for all s J and hence u r 4a, which gives x r 2. Thus we have proved that, if Q(r ) = {(z, v) Q : z r 2 and v r 2 }, then Ψ maps Q(r ) into itself. Now let us take t 1, t 2 J with t 1 < t 2 and, for all (x, u) Q(r), r >, we need to show that Ψ[Q(r) is equicontinuous. u(t 1 ) u(t 2 ) B i E α,α (A (T r i (t 1 )) α )ṙ i (t 1 ) Bi E α,α (A (T r i (t 2 )) α )ṙ i (t 2 ) [ W 1 x 1 E α (AT α )x G(T ) (T s) α 1 E α,α (A(T s) α ) ρ j ψ j (r)ds (4.3.11)

12 Chapter 4 57 and x(t 1 ) x(t 2 ) t2 Eα (At α 1 )x E α (At α 2 )x G(t 2, s) hi (t 1 ) i=m1 h i () h i () t1 i=m1 t1 hi (t 2 ) t 1 ρ j ψ j (r)ds, H(t 1, s) H(t 2, s) B i ṙ i (s) u (s) ds H(t 1, s) H(t 2, s) B i ṙ i (s) u (s) ds H(t 1, s) H(t 2, s) B i ṙ i (s) u(s) ds h i (t 1 ) H(t 2, s) B i ṙ i (s) u (s) ds G(t 1, s) G(t 2, s) t2 where H(t, s) = (t r i (s)) α 1 E α,α (A(t r i (s)) α ), G(t, s) = (t s) α 1 E α,α (A(t s) α ). ρ j ψ j (r)ds t 1 H(t 2, s) B i ṙ i (s) u(s) ds, (4.3.12) Moreover, for all (x, u) Q(r), u(t) Bi E α,α (A (T r i (t)) α ) ṙ i (t) W 1 [ x 1 E α (AT α )x i=m1 h i () hi (T ) (T r i (s)) α 1 E α,α (A(T r i (s)) α ) B i ṙ i (s) u (s) ds h i () (T r i (s)) α 1 E α,α (A(T r i (s)) α ) B i ṙ i (s) u (s) ds (T s) α 1 E α,α (A(T s) α ) ρ j ψ j (r)ds. (4.3.13) Thus the right hand sides of the equations (4.3.11) and (4.3.12) are independent of (x, u) Q(r) and tend to zero as t 1 t 2. Hence Ψ[Q(r) is equicontinuous, for all r > and by the regularity assumption on f, the operator is continuous and hence it is completely continuous by the application of Arzela-Ascoli s theorem. Since Q(r ) is closed, bounded and convex, the Schauder fixed point theorem guarantees that Ψ has a fixed point (z, v)

13 Chapter 4 58 Q(r) such that Ψ(z, v) = (z, v) (x, u). Hence x(t) is the solution of the system (4.2.2) and it is easy to verify that x(t ) = x 1. Further the control function u(t) steers the system (4.2.2) from the initial complete state y() to x 1 on J. Hence the system (4.2.2) is globally relatively controllable on J. 4.4 Examples In this section, we apply the results obtained in the previous section to the following fractional dynamical systems with multiple delays in control Example Consider the nonlinear fractional dynamical system with time varying multiple delays in control represented by the scalar fractional differential equation C D α x(t) = Ax(t) B 1 u(t) B 2 u(t h) f(t, x(t)), < α < 1, t J (4.4.1) x() = x, where A = 2, B 1 = 4, B 2 = 4 and f(t, x(t)) = t sin x(t). The Mittag-Leffler matrix function of the system is given by E α (At α ) = E α (2t α 2 k t αk ) = Γ(1 kα). Further E α,α (2(T s) α ) = k= k= 2 k (T s) αk Γ(α kα), E α,α(a(t (s h) α ) = By simple matrix calculation one can see that the controllability matrix W (, T ) = (T r i (s)) α 1 [E α,α (A(T r i (s)) α )B i ṙ i (s) = 16 k= [ k= l= k= [E α,α (A(T r i (s)) α )B i ṙ i (s) ds l= 2 kl (T s) α(kl1) 1 Γ(α kα)γ(α lα) 2 kl (T (s h)) α(kl1) 1 Γ(α kα)γ(α lα) ds [ 2 kl T α(kl1) = 16 [α(k l 1)Γ(α kα)γ(α lα) k= l= 2 kl [(T h)) α(kl1) h α(kl1) [α(k l 1)Γ(α kα)γ(α lα) k= l= 2 k (T (s h)) αk. Γ(α kα)

14 Chapter 4 59 is positive definite for any T > h. Further the nonlinear function f(t, x(t)) satisfies the hypotheses of the Theorem and so the fractional system (4.4.1) is globally relatively controllable on [, T. Example Consider the nonlinear fractional dynamical system with time varying multiple delays in control represented by the fractional differential equation C D α x(t) = Ax(t) B 1 u(t) B 2 u(t h) f(t, x(t)), t J, (4.4.2) x() = x, ( ) ( ) ( ) where < α < 1, A =, B 1 =, B 2 = and f(t, x(t)) = 1x 1 1x 2 1 (t)x2 2 (t) x 2 1x 2 2 (t) ( ) x 1 (t) Here x(t) = with x 1 (t) = x(t); D α 2 x 1 (t) = x 2 (t). The Mittag-Leffler matrix of x 2 (t) the system is given by E α ( t α ) E α (At α ) =. 3E α ( t α ) 3E α ( 2t α ) E α ( 2t α ). Further E α,α (A(T s) α ) = ( a c b ), E α,α (A(T (s h) α ) = ( ā c b ), where a = E α,α ( (T s) α ), b = E α,α ( 2(T s) α ), c = 3a 3b, ā = E α,α ( (T (s h)) α ), b = Eα,α ( 2(T (s h)) α ), c = 3ā 3 b. By simple matrix calculation, we see that the controllability matrix ( ) ( T W (, T ) = [(T s) α 1 a 2 ac (T (s h)) q 1 ac b 2 c 2 ā 2 ā c ā c b 2 c 2 ) ds

15 Chapter 4 6 is positive definite for any T > h. Further the nonlinear function f(t, x(t)) satisfies the hypotheses of the Theorem and so the fractional system (4.4.2) is globally relatively controllable on [, T. Example Consider the following fractional dynamical system with time varying multiple delays in control variable C D α x 1 (t) = x 2 (t) u 1 (t) u 1 (t h) 1x 1(t) 1x 2 1 (t)x2 2 (t) C D α x 2 (t) = x 1 (t) u 2 (t) u 2 (t h) x 2(t) 1x 2 1 (t)t for t [, T and < α < 1. It has the following matrix form C D α x(t) = Ax(t) B 1 u(t) B 2 u(t h) f(t, x(t)), < α < 1, (4.4.3) where A= ( x() = x ) ( 1, B 1 = ), B 2 = ( 1 1 ) and f(t, x(t))= ( 1x1 (t) 1x 2 1 (t)x2 2 (t) x 2 (t) 1x 2 1 (t)t ). The Mittag-Leffler matrix function of the system is given by ( 1) j t 2jα Γ(12jα) j= j= E α (At α ) = j= ( 1) j t (2j1)α Γ(1(2j1)α) j= ( 1) j t (2j1)α Γ(1(2j1)α) ( 1) j t 2jα Γ(12jα). By simple matrix calculation, we see that the controllability matrix W (, T ) = ( ) T {(T s) α 1 T1 2 T2 2 T 1 T 3 T 2 T 4 T 1 T 3 T 2 T 4 T1 2 T2 2 ( (T (s h) α 1 S1 2 S2 2 S 1 S 3 S 2 S 4 S 1 S 3 S 2 S 4 S1 2 S2 2 ) } ds is positive definite, for any T > h, where T 1 = T 4 = T 2 = T 3 = j= j= ( 1) j (T s) 2jα, Γ[(2j 1)α ( 1) j (T s) (2j1)α, Γ[2α(j 1)

16 Chapter 4 61 ( 1) j (T (s h)) 2jα S 1 = S 4 =, Γ[(2j 1)α j= ( 1) j (T (s h)) (2j1)α S 2 = S 3 =. Γ[2α(j 1) j= Further the nonlinear function f(t, x(t)) satisfies the hypotheses of the Theorem and so the fractional system (4.4.3) is globally relatively controllable on [, T. Remark It should be noted that the corresponding controllability for linear and nonlinear fractional dynamical systems without delays in control variable have been discussed in [28. For α = 1, the above fractional system (4.4.3) is reduced to the control problem of integer order dynamical system with time varying multiple delays in control as studied in [24.