Controllability of nonlinear fractional Langevin delay systems

. In this paper, we discuss the controllability of fractional Langevin delay dynamical systems represented by the fractional delay differential equations of order 0 < α, β (cid:54) 1 . Necessary and sufﬁcient conditions for the controllability of linear fractional Langevin delay dynamical system are obtained by using the Grammian matrix. Sufﬁcient conditions for the controllability of the nonlinear delay dynamical systems are established by using the Schauders ﬁxed-point theorem. The problem of controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays and distributed delays in control are studied by using the same technique. Examples are provided to illustrate the theory.


Introduction
The concept of controllability plays a major role in both finite and infinite dimensional spaces for systems represented by ordinary differential equations and partial differential equations. So it is natural to study this concept for dynamical systems represented by fractional differential equations and fractional delay differential equations. The controllability of delay differential systems is studied by Wiess [33]. Chyung [10] studied the controllability of linear time-varying systems with delay. The controllability of nonlinear delay systems is discussed by Dauer and Gahl [12]. Balachandran and Somasundaram [6] studied the controllability of a class of nonlinear systems with distributed delay in control. The constrained controllability of semilinear delay systems was studied by Klamka [17]. Yi et al. [34] discussed the controllability and observability of systems of linear delay differential equations via the matrix Lambert W function. A sliding mode control for linear fractional systems with input and state delays is investigated by Si-Ammour et al. [27]. The controllability of differential equations with delayed and advanced arguments is investigated by Manzanilla et al. [22]. Bhalekar and Gejji [8] studied the fractional ordered Liu system with time delay. An application of delay differential equations in life sciences is discussed by Smith [29]. Pinning-controllability analysis of complex networks and M -matrix approach is discussed by Song et al. [30]. Balachandran et al. [3][4][5]7] studied the relative controllability of fractional dynamical systems with multiple delay and distributed delay in control. A numerical method for delayed fractional-order differential equations is studied by Wang [32]. Zhang et al. [36] discussed the controllability criteria for linear fractional differential systems delay in state and impulse. Analysis and numerical methods for fractional differential equations with delay were analyzed by Morgado et al. [23]. Shu [26] studied the explicit representations of solutions of linear delay systems. Lu et al. [19,20] studied the pinning controllability of Boolean control networks. Controllability of fractional damped dynamical systems with delay in control is investigated by He et al. [16]. Nirmala and Balachandran [24] investigated the controllability of nonlinear fractional delay dynamical systems. The control problems involving the delay in state variable are challenging and are not much developed. Controllability of impulsive neutral evolution integro-differential equations with state-dependent delay in Banach spaces is studied by Chalishajar et al. [9]. Sikora and Klamka [28] studied the constrained controllability of fractional linear systems with delays in control. Govindaraj and George [14] studied the controllability of fractional dynamical systems of functional analytic approach.
The theory of Brownian motion is perhaps the simplest approximate approach to treat the dynamics of nonequilibruim systems. The fundamental equation is called the Langevin equation; it contains both frictional forces and random forces. It is used to describe the evolution of physical phenomena in fluctuating environments. However, for the systems in complex media, an integer-order Langevin equation does not provide the correct description of the dynamics. One of the possible generalizations of a Langevin equation is to replace the integer-order derivative by a fractional-order derivative in it. This gives rise to a fractional Langevin equation [31]. The fractional Langevin equation with Brownian motion is revisited by Mainardi and Pironi [21]. Fa [13] investigated the fractional Langevin equation and Riemann-Liouville fractional derivative. Lim et al. [18] and Baghani [2] studied the Langevin equation with two fractional orders. Nonlinear Langevin equation involving two fractional orders in different intervals are discussed by Ahmad et al. [1]. Guo et al. [15] addressed the numerics for the fractional Langevin equation driven by the fractional Brownian motion. Yu et al. [35] discussed the existence and uniqueness of solutions of initial value problems for nonlinear Langevin equations involving two fractional orders. This motivates us to study the fractional Langevin delay differential equations.
The aim of this paper is to study the controllability of fractional Langevin delay dynamical systems. The necessary and sufficient conditions for the controllability for linear systems are derived using controllability Grammian matrix, which is defined by means of Mittag-Leffler matrix function. Sufficient conditions for the controllability of nonlinear fractional Langevin delay dynamical systems are established by using the Schauders fixed-point theorem. Controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays in control and distributed delays are studied by using the same technique.

Preliminaries
In this section, we introduce the definitions and preliminary results from fractional calculus, which are used throughout this paper. Definition 1. The Riemann-Liouville fractional integral operator of order α > 0 is defined by where Γ(·) is the Euler gamma function.
Definition 2. The Caputo fractional derivative of order α ∈ C with n − 1 < α n, n ∈ N, for a suitable function f is defined as For brevity, the Caputo fractional derivative C D α 0+ is written as C D α . Definition 3. The Mittag-Leffler functions of various types are defined by , where (γ) n is a Pochhamer symbol, which is defined as γ(γ + 1) · · · (γ + n − 1), and (γ) n = Γ(γ + n)/Γ(γ). For an n × n matrix A, , α, β > 0, Definition 4. (See [25].) The formal definition of the Laplace transform of a function f (t) of a real variable t ∈ R + = (0, ∞) is given by The convolution operator of two functions f (t) and g(t) given on R + is defined for x ∈ R + by the integral The Laplace transform of a convolution is given by Let L{f (t)} = F (s) and L{g(t)} = G(s). The inverse Laplace transform of product of two functions F (s) and G(s) is defined by The Laplace transforms of Mittag-Leffler functions are defined as and also we have

Linear delay systems
Consider the linear fractional Langevin delay dynamical system of the form where α + β > 1, x ∈ R n , u ∈ R m , A, B are n × n real matrices, and C is an n × m real matrix with n > m; φ(t) is the initial function on [−h, 0]. By taking Laplace and inverse Laplace transform on both sides of (1) and, by simple calculations using convolution of Laplace transform, we have For simplicity of notation, let us take The function φ(t) is extended to (−h, ∞) by defining φ(t) = φ(0) for t 0; then the solution of (1) is given as [24] x where Definition 5. System (1) is said to be completely controllable on J if, for every initial function φ(t) and q 0 , x 1 ∈ R n , there exists a continuous control function u such that the solutions of (1) satisfy x(T ) = x 1 .
Define the controllability Grammian matrix by where the T denotes the matrix transpose. Proof. Assume that W is positive definite. Let φ(t) be continuous on [−h, 0], and therefore its inverse is well defined. Define the control function as Substituting t = T in the solution (1) and inserting (4), we have Thus, (1) is controllable. Now assume that W is not positive definite and there exists a vector y = 0 such that y T W y = 0. It follows that Consider the zero initial function φ, q 0 = 0 and the final point x 1 = y. Since the system is controllable, there exists a control u(t) on J that steers the response to x 1 = y at t = T , that is, It follows that and leading to the conclusion that y T y = 0. This is a contradiction to y = 0. Thus, W is positive definite. Hence, the proof is complete.

Nonlinear delay systems
Consider the nonlinear fractional Langevin delay dynamical systems of the form https://www.mii.vu.lt/NA where α + β > 1, the state vector x ∈ R n , the control vector u ∈ R m , and A, B are n × n real matrices, C is an n × m real matrix with n > m, and f : J × R n × R n × R m → R n is continuous. The solution of (5) is given by for t ∈ J and y(t) = φ(t), t ∈ [−h, 0]. For simplicity, let us take Now we prove the main result of the paper.
By hypothesis, the function f satisfies the following conditions in [11]. For each pair of positive constants c and d, there is a positive constant r such that, if |(x, u)| r, then c f (t, p) + d r for all t ∈ J.
Also, for given c and d, if r is a constant such that inequality (8) is satisfied, then any r 1 such that r < r 1 will also satisfy (8). Now take c and d as given above, and let r be chosen so that the aforementioned inequality is satisfied and Therefore, if x r/3, then u r/3 for s ∈ J. It follows that d + c sup |f (s, x(s), x(s − h), u(s)) r for s ∈ J. Therefore, |v(t)| r/(6b) for all t ∈ J and hence v(t) r/(6b), which gives z(t) r/3. Thus, we have proved that, if Q(r) = {(x, u) ∈ Q: x r/3 and u r/3}, then P maps Q(r) into itself. It is easy to see that P is completely continuous. Since Q(r) is closed, bounded and convex, the Schauders fixed-point theorem implies that P has a fixed-point (x, u) ∈ Q(r) such that (z, v) = P (x, u) = (x, u). It follows that for t ∈ J, and x(t) = φ(t) for t ∈ [−h, 0]. Hence, x(t) is a solution of system (5), and x(T ) = x 1 . So system (5) is completely controllable on J.

Systems with multiple delays
Consider the linear fractional Langevin delay dynamical systems with multiple delays in control of the form where α + β > 1, x ∈ R n , u ∈ R m , A, B are n × n real matrices, and C i are n × m matrices for i = 0, 1, 2, 3 . . . , M , φ(t) is initial function on [−h, 0]. The solution of (10) is Assume the following conditions as in [5]: . . . , M , such that r i (σ i (t)) = t for t ∈ J. Further, let also σ 0 (T ) = T . Then the following inequality holds: The following definitions of complete state of system (10) at time t and relative controllability are assumed.
is said to be the complete state of system (10) at time t. Definition 7. System (10) is said to be relatively controllable on [0, T ] if, for every complete state y(t) and every x 1 ∈ R n , there exists a control u(t) defined on [0, T ] such that the solution of system (10) satisfies x(T ) = x 1 . The solution of system (10) can be written as Using the time lead functions r i (t), we have By using inequality (14), we get For simplicity, let us write the solution as where Now let us define the controllability Grammian matrix as Proof. The proof is similar to that of Theorem 1. Hence, it is omitted.
Consider the nonlinear fractional Langevin delay dynamical systems with multiple delays in control of the form where 0 < α, β 1 and assume α + β > 1, x ∈ R n is a state vector, u ∈ R m is a control vector, A, B are n × n real matrices, C i for i = 0, 1, 2, . . . , M, are n × m real matrices and f : J × R n × R n × R m → R n is a continuous function. The solution of nonlinear system (15) using the time lead function r i (t) is given by Theorem 4. Let the continuous function f satisfies the condition uniformly in t ∈ J and suppose that system (10) is relatively controllable on J. Then system (15) is relatively controllable on J.
Proof. The proof is similar to that of Theorem 2. Hence, it is omitted.

Systems with distributed delays
Consider the linear fractional Langevin delay dynamical system with distributed delays in control represented by the fractional differential equation of the form where α + β > 1, x ∈ R n , and the second integral term is in the Lebesgue-Stieltjes sense with respect to τ . Let h > 0 be given. The solution of system (18) can be expressed as Now using the well-known unsymmetric Fubini theorem and changing the order of integration of the last term, we have where dC τ denotes the Lebesgue-Stieltjes integration with respect to τ in the function C(t, τ ). For convenience, let us introduce the notation Define the controllability Grammian matrix by Theorem 5. The linear fractional Langevin delay dynamical systems (18) with distributed delays in control are controllable on J if and only if the controllability Grammian matrix is positive definite, for some T > 0.
Proof. The proof is similar to that of Theorem 2. Hence, it is omitted.
Consider the nonlinear fractional Langevin delay dynamical systems with distributed delays in control of the form where 0 < α, β 1, and assume α + β > 1, x ∈ R n , A, B and C are as above and f : J × R n × R n × R m → R n is a continuous function. Now the solution of nonlinear system (25) can be expressed in the following form: It follows from the unsymmetric Fubini theorem that and d τ C t denotes the Lebesgue-Stieltjes integration with respect to the variable τ in the function C t (s − τ, τ ). For brevity, let us introduce the notation Define the control function by where the complete state y(0) and the vector x 1 ∈ R n are chosen arbitrary and T denotes the matrix transpose.