Linear conformable differential system and its controllability

This article deals with the sequential conformable linear equations. We have focused on the solution techniques of these equations and particularly on the controllability conditions of the time-invariant system. For the controllability conditions and results, we have defined the conformable controllability Gramian matrix, the conformable fundamental matrix, and the conformable controllability matrix.


Introduction
Fractional calculus has gained importance due to the adequate analysis approach in real world problems. It is generally used in medical sciences [27], material sciences, fluid mechanics, edge detection, and electromagnetics [12,16,25,26]. Numerous physical phenomena having memory and hereditary characteristics can be efficiently interpreted through the fractional calculus approach [21]. There are various publications which are based upon the study of fractional differential equations and have also highlighted their respective applications, see [4-6, 19, 23, 24].
There are various real world problems which are expressed as both linear and nonlinear system of fractional order differential equations. The solution technique, along with the applications of such systems, can be seen in [12,14,20].
Several definitions of fractional order derivative are introduced by famous researchers like Euler, Fourier, Abel, Sonin, Letnikov, Laurent, Nekrasov, and Nishimoto. Also, the most popular definitions for it are Riemann-Liouville, Caputo, and Grünwald-Letnikov definitions. Some other definitions for fractional derivatives and fractional integrals are also provided in [16,19] by Kilbas and Miller. Here are some of the well-known definitions on which we have focused throughout in our article: • Riemann-Liouville derivative: D α x f (x) = 1 Γ (nα) d n dx n x 0 (xt) n-α-1 f (t) dt, n -1 < α ≤ n; ( 1 ) • Caputo derivative: D α x f (x) = 1 Γ (nα) x 0 (xt) n-α-1 f (t) (n) dt, n -1 < α ≤ n; ( 2 ) • Grünwald-Letnikov derivative: A few properties of these fractional order derivatives are similar to those of the classical order derivative. However, there are few drawbacks, e.g., D α a (1) = 0 does not fulfill in Riemann-Liouville derivative. And for Caputo derivative, we have to assume that f is differentiable, otherwise we cannot apply this definition. It is also important to note that Liouville's theorem in fractional case does not hold as well. In short, all fractional derivatives are deficient in some mathematical properties, like the product, chain, and quotient rules. For a review about various properties of these fractional order derivatives, one can see [26].
Due to these issues, a new definition of fractional order derivative is required, which can fulfill more mathematical conditions as compared to the previous ones. Recently, Khalil and Horani have introduced a new well-defined and simple definition of fractional order derivative known as conformable fractional derivative [7,15,34]. This definition depends on the basic limit which is defined for classical order derivative. They have also established various properties like the product, quotient, and chain rules, and additionally mean value theorem of conformable fractional derivative. Some other new ideas on conformable derivative can be seen in [1,2,9,11,28,30,31,33]. However, the conformable fractional derivative is not considered to be the same as a fractional order derivative, it is a first-order derivative multiplied by an additional simple factor. Hence, this new definition appears to be a natural extension of the conventional order derivative to arbitrary order without memory affect. A recent new approach in defining fractional operators with nonsingular Mittag-Leffler kernel with memory affect can be found in [8].
With the popularity of fractional calculus approach in mathematical modeling, control theory has also utilized it in the controllability analysis. For a review on fractional controllability, one can see [10,18,22,32]. In these articles the controllability criterion and conditions with different fractional order derivatives are discussed.
With this motivation, in this article, we have used the definitions of conformable fractional derivative, conformable fractional integrals, and some fundamental results of conformable fractional calculus. By using the definition of conformable fractional derivative, we will demonstrate the Liouville's theorem, the controllability conditions for a timeinvariant system.
The methodology of our work in this manuscript has two strategies: the first is a direct technique and the second uses a transformation into state equations. We will also define a conformable fractional transition matrix and the fundamental matrix.
For obtaining the controllability conditions, we will study sequential linear conformable fractional differential equations of order nα. We will revise some results about the existence and uniqueness for order nα sequential linear conformable fractional homogeneous and nonhomogeneous fractional differential equations defined in the following form: T nα y + a n-1 T (n-1)α y + a n-2 T (n-2)α y + · · · + a 0 y = 0 and T n α y + p n-1 (t)T n-1 respectively [3,13]. There are several ways to solve the conformable fractional differential equations, some of which are the same as those involving the classical order conformable derivative.

Basic concepts
Here, basic results and a few definitions on conformable fractional calculus will be revised.
In particular, if a = 0 then (4) becomes If a given function f satisfies Definition 1 for all t > a, then f is called an α-differentiable function.
Remark 1 The α-derivative is a linear operator, that is,

Definition 2 ([15])
The α-fractional integral for a function f , is defined as where the integral is the usual improper Riemann integral and α ∈ (0, 1). .
Let us consider the following conformable fractional linear homogeneous differential equation: and conformable fractional nonhomogeneous differential equation where y ∈ C 1 , p(t) and q(t) are real-valued continuous functions. If we have a conformable fractional initial value problem where p(t) and q(t) are assumed to be continuous and defined for all t ∈ (a, b), then there exists a unique solution of the initial value problem (8).

Lemma 4 ([13])
Let t α-1 p(t), t α-1 q(t) ∈ C(a, b), and y be a continuously differentiable function. Then, the initial value problem has a unique solution on the interval (a, b) where t 0 ∈ (a, b).
Let us consider a second-order conformable fractional equation of the following form: where a 0 , a 1 are real constants and r(t) is a nonzero continuous function, and a homogeneous fractional differential equation of the form Definition 3 The conformable fractional Wronskian W α [y 1 , y 2 ] for two independent functions y 1 and y 2 is defined as

Lemma 5
If y 1 and y 2 are two linearly independent solutions of conformable fractional homogeneous equation (11), then the particular solution y p of corresponding the conformable fractional nonhomogeneous equation (10) is Let us consider the linear sequential conformable homogeneous fractional differential equation of order nα given by α y + a n-1 T (n-1) α y + a n-2 T (n-2) α y + · · · + a 0 y = 0, where the coefficients a 0 , a 1 , . . . , a n are real constants, α ∈ (0, 1) and T (n) α = T α T α · · · T α . If y is n times differentiable, then there exist n independent solutions y 1 , y 2 , . . . , y n for the homogeneous equation (13). For α = 1, equation (13) becomes an nth order linear homogeneous differential equation. Let us discuss the nonhomogeneous case Note that for α = 1, equation (14) becomes an nth order linear nonhomogeneous differential equation. Now, we will define the conformable fractional Wronskian for the particular solution of (14). (13) has n linearly independent solutions y 1 , y 2 , . . . , y n then the function
It is important to mention here that there are some problems with the previous technique. For example, it is a fact that for complementary solutions of (13), we have to change this equation into a classical order differential equation, which becomes a lengthy process.
Moreover, for the particular solution y p of (13) it is necessary that y 1 , y 2 , . . . , y n are linearly independent. Here we make the following changes of variables for equation (14): Therefore, we can write . . . , in the matrix form as: where Equation (16) is called the state representation of equation (14). The choice of state variables is by no means unique. In fact, the choice is limitless. So, the existence of the solution and its uniqueness of are firstly addressed for this state equation representation.
To solve the nonhomogeneous conformable fractional vector differential equation we first solve its corresponding homogeneous portion, that is, For this, we need a linear conformable fractional differential operator.

Definition 5 Consider a set
for t ∈ I, then L α is called a conformable fractional vector differential operator.

Lemma 6
The conformable fractional vector differential operator L α is linear, that is, where x and y both are n by 1 continuously differentiable vector-valued functions and λ 1 , λ 2 are arbitrary real constants.
Theorem 2 If (λ 0 , ξ 0 ) be an eigenpair for the matrix A of order n × n, then Proof Let us consider Applying the α-conformable derivative on both sides, we have As (λ 0 , ξ 0 ) is an eigenpair, for t ∈ R, and the proof is completed.
Example 1 We will solve the conformable fractional differential equation given as x.
The characteristic equation of A can be written as and the eigenvalues are λ 1 = -1 and λ 2 = -4. The first eigenpair for A corresponding to Similarly, the eigenpair for λ 2 = -4 is Hence from Theorem 2, the vector functions ψ 1 and ψ 2 are of the form Since ψ 1 and ψ 2 are linearly independent on R, the general solution x(t) is of the following form: Let us define the matrix conformable fractional equation as where both X(t) and A are n by n continuous matrix functions on an interval I. We say that a matrix function Ψ is a solution of (19) on I if Ψ is a continuously differentiable n by n matrix function on I such that (19) holds for all t ∈ I; and then Ψ is called the conformable fractional fundamental matrix for conformable fractional vector differential equation (18) for t ∈ I.
The next theorem establishes the relationship between (18) and (19).

Theorem 3 Let A be a continuous n by n matrix function, and Ψ (t) be defined on an interval I as
Then Ψ (t) is the solution of the conformable fractional differential matrix equation (19) on I if and only if each column ψ 1 (t), ψ 2 (t), . . . , ψ n (t) is a solution of the conformable fractional vector differential equation (18) is the solution of the conformable fractional differential matrix equation (19) then x(t) = Ψ (t)c is a solution of the conformable fractional differential vector equation (18) for any constant vector c of order n × 1.
Proof Assume that ψ i , i = 1, 2, . . . , n are solutions of (18) on I. Then the n by n matrix function Ψ is of the following form: As Ψ is a continuously differentiable matrix function, we can easily obtain T α Ψ (t) = AΨ (t) for t ∈ I. Hence it is proved that Ψ is a solution of conformable matrix differential equation (19). Next, for the converse part, suppose that Ψ is the solution of conformable matrix differential equation (19). We have to prove that ψ i s are solutions of (18). As Ψ is a solution of conformable matrix differential equation (19), T α ψ 1 (t), ψ 2 (t), . . . , ψ n (t) = Aψ 1 (t), Aψ 2 (t), . . . , Aψ n (t) .
This implies that . . . , and this shows that ψ i s are solutions of (18). Next assume that Ψ (t) is the solution of conformable fractional differential matrix equation (19) and let t ∈ I. Consider x(t) = Ψ (t)c. Then for t ∈ I, which proves the theorem.

Theorem 4 (Fractional Liouville's theorem)
If Ψ (t) = [ψ 1 (t), ψ 2 (t), . . . , ψ n (t)] is the matrix function of n solutions of the conformable differential vector equation (18), then for t 0 ∈ I, Proof We will prove this result for n = 2. Suppose ψ 1 (t) and ψ 2 (t) are two solutions of (18) on I and Ψ is a matrix function of the form where Let Taking the conformable fractional derivative of both sides, we get After simplifying, we have

Solving this conformable differentiable equation results in
Multiplying equation (20) and at t = t 0 we can write which implies that h(t 0 ) = c 0 . Substituting the value of c 0 into equation (21), we get , which can be written as which completes the proof.
With the above theorem, we can write a corollary that helps us observe the behavior of det Ψ (t) for all t ∈ I. Corollary 1 If Ψ is the matrix function with columns ψ 1 (t), ψ 2 (t), . . . , ψ n (t) of n solutions of the conformable differential vector equation (18) on I then either 1. For all t ∈ I, det Ψ (t) = 0, or 2. For all t ∈ I, det Ψ (t) = 0.
Next, for the converse part, we assume, for all t ∈ I, thatdet Ψ (t) = 0. By Theorem 4, By the fact which we have already used that the determinant of any matrix is actually equal to the product of its eigenvalues, we have zero as an eigenvalue of Ψ and we can write This implies that columns ψ 1 (t 0 ), ψ 2 (t 0 ), . . . , ψ n (t 0 ) of Ψ (t 0 ) are linearly dependent. As t 0 ∈ I is arbitrary, so this part is true for all t 0 ∈ I, proving that ψ 1 (t), ψ 2 (t), . . . , ψ n (t) are linearly dependent for all t ∈ I. Now if ψ 1 (t), ψ 2 (t), . . . , ψ n (t) are linearly independent, by the definition of linear independence of vectors, we have c 1 ψ 1 (t) + c 2 ψ 2 (t) + · · · + c n ψ n (t) = 0 only when all c i s are zero.
Definition 6 A matrix function Ψ n×n is said to be a conformable fractional fundamental matrix for the conformable fractional vector differential equation (18) if Ψ is a solution of the conformable fractional matrix equation (19) on I such that det Ψ (t) = 0 on I.
Theorem 5 A matrix function Ψ n×n is a conformable fractional fundamental matrix for the conformable fractional vector differential (18) if and only if the columns of Ψ are n linearly independent solutions of (18) on I. If Ψ is a conformable fractional fundamental matrix for the conformable differential vector equation (18), then the general solution x of conformable fractional vector differential equation (18) for t ∈ I is given by where c 1 is an arbitrary constant vector of order n × 1. There are infinitely many conformable fundamental matrices for the conformable differential vector equation (18).
Proof Consider that the columns of Ψ n×n are linearly independent solutions of (18) on I.
Since the columns of Ψ are the solutions of (18), then, by Theorem 3, Ψ is a solution of (19).
Since the columns of Ψ are linearly independent solutions for (18), from Corollary (1) we have det Ψ (t) = 0 on I. Hence Ψ is the conformable fundamental matrix for conformable differential vector equation (18). Next, for the converse part, we assume that Ψ is the conformable fundamental matrix for the conformable differential vector equation (18). Then by the definition of conformable fundamental matrix, Ψ is the solution matrix of equation (19) on I such that det Ψ (t) = 0. From Theorem 3, columns of Ψ are solutions of (18). Since det Ψ (t) = 0 and columns of Ψ are solutions of (18), by Corollary 1, columns of Ψ are linearly independent. Since for any nonsingular n by n matrix X 0 , the solution of the IVP is a fundamental matrix for (18). Next assume that Ψ is a conformable fractional fundamental matrix for (18). Then by Theorem 3, x(t) = Ψ (t)c is a solution of (18). Now consider that z is an arbitrary, but fixed solution of (18). Here we define Then z and Ψ (t)c 0 are solutions of (18) which have the same vector value at t 0 . Hence, by the uniqueness theorem, we have z(t) = Ψ (t)c 0 . Therefore, for t ∈ I, where c is an arbitrary constant vector, defines a general solution of (18).
Example 2 For the conformable fractional vector differential equation find the conformable fractional fundamental matrix Ψ and write the general solution of this equation (24) in the term of Ψ . Solution. For the matrix A, the characteristic equation is as follows: are solutions of (24), and the matrix function Ψ (t), given by is a matrix solution of conformable fractional matrix equation corresponding to (19). Since det Ψ (t) = 7e t α α = 0, this implies that Ψ is a conformable fractional fundamental matrix of (24) on R. Also, from Theorem 5, the general solution of (24) is given by for t ∈ R, where c is any 2 × 1 constant vector.

Theorem 6
If Ψ is a fundamental matrix for (18), then Φ = Ψ C is a general conformable fractional fundamental matrix of (18), where C is a nonsingular constant matrix of order n × n.
Proof Consider that Ψ is a conformable fractional fundamental matrix for (18) and let Φ = Ψ C, where C is any n × n constant matrix. Then Φ is a continuously differential function on I and Hence Φ = Ψ C is a solution of the conformable fractional matrix equation (19). Now for the general conformable fractional fundamental matrix, assume that C is a nonsingular matrix. Since for t ∈ I, Φ = Ψ C is a conformable fractional fundamental matrix of (18). Now the only remaining thing to prove is that any conformable fractional fundamental matrix is of the correct form. suppose that Φ is an arbitrary, but fixed matrix of (18). Let t ∈ I and C 0 = Ψ -1 (t 0 )Φ(t 0 ). Then C 0 is a nonsingular constant matrix For the solution of conformable fractional time-invariant system, we need to define conformable fractional matrix exponential function and its properties.

Definition 7
For the matrix A of n × n order, the conformable fractional matrix exponential function defined by e At α α is the solution of conformable fractional initial value problem In the following theorem we will give some properties of the conformable fractional matrix exponential function.

Theorem 7
Assume that A and B are n × n constant matrices. Then 6. If A has n distinct eigenvalues, say, λ 1 , λ 2 , . . . , λ n , then there exists a non-singular matrix P such that Proof We will show that all these properties hold, as e A (t-a) α α is differentiable. From Lemma 1, it is obvious that property 1 holds. Since e A (t-a) α α is an identity matrix and at t = a, det(I) = 1 = 0, from [29,Corollary 2.24]. So, det[e A (t-a) α α ] = 0 for all t ∈ [a, b). Hence property 2 is satisfied. To establish property 3, let us consider Now, when applying T a α to equation (25), we have So Ψ becomes a solution of the conformable fractional matrix equation T a α X = AX. Also Ψ (a) = 0, so by [13,Theorem 4.2], Ψ (t) = 0. Hence For property 4, we assume that AB = BA and consider Then, by using equation (25), for t ∈ [a, b). Also, Ψ (a) = 0, so by the uniqueness theorem [13], Ψ (t) = 0. Hence We know that e At = ∞ k=0 At k k! . Replacing t by (t-a) α α , we obtain This implies that property 5 is satisfied. Property 6 can be simply proved as in the classical order case.
By using the above definition and properties, we prove the following theorem.

Theorem 8 Consider fractional nonhomogeneous system
where C is a constant vector and Proof Here the fractional nonhomogeneous system T a α y(t) = Ay(t) + f (t) can be written as Multiplying equation (27) with integrating factor e -A (t-a) α α on both sides, we can write Applying fractional integration on the both sides, we have Lemma 3 now yields where C is a constant vector and e A (t-a) α Now, we move towards the solution of the linear conformable differential equation For this, we construct a Peano-Baker series. By the definition of conformable integral, the Peano-Baker series becomes Theorem 9 For t 0 ∈ I and vector x 0 , the conformable linear state equation (28) with continuous A(t) has the unique and continuously differentiable solution If we have the equation where A(t) and B(t) are n × n matrices and x(t), F(t) are n × 1 vectors, then we can guess the solution of the form For time-invariant conformable fractional linear state equation, we can write the approximating sequence (31) as Proof We know that the solution of T t 0 α x = Ax with X(0) = I is e A (t-t 0 ) α α . So, we can prove that there exist analytic scalar functions λ 1 (t), λ 2 (t), . . . , λ n-1 (t) such that According to Cayley-Hamilton theorem, where a 0 , a 1 , . . . , a n-1 are the coefficients of characteristic polynomial of A. Then (38) implies that By equating the coefficients of like powers of A, we get the time-invariant linear state equation 0 · · · 0 -a 0 1 · · · 0 -a 1 . . . . . . . . . . . . 0 · · · 1 -a n-1 Thus there exists an analytic solution of this linear state equation, which shows that the existence of analytic scalar functions λ 1 (t), λ 2 (t), . . . , λ n-1 (t) satisfying (39), and hence (38). So, we can write For the necessary and sufficient condition, we also need a matrix of order n × nm of the following form: B AB · · · A n-1 B .
(40) (30) is controllable for t ∈ [t 0 , t f ] if and only if the n × nm controllability matrix satisfies the following condition:
Therefore x T a B AB · · · A n-1 B = 0, proving that the rank condition (41) fails. This completes the proof.
Example 3 Consider the differential equation of an LC circuit where L is inductance, C is capacitance, q(t) is current, and e 0 is a constant driving force.
Let ω 2 0 = 1 LC be the natural angular frequency. Then the conformable fractional transform operator [17] is By using relation (45), equation (44) becomes We want to check whether the system is controllable or not. For this, we introduce the state variables Then we see that the state representation of equation (45) is where A = Since ω 2α 0 = ( 1 LC ) α = 0, controllability matrix (48) has full rank, that is, Hence, by Theorem 11, system (47) is controllable.

Conclusions
In this article, we have studied the sequential conformable linear equations. Using the conformable fractional derivative approach, we have developed the conformable controllability Gramian matrix, the conformable fundamental matrix, and the conformable controllability matrix. Our results are innovative compared to all the previous results obtained in the conformable case and are application-based as well. They will be highly helpful for the researchers in the future.