Controllability of conformable differential systems

Xiaowen Wang, JinRong Wang, Michal Fečkan Department of Mathematics, Guizhou University, Guiyang 550025, Guizhou, China xwwangmath@126.com; jrwang@gzu.edu.cn School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia michal.feckan@fmph.uniba.sk Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia


Introduction
Conformable calculus and equations has a rapid development in basic theory and application in many fields. For example, Khan and Khan [9] concerned the open problem in Abdeljawad [1] and introduced the generalized conformable operators, which are the generalizations of Katugampola, Riemann-Liouville, and Hadamard fractional operators. Bendouma and Hammoudi [3] established the conformable dynamic equations on time scales with nonlinear functional boundary value conditions and obtained the existence of solutions. Bohner and Hatipoǧlu [4] used conformable derivatives to establish thenew dynamic cobweb models and obtained the general solutions and stability criteria. Abdeljawad et al. [2] proposed conformable quadratic and cubic logistic models and obtained existence theorems and stability of solutions. Jaiswal and Bahuguna [7] proposed conformable abstract Cauchy problems via semigroup theory, introduced the concept of mild and strong solution, and obtained existence and uniqueness theorem. Bouaouid et al. [5] investigated nonlocal problems for second-order evolution differential equation in the frame of sequential conformable derivatives and presented Duhamel's formula and existence, stability, and regularity of mild solutions. Martínez et al. [11] applied this new conformable derivative to analyze RC, LC, and RLC electric circuits described by linear differential equations with noninteger power variable coefficients derivative. However, there are quite a few papers on controllability of systems governed by conformable differential equations.
In this paper, we study controllability of linear and semilinear conformable control systems governed by where D 0 α (0 < α < 1) denotes the conformable derivative with lower index zero (see Definition 1), M ∈ R n×n and Q ∈ R n×r , f : J × R n → R n . The state x(·) take values from R n , the control functions u 1 (·) and u(·) belong to L 2 (J, R r ).
The main contributions are stated as follows: (i) We establish conformable Gram criterion and rank criterion to give the necessary and sufficient conditions to guarantee (1) is null completely controllable. The corresponding control function is also presented. (ii) We construct a suitable control function and apply Krasnoselskii's fixed point to derive complete controllability of (2).

Preliminaries and notation
Let R n be the n-dimensional Euclid space with the vector norm · and R n×n be the n×n matrix space with real value elements. Denote by C(J, R n ) the Banach space of vectorvalue continuous functions from J → R n endowed with the norm x C = sup t∈J x(t) for a norm · on R n . Let X, Y be two Banach spaces, L b (X, Y ) denotes the space of all bounded linear operators from X to Y , and L p (J, Y ) denotes the Banach space of all the Bochner-integrable functions endowed with · L p (J,Y ) for some 1 < p < ∞. For M : R n → R n , we consider its matrix norm M = sup x =1 M x generated by · . 0 denotes the n-dimensional zero vector.
Remark 1. If D a β x(t 0 ) exists and is finite, we consider that x is β-differentiable at t 0 . If x : [a, ∞) → R is a once continuous differential function, then D a β x(t) = (t − a) 1−β × x (t).
Definition 2. (See [10,Thm. 3.3].) A solution x ∈ C(J, R n ) of system (1) has the following form: Obviously, a solution x ∈ C(J, R n ) of system (2) has the following form: Definition 3. System (1) is called null completely controllable on J if for an arbitrary initial vector function x 0 , the terminal state vector 0 ∈ R n , and terminal time t 1 , there exists a control u 1 ∈ L 2 (J, R r ) such that the state x ∈ C(J, R n ) of system (1) satisfies Definition 4. System (2) is called completely controllable if for an arbitrary initial vector function x 0 , for the terminal state of vector x 1 ∈ R n and time t 1 , there exists a control u ∈ L 2 (J, R r ) such that the state x ∈ C(J, R n ) of system satisfies x(t 1 ) = x 1 .
Lemma 1 [Krasnoselskii's fixed point theorem]. Let B be a bounded closed and convex subset of Banach space X, and let F 1 , F 2 be maps of B into X such that F 1 x + F 2 y ∈ B for every pair x, y ∈ B. If F 1 is a contraction and F 2 is compact and continuous, then the equation F 1 x + F 2 x = x has a solution on B.

Linear systems
In this section, we are going to investigate the null completely controllable of system (1). We introduce a notation of a conformable Gram matrix as follows: where the denotes the transpose of the matrix. Then we will give the first controllability result. Proof. Sufficiency. Owing to W c [0, t 1 ] is nonsingular, its inverse W −1 c [0, t 1 ] is well defined. For any nonzero state x in the state space, the corresponding control input u 1 (τ ) can be constructed as: According to (3), for all x 0 ∈ R n , one can get Consequently, we can get Owing to system (1) is relatively controllable, according to Definition 3, there exists a control u(τ ) that drives the initial state to zero at t 1 , that is, According to (8), we can havē Then we get By (7) and (9), one can get x 0 2 = 0, that is which contradicts the conditions of thex Now, we introduce a notation of a rank criterion as follows: Then we are ready to give the second controllability result.
Theorem 2. The necessary and sufficient condition for null complete controllability of (1) is rank Γ c = n.
Proof. Sufficiency. Assuming that system (1) is not controllable. By Theorem 1, W c [0, t 1 ] is nonsingular. Namely, there exists at least one nonzero state vector β such that For (12), find the derivative of z = τ α /α to n − 1 times and then take τ = 0. We have According to β = 0, we can know Γ c < n, which contradicts to our assumption. So system (1) is controllable.
Necessity. Assume rank Γ c < n. Namely, there exists at least one nonzero state β in R n such that From (14) we obtain According to Cayley-Hmilton theorem, M n , M n+1 can be expressed as a linear combination of I, M , . . . , M n−1 . Then the upper form can be expanded to That is, From (16) we can know W c [0, t 1 ] is singular, namely, the system is not controllable, which is contradictory to what is known. Thus, rank Γ c = n. The proof is complete.

Semilinear systems
We introduce the following assumptions: has an inverse operator W −1 , which takes values in L 2 (J, R r ) \ ker W .
Then we set

Remark 2.
Obviously, W must be surjective to satisfy (A1). We recommend the reader to see the demonstrated examples in [6,12]. On the other side, if W is surjective, then we can define an inverse W −1 : R n → L 2 (J, R r )\ker W . We present its natural construction as follows. Let (·, ·) denote the Euclid scalar product in R n . Since L 2 (J, R r ) is a Hilbert space, we can use ker W = im W * ⊥ . We need to find W * . LetW (τ ) = e M (t α 1 −τ α )/α Q, and for any w ∈ R n and u ∈ L 2 (J, R r ), we derive So the surjectivity of W is equivalent to the regularity of W c [0, t 1 ], and we assume this. To solve W u = v, u ∈ ker W ⊥ = im W * , we take u(t) = W (t) w and then solve In addition, by (17), we derive We note that (17) also implies In viewing of (A1), for arbitrary x(·) ∈ C(J, R n ), consider a control function u x (t) given by Next, we state our main idea to prove our main result via fixed point method. We firstly show that, using control (19), the operator P : C(J, R n ) → C(J, R n ) defined by has a fixed point x, which is just a solution of system (2). Then we check (Px)(t 1 ) = x 1 and (Px)(0) = x 0 , which means that u x steers system (2) from x 0 to x 1 in finite time t 1 . This implies system (2) is relatively controllable on J.
For each positive number r, we define B r = {x ∈ C(J, R n ): x C r}, which is obviously a bound, closed and convex set of C(J, R n ). For the sake of brevity, we set R f = sup t∈J f (t, 0) and N = M .
Step 1. We attest that there exists a positive number r such that P(B r ) ⊆ B r . Note In consideration of (19), using (A1), (A2), and e At e A t , t ∈ R, we have , and H 2 is defined in the above. According to (A1) and (A2), we have . Therefore, we obtain P(B r ) ⊆ B r for such an r. Next, we split P into two operators P 1 and P 2 on B r as, respectively, Step 2. We prove that P 1 is a contraction mapping. Let x, y ∈ B r . In light of (A1) and (A2), for each t ∈ J, we obtain From the above fact we get http://www.journals.vu.lt/nonlinear-analysis which gives that According to (20), we conclude that L < 1, which implies P 1 is a contraction.
Step 3. We show that P 2 is a compact and continuous operator. Let x n ∈ B r with x n → x in B r . Denote F n (·) = f (·, x n (·)) and F (·) = f (·, x(·)). Using (A2), we have F n → F in C(J, R n ) and thus uniformly for t ∈ J, which implies that P 2 is continuous on B r . In order to check the compactness of P 2 , we prove that P 2 (B r ) ⊂ C(J, R n ) is equicontinuous and bounded.
In fact, for any x ∈ B r , t 1 t + h t > 0, it holds where Combining the previous derivations, we have Then we check I i → 0 as h → 0, i = 1, 2, uniformly for t. For I 1 , using (A2), For I 2 , it is easy to get that From above we obtain uniformly for all t and x ∈ B r . Thus, P 2 (B r ) ⊂ C(J, R n ) is equicontinuous. According to the above computations, one can get Thus P 2 (B r ) is bounded. By Arzela-Ascoli theorem, P 2 (B r ) ⊂ C(J, R n ) is relatively compact. Hence, P 2 is a compact and operator. Then Krasnoselskii's fixed point theorem gives that P has a fixed point x on B r . Apparently, x is a solution of system (2) satisfying x(t 1 ) = x 1 . The proof is completed. where and α = 0.9. By elementary calculation, the conformable Gram matrix of system (21) with (22) via (5) can be written into: Obviously, W c [0, 1] is nonsingular. Therefore, according to (6), we obtain Finally, by Theorem 1, system (21) with (22) is null completely controllable on [0, 1].