BOUNDARY CONTROLLABILITY FOR THE TIME-FRACTIONAL NONLINEAR KORTEWEG-DE VRIES (KDV) EQUATION∗

In this paper, we study the time-fractional nonlinear Kortewegde Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.

The Korteweg-de Vries equation was first introduced by Korteweg and de Vries [24] in 1895 as a model for propagation of some surface water waves along a channel. In recent years, it attracted much attention and appeared in several areas such as the models for some water waves, the unidirectional propagation of small-amplitude long waves, the blood pressure waves in large arteries and acoustic-gravity waves in a compressible heavy fluid, see e.g. [3, 5-7, 11, 11, 13-16, 20, 21, 26, 33].
In the present paper, we present other results (the well-posedness and boundary controllability) for problem (1.1). In the process of our study, we face three main difficulties. Firstly, as we all know that, applying the theory of semigroups is a crucial method to investigate the linear estimates and properties of solution for the partial differential equations. But the method only applies to the integer order equations. To solve the time-fractional nonlinear problem (1.1), we resort to the so-called the Laplace transform such that time-fractional nonlinear problem (1.1) has good linear estimates and properties. Secondly, due to the presence of the nonlinear boundary condition, the Kato smoothing property is not strong enough to enable us to establish the controllability for the time-fractional nonlinear problem (1.1) via the contraction mapping principle. Instead, we consider the sharp Kato smoothing property of the backward adjoint system of the linear system associated to the time-fractional nonlinear problem (1.1) (1.5) But it is very difficulty to show that the solution of system (1.5) possesses the hidden regularities for any ϕ T ∈ L 2 (0, L). To get around, we will invoke some harmonic analysis tools (see [8]) to solve our difficulty. Finally, we encounter some difficulties that how to treat the extra term which derived from the process of investing the control of the linear system of problem (1.1) by the usual multiplier method and compactness arguments. To achieve our purpose, we resort to the hidden regularity, again. Finding the observability inequality for the adjoint system (1.5) to overcome the problem. This paper is organized as follows. In Section 2, we give some linear estimates. In Section 3, we present the linear result. In Section 4, we prove the nonlinear results. In the final section, we give some conclusions.

Linear estimates
In this section, we first consider the linear problem of the system (1.1) with nonhomogeneous boundary datas of the form (2.1) Applying the Laplace transform with respect to t in both sides of in (2.1), we have As we all known that, whenĥ 1 (t) =ĥ 2 (t) =ĥ 3 (t) = 0, the solution u of (2.2) can be denoted by where W 0 (t) is the C 0 -semigroup in the space L 2 (0, L) (see [29]) generated by the dissipative linear operator And its definition domain is If u 0 = 0, the solution u of (2.2) can be written as The operator W bdr (t) is the boundary integral operator of Eq.(2.2). In the following, we first look for the explicit representation formula of W bdr (t).
The characteristic equation of (2.2) is s α + aλ 3 = 0, and its solutions are Then, the solutions u(x, s) of (2.2) can be written as and c j = c j (s), j = 1, 2, 3 solve the linear system Using the inverse Laplace transform of u for any c > 0, we obtain that , △(s) is the determinant of the coefficient matrix and △ j (s), j = 1, 2, 3 are the determinants of the matrix by replacing the jth-column of △(s) by the column vector ( h 1 , h 2 , h 3 ) T . Taking the same arguments as those in [30], we infer that the solution u can be written as the following representation Here, when k ̸ = m (k, m = 1, 2, 3), △ j,m (s) is obtained from △ j (s) by letting h m (s) = 1 and h k (s) = 0.
Next, let s = iρ 3 , we have Then, we turn to estimate the solution u(x, t) of Eq.(2.2). The following technical Lemmas due to Bona, Sun and Zhang [2,3] are needed which play a similar role as the Plancherel theorem in estimating u(x, t).
Lemma 2.1 (see [8]) . For any f ∈ L 2 (R + ), let Kf be the function defined by where γ(µ) is a continuous complex-valued function defined on (0, ∞) satisfying the following two conditions: (2) There exist a complex number α + iβ such that Then there exists a constant C such that for all f ∈ L 2 (0, ∞),

Lemma 2.2.
Let T > 0 be given and 0 ≤ s ≤ 3. For any given Proof. Note that as stated above, the solution u can be written as Now, we only prove the result for u 1 , the proofs for u 2 , u 3 are similar. Note that When ρ → ∞,
Which ends the proof of Lemma (2.2) for u 1 .
In the following, we prove the continuity of ∂ k x u + 1 (k = 0, 1, 2) from the space (0, L) to the space H s+1−k 3 (0, T ), for any x, x 0 ∈ (0, L), we have Taking the Plancherel's Theorem in time t yields In views of Fatou's Lemma, This leads to the continuty of ∂ k x u + 1 (x, t).

Linear control results
In this section, we study the boundary controllability of the following linear problem where f ∈ L 1 (0, T ; L 2 (0, L)).
Arguing as before, the solution of Eq.(3.1) can be written as According to Lemma (2.2), we have

(3.3)
In order to the convenience of the readers, we first introduce the following Lemmas.  Now, let µ 1 , µ 2 , µ 3 be three roots of Eq.(3.5), we yield that (3.6) Suppose that there exist double roots. Without loss of generality, we assume that From Eq.(3.6), we have σ = 0. This contradicts with our assumption.
where c i (i = 1, 2, 3) are the solutions of the system  In views of Eq.(3.6), one has By reducing the rows of the matrix Eq.(3.7), we obtain 
In the following, we consider the linear system of Eq.(1.1) 11) we can get the following linear control result. Theorem 3.1. Suppose that T > 0, L ̸ ∈ F hold, there exists a linear and bounded operator Ψ : In the following, set Ψ be the linear bounded map from L 2 (0, L) → L 2 (0, L), and let u be the corresponding solution of Eq.(3.11). From Lemma 3.2, we infer that Using the Lax-Milgram theorem, we obtain that Ψ is invertible. For given u T ∈ L 2 (0, L), one can define ϕ T = Ψ −1 1 s α u(., T ) such that the solution u T ∈ X T of Eq.(3.11) satisfies u| t=0 = 0, u| t=T = u T .
And the following Lemma.
Lemma 4.1 (see [8]). Let 0 ≤ s ≤ 3, T > 0 be satisfied. Then there exists a constant C such that for any u, v ∈ Y s,T , Then, we consider the following time-fraction nonlinear system and get the results. Proof. For given (u 0 , h 1 , h 2 , h 3 ) ∈ S T , we define a set where k > 0, θ > 0. It is clear that the set S s θ,k is closed, conves and bounded. In the following, set a map Ψ 1 on S s θ,k by u( By Lemma 2.1 and Lemma 4.1, there exist constants c 1 , c 2 such that Setting k > 0 and θ > 0 such that then, we infer that Moreover, for any v 1 , v 2 ∈ S s θ,k , we obtain that ω( Applying Lemma 2.1 again yields, When v 1 → v 2 , one has Therefore, Ψ 1 is a contraction mapping of S s θ,k and its fixed point Ψ 1 (u) = u is the unique solution of Eq.(4.1).
Proof. As before, the solution of Eq.(4.1) can be written as Arguing as in the proof of Theorem 4.1, for v ∈ Y s,θ , we have According to Theorem 3.1, for any u 0 , u T ∈ L 2 (0, L), we obtain that (h 1 , h 2 , h 3 ) = (u 0 , u T ) satisfies v(t) = W 0 (t)u 0 + W bdr (u 0 , u T ) − In the following, we consider the map Φ as form Similar to the discussion of Theorem 4.1, we infer that the map Φ is the contraction mapping. Moreover, it is continuous. Hence, its fixed point Φ(u) = u is a solution of Eq.(4.1) with (h 1 , h 2 , h 3 ) = (u 0 , u T ), and satisfying u| t=0 = 0, u| t=T = u T .

Conclusion
As we know, the results of boundary controllability seem to be considered by few authors. In particular, Caicedo and Zhang (2017) dealt with the boundary controllability of the Korteweg-de Vries (KdV) equation on a bounded domain. Motivated by the works described above, in this paper, we study the following time-fractional nonlinear Korteweg-de Vries (KdV) equation where a > 0, L > 0, T > 0, α ∈ (0, 1), D α t u = 1 Γ(1−α) d dt t 0 (t − ξ) −α (u(ξ) − u(0))dξ and Γ represents the Euler Gamma function. Firstly, for the fractional order, we give a proper treatment. Then, we obtain that the control result for the linear system of problem (SP). Finally, we prove the well-posedness and boundary controllability of problem (SP) posed on a finite domain (0, L) with nonhomogeneous boundary conditions.