Boundary Controllability of Riemann-Liouville Fractional Semilinear Equations

We study the boundary regional controllability of a class of Riemann-Liouville fractional semilinear sub-diffusion systems with boundary Neumann conditions. The result is obtained by using semi-group theory, the fractional Hilbert uniqueness method, and Schauder's fixed point theorem. Conditions on the order of the derivative, internal region, and on the nonlinear part are obtained. Furthermore, we present appropriate sufficient conditions for the considered fractional system to be regionally controllable and, therefore, boundary regionally controllable. An example of a population density system with diffusion is given to illustrate the obtained theoretical results. Numerical simulations show that the proposed method provides satisfying results regarding two cases of the control operator.


Introduction
Mathematical control theory may be seen as an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamic systems through many perceptions, one of them being controllability.In short, controllability means that one can govern the system to any desired state of the dynamical system, in finite time, using a suitable control function into the system.The need to control a system has been visualized in many phenomena, such as the design of electric circuits, manufacturing processes, communication systems, and biological networks [1][2][3][4].
In many phenomena, we are interested to govern the system into a desired state in a subregion of the whole domain and, in some cases, we are also interested to impose some computational requirements.For this reason, an increasing of interest related to regional controllability can be observed in the literature [9][10][11][12].
Regional and approximate controllability of partial differential equations (PDEs) has been the object of intensive researches during the last decade [13][14][15][16].In particular, fractional PDEs become more and more interesting for modeling anomalous phenomena in complex systems theory.Indeed, memory effects, that appear in some complex processes, can be better described by using time-fractional systems than using integer-order equations [17][18][19].This is in particular true for population models, where the fractional operators make the increasing behavior of a population slower.This has been well documented, for instance, by Mirzazadeh et al., who presented a nonlinear time fractional biological population model and established an efficient method to find its exact solution [20].
Due to the large number of applications of fractional-order systems, several papers on controllability of fractional control systems have appeared in the literature [21][22][23][24].For the regional controllability problem, Ge et al. studied the regional controllability of a linear diffusion system with a Caputo derivative of order α ∈ (0, 1] in two cases: when the control operator is bounded, by using the Hilbert Uniqueness Method (HUM) approach; and when one has an unbounded operator, by using a compactness condition [25].Tajani et al. established sufficient conditions for regional controllability of Riemann-Liouville fractional semilinear sub-diffusion systems using two approaches: an analytical approach, based on the contraction mapping theorem; and using the HUM approach, where the regional approximate controllability of the associated linear system is assumed to hold [26,27].
Our goal is to establish the regional boundary controllability, that is, to provide a suitable control that is able to bring the state of the system to a desired one in the boundary of the whole domain.That is the case, for example, of a biological reactor in which the concentration regulation of a substratum at the bottom of the reactor is expected.A fractional system of Riemann-Liouville type can effectively capture the complex behavior of the biological reactor, where the fractional derivative represents the asymmetric heat transfer and the hysteresis effect of temperature variation.Tusset et al. studied the controllability of a cooling fluid flow that passes through the fermented jacket, in order to maintain the ideal temperature in the biological reactor in two cases: integer-order system and Riemann-Liouville type fractional system, in addition to some successful numerical results in both cases [28].To do the regional boundary controllability, our approach consists to study the regional controllability in an internal subregion that contains a part of the boundary, based on a generalization of the Gronwall-Bellman inequality, and then to bring the state to the boundary by projection.
Motivated by the above arguments, and to see the deviation from the classical behavior, which is studied numerically for some real phenomena, for example the cholera outbreak studied in [29], when Baleanu et al. demonstrate that the fractional model of order approximated 1 is more consistent with the real data when compared to the classical model.Our new techniques allow us to establish the regional boundary controllability of a wider class of fractional Riemann-Liouville systems with fractional order in ] 2  3 , 1], under a more general assumption in the nonlinear part, that allows one to cover new problems with relevance in some real problems, for example processes with impulsive type initial condition to describe certain characteristics of viscoelastic materials [30].
The text is organized in the following way.In Section 2, we present the considered problem, we recall some definitions of fractional operators involving Riemann-Liouville and Caputo derivatives, and we also recall the definition of mild-solution, respectively for Riemann-Liouville fractional systems (with a left sided derivative) and Caputo systems with right-sided fractional derivatives.Some useful properties are also introduced, as the regional controllability concept for the kind of systems under investigation.In Section 3, after extending the notion of regional controllability to the boundary case, we establish a relation between regional and boundary regional controllability.
We prove our main result under an assumption in the nonlinear term that often is satisfied in applications of Biology.The result is proved by extending the steps of the HUM method to the fractional-order case under study and by using Schauder's fixed point theorem.To illustrate the obtained result numerically, in Section 4 we apply the proposed method to study the controllability on a part of the boundary of a Riemann-Liouville fractional version of the diffusion logistic growth law model.Our example shows the importance of choosing an appropriate region ω and the location of the actuator to obtain more efficient results on Γ with Γ ⊂ ∂ω.We end with Section 5 of conclusions, pointing also some possible directions of future research.

Problem Setting and Preliminaries
We start by recalling some necessary definitions from fractional calculus and semigroup theory.Definition 2.1 (See [31]).The left sided Riemann-Liouville integral of order α > 0 for a given integrable function y is defined as where Γ is the gamma function.
Definition 2.2 (See [31]).The left sided Riemann-Liouville derivative of order 0 < α < 1 is defined as Definition 2.3 (See [31]).The right sided Caputo derivative of order 0 < α < 1 for a given differentiable function y is defined as Definition 2.4 (See [32]).Let X be a Banach space.We define a C 0 -semigroup (a strongly continuous semigroup) to be a family (S(t)) t∈R + of bounded linear operators from X into X that satisfies: i. S(0) = I, where I is the identity operator of X; ii. S(t + s) = S(t) + S(s) for all t, s ∈ R + ; iii. lim Definition 2.5 (See [32]).The infinitesimal generator of a C 0 -semigroup (S(t)) t∈R + is the following linear unbounded operator: defined for every x in its domain given by Now we introduce the considered problem in this work.Let Ω be an open bounded and regular subset of R n , n ≤ 3, with a smooth boundary ∂Ω.For T > 0, we denote Q = Ω × [0, T ] and Σ = ∂Ω × [0, T ].We consider the following semilinear time-fractional sub-diffusion system with Riemann-Liouville fractional derivative of order α ∈ ]1/2, 1]: where RL D α 0 + and I 1−α 0 + denote, respectively, the Riemann-Liouville fractional derivative of order α and the Riemann-Liouville fractional integral of order (1 − α).The operator A is an infinitesimal generator of a C 0 -semigroup (S(t) t>0 ) on the Hilbert space X = H 1 (Ω).In addition, y ∈ L 2 (0, T ; X), the initial datum y 0 is in X, ν A is the outward unit normal to ∂Ω of the operator A, and B is the control operator, which is a bounded (possibly unbounded) linear operator from R m into X, where m is the number of actuators, u ∈ U := L 2 (0, T ; R m ), and N is a Lipschitz continuous nonlinear operator.The Riemann-Liouville fractional semilinear system (1) is defined for α ∈ ]1/2, 1] because this is a condition for existence and uniqueness of a mild solution.
We proceed by recalling the necessary results and notions to be applied throughout the paper.
Now, consider the following Caputo fractional differential equation: We consider the following definition of a mild solution for (3).
Definition 2.8 (See [26]).The function p(t) ∈ L 2 (0, T ; X) defined by is a mild solution of system (3), where At this point, we can define the mild solution y(x, t, u) := y u (t) of system (1) by the following integral equation: Lemma 2.9 (See [34]).For α ∈ (0, 1) and t ∈ [0, T ] one has where P * α and S * α are the adjoint operators of P α and S α , respectively.Lemma 2.10 (See [35]).For any t > 0, K α (t) := t 1−α P α (t) is a continuous linear bounded operator, i.e., there exists M > 0 such that Lemma 2.11 (See [36]).For γ ∈ (0, 1] and 0 < t 1 ≤ t 2 , the following inequality holds: Let us consider ω ⊆ Ω and denote the projection operator on ω by the following map: We denote by χ * ω its adjoint operator.We finish this section with the notion of internal regional controllability.Definition 2.12 (See [35]).(i) System (1) is said to be ω-exactly controllable if for all y d ∈ X there exists a control u ∈ U such that χ ω y u (T ) = y d .(ii) System (1) is said to be ω-approximately controllable if for all y d ∈ X, given ε ≥ 0, there exist a control u ∈ U such that The main result of our work is to study the regional boundary controllability in the semilinear case, which is the subject of Section 3. To obtain our results for the semilinear case, we assume that the linear part of our semilinear system is regionally controllable and we use available results in the literature about the linear controllability.According to the choice of the supports of actuators, for the controllability of the associated linear part in a subregion ω we can use Proposition 2.13.Proposition 2.13 (See [37]).If we denote then the following two properties are equivalent: 1. the linear system associated to (1) is approximately ω-controllable at time T ; 3 The Boundary Regional Controllability Problem In this section, we explore the Hilbert Uniqueness Method (HUM), first introduced by Lions in [38,39], and further developed in 2016 and 2017 to cover the case of fractional linear systems [25,40].Our goal is to find a control that ensures the boundary regional controllability.For that, we make use of a relation between internal and boundary regional controllability.
Let us consider the trace operator of order zero γ 0 : H 1 (Ω) → H The solution of (6) exists in general.However, an explicit expression for γ * 0 is not available.For more details and properties about the trace and adjoint operators, we refer the reader to [41].
If Γ ⊆ ∂Ω, then we consider χ Γ given by and we denote by χ * Γ its associated adjoint operator.Definition 3.1.We say that system (1) is exactly (respectively approximately) boundary controllable on Γ (B-controllable on Γ) if for all y d ∈ H 1 2 (Γ ) and for all ε > 0, there exists a control u ∈ U such that The following lemma gives a useful relation between internal and boundary regional controllability on Γ .Lemma 3.2 (See [26,37]).If Γ ⊆ ∂ω and system (1) is ω-exactly (respectively ω-approximately) controllable, then it is exactly (respectively approximately) B-controllable on Γ .
Throughout the rest of the paper we assume that Γ ⊆ ∂ω, so that it is sufficient, by Lemma 3.2, to prove the controllability results on a constructive internal part ω of the evolution domain Ω such that Γ ⊆ ∂ω.
The condition α ∈ (2/3, 1] of hypothesis (H 1 ) is necessary in order to deal with the class of semi-linear systems that we are studying (see the proof of Theorem 3.3).Moreover, the interval (2/3, 1] represents a region that includes the classical model (α = 1) and is large enough to describe well the deviation of the classical behavior in several applications [29].
While system (1) is well defined for 1/2 < α ≤ 1, we are only able to prove our regional controllability result under hypotheses (H 0 ) and (H 1 ).We can see that (H 0 ) is a kind of generalization of the conditions used in the HUM approach, and that there are many nonlinear operators satisfying the (H 0 ) condition, which means that several real phenomena are included in our model.For example, (H 0 ) holds for some important nonlinear operators that appear in biological systems, including the logistic growth law model [20,42].Theorem 3.3.Assume that (H 0 ) and (H 1 ) hold.In addition, let the associate linear system to (1) be approximately ω-controllable at time T .Then the semilinear system (1) is exactly controllable in ω by the control function , where ϕ is the mild solution of the following retrograde system: In order to prove Theorem 3.3, we recall the steps of the fractional HUM approach, that transfers our controllability problem to a solvability problem (a fixed point problem to obtain ϕ 0 ).
Let us consider G = {y ∈ X such that y = 0 in Ω\ω} , in which we define the norm [26]: Consider system (1) controlled by u where ϕ(t) = S * α (T − t)ϕ 0 is the mild solution of system (8).We decompose system (9) into three parts: the first being linear, the second linear controlled by u * , (11) and the third being semilinear, such that φ = φ 0 + φ 1 + φ 2 .Let us now define the operators Thus, the controllability problem reduces to solve Since the associated linear system of ( 1) is ω-approximately controllable, then, by Lemma 3.5 in [37], C l is an invertible operator.If we denote then the equivalent solvability problem is to find a fixed point of the operator K.We are now ready to prove Theorem 3.3.
Proof.In order to prove Theorem 3.3, we use Schauder's fixed point theorem.For that we need to prove that K defined by ( 14) maps B(0, r) := B r ⊆ G into itself for r > 0 and, moreover, K is a compact operator.According to (14), tell us that it is sufficient to show that E N is relatively compact.Using the Arzelà-Ascoli theorem, we can prove the relative compactness of E N in two steps: (i) E N is uniformly bounded.For simplify, we denote P ω = χ * ω χ ω and, by the linearity and continuity of P ω , we have that Moreover, φ 2 (t) is the mild solution of (12), which is written as From Lemma 2.10 and hypothesis (H 0 ), we have and, by the same argument used in [26], we get that where Recalling that φ 0 (t) = t α−1 K α (t)y 0 , and we obtain the following inequalities: which are only valid under assumption (H 1 ).
By applying the generalized Gronwall's lemma in [43] with and n = 2, we obtain that , such that T satisfies the following condition: It follows that Thus, E N is uniformly bounded.
(ii) E N is equicontinuous.For ϕ 0 ∈ E N , let us consider t i , t s such that 0 < t i < t s ≤ T .Then, where The goal in this step is to show that J 1 , J 2 and J 3 tend to zero as t s − t i −→ 0, independently of the choice of ϕ 0 in E N , so that we get that E N is equicontinuous.According to Lemmas 2.10 and 2.11 and assumption (7), we obtain that . On the other hand, for t i > 0 and ν > 0 small enough, we have Therefore, by the continuity of the operator K α , it can be seen that J 2 tends to zero as t s − t i −→ 0, ν −→ 0 for all ϕ 0 in E N .For J 3 , we can also obtain the following inequality, which ensures that the limit of J 3 when t s − t i tends to zero is zero: Finally, we prove that K(B r ) ⊆ B r .If this is not the case, then, for each r > 0, it is possible to find a ϕ 0 ∈ B r such that K(ϕ 0 ) G > r.From the definition of K, we have that .
Dividing both sides by r, and taking the lower limit as r → +∞, we obtain a contradiction.We conclude that K(B r ) ⊆ B r .
It should be noted that Theorem 3.3 is not a consequence of known results and that its proof employs a novel method.Here we study the boundary regional controllability for a class of Riemann-Liouville fractional semilinear systems under a condition on the nonlinear part of the system that appears in some real models, for example in the nonlinear growth population model, that is not covered by the results of [26,27].To do so, we use in the proof of Theorem 3.3 the HUM approach and a generalization of the Gronwall-Bellman inequality, which contrasts with the method used in papers [26,27] that is based on the standard Gronwall's lemma.

Algorithm and Simulations
Now we consider the two-dimensional diffusive logistic population growth nonlinear model with a Riemann-Liouville time derivative of fractional order 0 < α < 1: C and K are positive constants, and where we consider the initial data N 0 to be non negative and nonzero.
Note that the Neumann boundary conditions signify that the number of individuals is fixed in the domain.Moreover, the logistic growth operator satisfies our hypothesis.In this case, the operators K α and S α have the following form: where (ξ jk ) j≥1,1≤k≤rj is a complete system of eigenfunctions in X of the Laplace operator and λ j are the associated eigenvalues with the multiplicity r j .
By applying Algorithm 1, we show, through a simulation study of the diffusion logistic population growth law model, the effectiveness of our proposed method to reach the steady state at time T by means of a suitable control function (zonal or pointwise actuator).
We can calculate the expression of ϕ using the formula ϕ(t) = S * α (T − t)ϕ 0 .Consequently, after the decomposition of the mild solution N (x, y, t), we obtain the expression of ψ 0 (t) using the formula of K α .For obtaining ψ 1 (t), we use the expression of K α and B * according to the choice of the actuators, while for ψ 2 (t) we solve the system using the predictor-corrector method and the explicit expression of ψ 0 and ψ 1 .

Example 1: Zonal Actuator
We consider system (15)    Precisely, Figures 1a and 1b display the desired and the reached states in ω and also on Γ .They show that the reached state is very closed to the desired state with an error in ω equal to 8.0 × 10 −3 and an error of order 10 −5 on Γ .Our simulation results, including the error on Γ as a function of the chosen subregion and the actuator location, are summarized in Table 1, which shows that the choice of the region ω is crucial to obtain the boundary regional controllability.
Table 1: Relation Error-Actuator-Region for the example of Section 4.1.

Actuator
Region ω Error on Γ [0.The results obtained by the application of our method are reported in Figures 2a and 2b.They show that the desired state and the steady state are close in ω, differing by 9.31 × 10 −2 , and on Γ by only 6.95 × 10 −5 .Table 2: Relation Error-Actuator-Region for the example of Section 4.2.

Conclusion
The Fractional Hilbert Uniqueness Method (FHUM) has been proposed for solving a boundary regional controllability problem for a new class of semilinear Riemann-Liouville fractional systems.We studied the regional controllability on a subregion of the boundary evolution domain under some assumptions on the nonlinear operator and the order of the derivative.We developed a numerical simulation to illustrate the obtained result through an example of the logistic growth law model with different actuators.An interesting open question concerns the study of the gradient regional controllability of semilinear fractional systems for fractional operators with respect to another function.This is under investigation and will be the objective of our future work.
The desired and reached states in Ω for the example of Section 4.1.The desired and reached states on Γ for the example of Section 4.1.