Regional Reconstruction of Semilinear Caputo Type Time-Fractional Systems Using the Analytical Approach

The aim of this paper is to investigate the concept of regional observability, precisely regional reconstruction of the initial state, for a semilinear Caputo type time-fractional di usion system. The approaches attempted in this work are both based on xed point techniques that leads to a successful algorithm which is tested by numerical examples.


Introduction
"An apparent paradox from which one day useful consequences will be drawn", the answer that Leibniz gave when l'Hospital had asked him about the meaning of d n dx n f (x) when n = 1 2 , and then fractional calculus was born. Even-though fractional calculus is approximately 300 years of age, it only caught so much attention in the last 25 years or so, the main reason being, its capability of a better describing of real world phenomena. In fact a lot of works show that non-integer order ordinary and partial dierential equations present, most times, better modeling systems than integer order ones, for instance in [6] the authors presented a model of beam heating process, and with experimental setup (thermo-electrical module), theoretical results were Email addresses: zguaid.khalid@gmail.com (Khalid Zguaid), fzelalaoui2011@yahoo.fr (Fatima-Zahrae El Alaoui), (Ali Boutoulout) veried and a high degree of accuracy was obtained in the experimental ones. Also the modeling of the ultra-capacitor is given in [8], and in [7] both examples of the heat beam process and the ultra-capacitor are given. More information on fractional calculus can be found in [1,12,13,14,22,24,32,35,36,37,39,40] Control theory is an important and very active branch of mathematics which serves as a link between theoretical mathematics and its applications in the real world where most processes are modeled by nonlinear distributed parameter systems, which explains the big interest of researchers in the study (Controllability, Observability, Stability...) of nonlinear and semilinear systems. As for the study of linear systems, there exists a very wide literature for integer order system, see [9] - [34] and the references therein, whereas for fractional order systems there is a much less literature, see [17,26,29,30,38] and the references therein.
Most works in control theory precisely in observability deal with state estimation in the whole evolution domain, namely Ω, of the considered system (global observability) [34], but in the 90's the regional observability concept was born by professor El Jai et al. [31], and was after that developed by others, its purpose is to estimate the initial state of a given system only in a subregion ω ⊂ Ω, with positive Lebesgue measure, several works have treated this notion for various kind of systems [3,4,5,18,20,23]. The main reason for introducing such a concept is its applicability to non observable systems in the whole domain.
Here we give an extension of the results of regional observability of semilinear systems to time-fractional semilinear systems. We make two approaches in order to reconstruct the initial state in ω, the rst one consists of reconstructing the trajectory of the system in ω, "y(t)| ω ", than substituting t with 0, we get the wanted result (the direct approach), while the second one, where we make some assumptions so that the dynamic of the system generates an analytical semigroup on the state space, gives directly the initial state in ω as a xed point of a function to be dened later.
This manuscript is organized as follows. In the second section we give some introductory notions and useful tools for a better comprehension of the manuscript, as for the third section we introduce the considered system, the denition of its mild solution and we also talk about regional observability. The fourth section deals with the direct approach whereas in the fth one we present the analytical approach , and just before the conclusion, we give, in section six an algorithm for the regional reconstruction of the initial state also as two numerical results.

Preliminary Notes
We layout, in this section, some denitions and properties, which will be used all over this manuscript. We will start with the denition of the Caputo left fractional derivative. Denition 2.1. [2] The left sided fractional derivative of order α ∈ ]0, 1[ in Caputo's sense of y(x, t) with respect to t, is given by the following expression For two Banach spaces E and F we denote the space of all linear bounded mappings dened from E to F by L (E, F ) and L (E) := L (E, E), also we mean by L p (0, T ; E) (resp. L p loc (0, T ; E)), the space of all vector-valued functions f going from the interval [0, T ] to E, which are measurable, such that f (.) E is in L p (0, T ) (resp. L p loc (0, T )). The two following propositions show, respectively, the existence of the convolution between an operator and a vector-valued function, and the Young inequality for this convolution [41]. Proposition 2.2. [41] Let E and F be two Banach spaces. Let's consider v ∈ L 1 loc (0, T ; E) and T : [0, T ] → L(E, F ) be strongly continuous. Then the convolution exists (in the Bochner sense) and is a continuous function (T * v : [0, T ] → F ).
The previous convolution ( * ) is called the Laplace convolution operator. Proposition 2.3. [41] With the same considerations as in the previous proposition. Let's consider p, q, r ≥ 1 such that For more information about vector-valued analysis see [21,41]. An important type of C 0 -semigroup is the analytic one, especially in nonlinear systems. Before we give its denition, we need to dene the following sector ∆ θ := z ∈ C |arg(z)| < θ , z = 0 , where θ ∈]0, π[.
can be extended to the sector ∆ θ and it satises the following properties : Here is a useful characterization of an analytic semigroup.
Proposition 2.5. [28] Let A be a densely dened, linear closed operator from E to itself. If we assume that A enjoys the following conditions : , π[, we have that ∆ θ ⊂ σ(A).
ii-∃M 0 > 0 such that the resolvant R(λ, A) satises the estimate Then, A generates an analytic semigroup on E.
Before we nish the current section, let's introduce the Following weighted Lebesgue space, for all q ≥ 1 and α ≤ 1 which is a Banach space endowed with the norm Remark 2.7. For all q ≥ 1, we have the following inclusions,

Considered System
Let Ω be an open bounded subset on R n , with smooth boundary Γ = ∂Ω and ]0, T ] a time interval. Let's denote Q = Ω × ]0, T ] and Σ = Γ × ]0, T ]. Consider the following fractional semi-linear evolution equation : in Ω, together with the output function : Where : * C D α 0 + is the left sided fractional derivative in Caputo's sense. * y 0 is the initial state. * A is a linear, second order, dierential operator which generates a C 0 -semigroup {S(t)} t≥0 on the state Space L 2 (Ω).
* C is a linear operator (operator of observation) from L 2 (Ω) into O (the observation space).
* N is a nonlinear operator, assumed to be dened to ensure the existence and uniqueness of a mild solution of the system (2) in L 2 (0, T ; L 2 (Ω)) see [25,42,43].
Without loss of generality we denote y(., t) = y(t).
We associate with the system (2), the following linear system We call a mild solution of (2) any function y ∈ C 0, T ; L 2 (Ω) which is written as follows [25,42,43]: The function W α is called "The Mairandi function", and it is written as follows : where α is a probability density function dened on ]0, +∞[ by : Remark 3.1. The Mairandi function is an alternating series, hence its sign is the same as its rst term, which is The Mairandi function satises the following Proposition 3.2. [25,43] For all v ≥ −1, we have : Let ω be a subregion of Ω with positive Lebesgue measure, we dene the restriction operator χ ω : (Ω) by χ ω y = y |ω and we denote by χ * ω its adjoint. We decompose the initial state to two parts :ỹ 0 = χ ω y 0 , the restriction of y 0 in ω or the initial state in ω (to be reconstructed), and y 0 , the residual (undesired) part of y 0 , we then have y 0 = χ * ωỹ 0 + χ * Ω\ω y 0 .
We introduce the operator L α (.) : L 2 (0, T ; L 2 (Ω)) −→ L 2 (0, T ; L 2 (Ω)), dened as follows : The solution of the system (2) can be written The observability operator can be given by If C is unbounded, we assume that it is an admissible observation operator for S α , that is : , ∀v ∈ L 2 (Ω).
Remark 3.3. If C is bounded, then it is an admissible observation operator.
The condition of admissibility on C gives us the wright to extend the operator CS α (t) to a bounded linear operator from L 2 (Ω) to O, see [11,16]. In both cases (bounded or not) the adjoint operator of K α , can be written This denition is equivalent to Denition 3.5. We say that the system (2), augmented with the measurements (3), is initially continuously observable in ω, if it is possible to reconstruct y(0)| ω , depending on z in a continuous way.

Direct Approach
For ω ⊂ Ω, we assume that (4)-(3) is approximately ω-observable. We dene the following mapping Proposition 4.1. The initial state of (2) in ω is the restriction in ω of a xed point of Φ at t = 0.
Proof. We have By applying the operator C, we get (4)-(3) is approximately ω-observable, by applying the pseudo inverse of K ω α , we havẽ Substituting (12) in (11), we get Thus y(.) is a xed point of Φ, and For the next result we suppose that Φ has a unique xed point y * (.), for example if Φ is a strict contraction. Proposition 4.2. If the following condition is satised, Then y * (0)| ω is the estimated initial state of (2) in ω.
Proof. We have then Cy * (.) = z(.), In all the previous results we worked with the residual part being any function in L 2 (Ω \ ω), so we can take y 0 = 0 for the rest of this work.

Analytical Approach
In this section, we shall use another approach where we make some assumptions that will allow our dynamic, A, to generate an analytic semigroup and −A to have a fractional power of order α ∈]0, 1[. Both of these consequences play an important role in the resolution of semilinear evolution systems. In fact the benet of working with a dynamic that generates an analytic semigroup is that it provides good information one has on the behavior of the solution at time t −→ 0 + , whereas, fractional powers of −A allow us to dene interpolation spaces ,between D(A) and L 2 (Ω), in which might lie the solution of our system, since, for semilinear systems, the solution might not live in the evolution space, in our case L 2 (Ω).
We make the following assumptions on the operator A : i-∃θ ∈] π 2 , π[, ∃b > 0, such that ii-∃M 1 > 0, such that R(λ, A) satises The conditions (i) and (ii) provide us with some useful consequences, see [28]. The rst consequence is that the fractional power of the operator −A, of order α ∈]0, 1[, is well dened and D((−A) α ) is dense in L 2 (Ω). The second one is that, using proposition (2.5), A generates an analytic semigroup, denoted again by {S(t)} t≥0 , on the state space L 2 (Ω), in fact one can see that if (i) is satises then ∆ θ ⊂ σ(A) and if (ii) is veried, (1) is also true. For any α in ]0, 1[ we denote, X α := D((−A) α ) in which we dene the following norm x X α := (−A) α x . Since (−A) α is bounded then . X α is equivalent to the graph norm. In addition to the fact that (−A) α is closed, we get that X α , endowed with the norm . X α , is a Banach space.  (2), one can reprove the upcoming results with a slight change in the expression of some constants, we kept the same order to simplify the calculations.
For the operator N we need to make the following hypotheses : iii-∃ q, p, r ≥ 1 verifying 1 q The nonlinear operator N : L r (0, T ; X α ) −→ L p (0, T ; L 2 (Ω)) is well dened and satises Even-though the condition (iii)-(**) on the operator N might seem harsh, yet, it can actually be achieved as shown in [27], in fact it was used to obtain the exact global observability of semilinear classical systems, also in [15] we nd that this condition on N is valid for an important classes of systems, such as the Burgers' equation.

As for the second
again by (8), we deduce that ||H α (t)|| The goal here is to study the regional reconstruction problem for (2)-(3) in V = Im(χ ω K * α ), which is a Banach space endowed with the norm ||.
We shall now, in the following proposition, show the existence of a set (ball) of admissible initial states in ω, in the sense that they give a unique solution of (2) in a ball of L r (0, T ; X α ).

. The mapping
"which for every initial stateỹ 0 in ω gives us the corresponding unique solution of (2)", satises the Lipschitz condition.
Remark 5.7. The constants a and m(a) are not unique, in fact ∀b ≤ a, ∃ m(b) such that the pair (b, m(b)) satisfy the last proposition.
Proof. : which gives , then ∀x, y ∈ B(0, a), we have :  0).a, which leads to hence m > 0. We deduce, from the Banach xed point theorem, that φỹ 0 has a unique xed point in B(0, a).
2 . Let x and y be two solutions of (2) with initial states in ω, respectively,x 0 andỹ 0 , we have : , which gives Finally, h is Lipschitz continuous.
In the next result we show that the initial state in ω (ỹ 0 ) is a solution of a xed point problem, keeping in mind that the measurements are in a ball of L 2 (0, T ; O). The solution of (2) can be written as applying the observation operator, we get or equivalently, K ω αỹ 0 = z(.) − CL α (.)N y, and since the system (4)- (3) is approximately ω-observable, we obtaiñ thenỹ 0 can be seen as a xed point of Φ z (.). (4)  Then we have the following assertions :

Numerical Approach
For this section we adopt the same assumptions of the fourth section. Let's consider the following sequence, , and for every n in N, we consider the following fractional system, with the output equation, z n (t) = Cy n (t).
Theorem 6.1. The sequence {y n 0 } n≥0 converges to the desired initial state y 0 in ω. Proof. All we need to show is that y n 0 converges to h (z) in V . Firstly we will show that (y n 0 ) n≥0 is a Cauchy sequence. We have, which gives that (y n 0 ) n≥0 is a Cauchy sequence and eventually convergent. Remark that, from the sequence denition, we have Thus we obtain the following algorithm, .

Simulations
We give here numerical illustrations for the obtained algorithm. We show, with the same fractional system, both types of sensors, zonal and pointwise. Remark 6.2. The results are related to the choice of ω, the sensor's location also as the initial state of the system.

Pointwise Sensor
For this case we consider the order of derivation α = 0.2 and that : The subregion ω = [0.4 0.75]. The measurements are given by means of a pointwise sensor located in b = 0.8, which means z(t) = y(0.8, t).
The reconstruction error is : y 0 −ỹ 0 L 2 (ω) = 1.21 × 10 −2 . We remark in gure (1) that the reconstructed initial state is very close to the initial one in the desired subregion ω.
The gure (2) shows the evolution of the reconstruction error in function of the sensor's location, and it is very clear that the reconstruction error is sensitive to the position of the sensor.
The reconstruction error is : y 0 −ỹ 0 L 2 (ω) = 9.4 × 10 −3 . As we can see in gure (3), the estimated initial state is quite near the actual one. In order to show that the error changes with the choice of the geometric domain of the sensor, we give the following

Conclusion
In this paper we shed light on the concept of regional observability of semilinear Caputo type timefractional diusion systems, of order α ∈ ]0, 1]. The two dierent methods that we gave are both based on xed point techniques, and regarding future works, we intend to investigate the same problem with the Hilbert Uniqueness Method (HUM), we also plan to study the regional boundary observability for the same class of systems.