Regional gradient observability for fractional differential equations with Caputo time-fractional derivatives

We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region, which is contained in the spatial domain. After giving necessary notions and definitions, we prove some useful characterizations for exact and approximate regional gradient observability. An example of a fractional system that is not (globally) gradient observable but it is regionally gradient observable is given, showing the importance of regional analysis. Our characterization of the notion of regional gradient observability is given for two types of strategic sensors. The recovery of the initial gradient is carried out using an expansion of the Hilbert uniqueness method. Two illustrative examples are given to show the application of the developed approach. The numerical simulations confirm that the proposed algorithm is effective in terms of the reconstruction error.


Introduction
The investigation of distributed parameter systems (DPS) drives many useful concepts in science and engineering, including the well-known notions of stability, controllability and observability [10,11,14,15,25,35].These concepts allow one to have a better understanding of the investigated system, enhancing the ability to control it.All these notions are important and have their particularities, but here we only focus on the concept of observability, which was firstly introduced by Kalman, for finite dimensional systems, in 1960 [21].The goal is to recover an initial unknown state using the output parameters or the measurements of the considered system.After the pioneer work of Kalman, the concept of observability was also developed to cover infinite dimensional systems [27,36].
In the nineties of the 20th century, El Jai and others introduced a more general notion called regional observability [2,17,42].Its main objective is to find and recuperate the unknown initial vector of the studied distributed parameter system, but only in a partial region of the spacial domain.The key advantage of regional observability becomes clear when the considered system is not (globally) observable in the whole spatial domain.In such cases, the studied system can be regionally observable in some well chosen sub-region.Thus, we can at least partially recover the initial state, which might be useful in many areas of science [12].After the regional observability concept has been introduced, Zerrik, Badraoui and El Jai proposed the notion of regional boundary observability, which has the same goal of regional observability but where the desired sub-region is a part of the boundary [37,38].Although important, all such notions and results were not enough to get all possible characterizations of DPS.For this reason, in the 21th century the notion of regional gradient observability has been introduced and investigated, with the goal of finding and recover the initial gradient vector in a suitable region [39,40].We adopt here this notion of gradient observability, which has been subject of a recent increase of interest [19,22,23].This is due to the fact that the concept of gradient observability finds applications in real-life situations.For instance, consider the problem of determining the laminar flux on the boundary of a heated vertical plate developed in steady state: see Figure 1 for the profile of an active plate.In this case, the notion of gradient observability is associated to the problem of determining the thermal transfer that is generated by the heated plate.For more on the subject, and for applications of the different observability concepts for various kinds of systems, we refer the reader to [5][6][7][8].
Fractional calculus is one of the most rapidly spreading domains in mathematics nowadays, especially the use of fractional-order systems to model real-world phenomena [3,4,29,31,32].It is well known that fractional operators, non-integer order differentiation and non-integer order integration operators, have many outstanding properties that make them fruitful and suitable for describing and studying the characteristics of certain real-world problems.The non-local fractional operators, not only consider the local points to calculate the (fractional) derivative of some function but also consider the past states, as is the case with left-sided fractional operators, or the future states, as happens with the right-sided operators.We also mention that fractional operators have hereditary properties [33,34].Moreover, the diversity of fractional operators can also be seen as an advantage of fractional calculus because having many different types of fractional integrals and derivatives lead to more choices in the modeling of real-world phenomena.This explains why fractional calculus has been used with success and benefit in various different domains.For more details, we refer the interested reader to the books [28,30].
In the subject of regional observability, we can already find several works dealing with fractional systems [9,13,[43][44][45][46][47].However, investigations of regional gradient observability for time-fractional diffusion processes are scarce.We are only aware of [19], where Ge, Chen and Kou propose a reconstruction procedure for Riemann-Liouville fractional time diffusion processes.The main goal of the present paper is the investigation of regional gradient observability for time-fractional diffusion systems described by the Caputo derivative, where the purpose is to find and reconstruct the gradient of the initial state of the considered system in a desired subregion of the evolution domain.This is in contrast with [19], where non-integer order systems are written with the Riemann-Liouville derivative and where it is mentioned that their approach fails to cover systems described by the Caputo derivative (cf.Lemma 7 of [19]).Here we prove an alternative lemma that fixes the drawbacks mentioned in [19].Our contribution consists of giving several characterizations for regional exact and approximate gradient observability of the considered linear system.We present a method that allows the regional reconstruction of the initial gradient vector in the desired subregion.Moreover, we provide some simple numerical simulations that back-up our theoretical results.
The organization of our paper is done in the upcoming manner.In Section 2, the necessary background information about regional gradient observability, as well as some of its useful properties and characterizations, are given.Section 3 is devoted to illustrate, throughout a counterexample, that we can have a system that is not gradient observable but it is regionally gradient observable in some suitable region included in the evolution space.A full characterization of the notion in hand, via gradient strategic sensors, is then given in Section 4, while in Section 5 we develop the steps to be followed in order to achieve the regional flux reconstruction.In Sections 6 and 7 we present, respectively, two applications of the obtained results and two successful numerical simulations.Finally, we end with Section 8 of conclusions and some future directions of research.

Problem Statement and Regional Gradient Observability
We now present a general formulation of the considered problem of initial gradient reconstruction.We also layout all the needed preliminary results and ingredients to make it easy for the reader to follow smoothly throw the manuscript.
Let Ω be a connected, open, and bounded set in R  ,  ≥ 1, possessing a Lipschitz-continuous boundary Ω.For any final time T ∈ R * + , we designate Q T := Ω × [0, T] and Σ T := Ω × [0, T].Let us take the dynamic of the considered system to be the following operator defined in the state space E = L 2 (Ω) as where   stands for
The operator A is an infinitesimal generator of a C 0 -semigroup {S()}  ≥0 on E, written as: ( Here we study fractional systems possessing the form: (•, ) is the fractional derivative of , in Caputo's sense, and is the Euler gamma function.For  0 ∈ H 1 0 (Ω), both its value and its gradient are supposedly unknown.System (3) has one and only one mild solution in C(0, T; E) ∩ L 2 (0, T; D (A)) of the following form: where stands for the one parameter Mittag-Leffler function [26].
Without any loss of generality, we take () := (•, ).The output function, which provides measurements and information on the consider system, is: The operator C : E → O satisfies the following admissibility condition for S α [48]: The Hilbert space O is called the observation space.The operator S α () defined in ( 4) is a linear bounded operator, see [48], which describes the evolution of the considered time-fractional system in function of its initial state.Moreover, if the operator C is bounded, then the admissibility condition (6) is always satisfied, which means that any bounded observation operator is an admissible observation operator.
Let ω ⊂ Ω be the desired sub-region.We introduce the following restriction operators: and their adjoint, Substituting (4) in (5) gives, where K α : D (K α ) ⊂ E −→ L 2 (0, T; O) is the observability operator, which has an important contribution in defining and characterizing both regional and regional gradient observability.In [20], the admissibility hypothesis on C makes it possible to express the adjoint of K α as: Let ∇ : As shown in [24], the adjoint of ∇ is minus the divergence, that is, ∀V ∈ D (∇ * ), The initial state can be decomposed as follows: The purpose of regional gradient observability is to reconstruct the gradient vector ∇ 1 0 in ω.We recall that system (3) augmented with ( 5) is called exactly (respectively, approximately) ω-observable if, and only if, I Based on the discussion in [39], we denote H α := χ  ω ∇K * α and we enunciate the following definitions.
Definition 1. System (3), augmented with (5), is exactly G-observable in ω (G stands for Gradient) if Definition 2. System (3), augmented with (5), is approximately G-observable in ω if Remark 3. Definitions 1 and 2 for fractional systems coincide with the standard notions for the classical systems in the particular case α = 1 [39].
1) ⇐⇒ 2) This is a direct consequence from the fact that K ( Moreover, we get that, and, using 2), we have: The proof is complete.
Before proving our second result, we recall the following lemma.
Lemma 5 (See [10]).Let F, G and E be three reflexive Banach spaces.Let us consider  ∈ L (F, E) and  ∈ L (G, E).The upcoming assertions are equivalent: The next proposition characterizes the exact regional gradient observability.
The proof is complete.
Remark 7. Our Propositions 4 and 6 generalize the main results of [39], which are only valid for the classical integer-order case α = 1.

A Counterexample
To show the importance of regional gradient observability, we now give an example of a system that is not approximately gradient observable, but it is approximately G-observable in ω.

Gradient Strategic Sensors
In this section, we give a characterization of strategic gradient sensors whenever the considered system is approximately G-observable in the desired subregion.
Definition 9. We call a sensor any element (D,  ), where D is the geometrical placement of the sensor, which is included in Ω, and  : D → R is its distribution.
We introduce here two types of sensors: Remark 10.When we consider a zonal sensor, then the observation operator is bounded; if we take a pointwise sensor, then the observation operator is unbounded but it is an admissible observation operator.
For more information about sensors and their characterizations see [16,20,41].Let us reconsider system (3).We take the measurements to be given by means of  sensors (D  ,   ) 1≤ ≤  .The observation space is O = R  and the output equation is written as: where   () = (),   L 2 (D) , ∀ ∈ 1 ,  .The adjoint of the observation operator C is expressed for all  = ( 1 , . . .,   ) ∈ R  by: for the case of zonal sensors, and by for the case of pointwise sensors.

Definition 11.
A sequence of sensors (or a sensor) is said to be gradient ω-strategic if (3), augmented with (13), is approximately G-observable in ω.
In [19], it is given that a lemma (Lemma 7 of [19]) that fails to be valid when the considered system is written in terms of the Caputo derivative, as we do here.Now we present an alternative new lemma that allows to deal with the problem.

Remark 12. The problem of this article is formulated with Caputo-type fractional derivatives only. However, Riemann-Liouville fractional derivatives appear naturally due to fractional integration by parts and Green's formulas (cf. Lemmas 14 and 15 below).
Lemma 13.Let  be a function that satisfies where: is the right-sided fractional derivative in the sense of Riemann-Liouville, and, is the right-sided Riemann-Liouville fractional integral.Then the following equality holds: The solution of ( 16) can be written as: where N α is a linear and bounded operator defined in terms of a probability density function [48].From Proposition 3.3 of [48], we have that: Hence, from (7), we have that: and the result is proved.
The following fractional integration by parts formula will be useful in the sequel.
Our next result (Theorem 16), provides a useful characterization of gradient ω-strategic sensors.To prove it, we make use of (17) and also the following fractional Green's formula.
Lemma 15 (Fractional Green's formula [48]).For any  ∈ H 2 (0, T; E) one has which means, by using Lemma 13, that: where  is the solution of ( 16).Let us now find the exact expression of χ * ω   ,    I 1−α T −  (0) E .Let  be an element in 1 ,  .We introduce the system: Its unique mild solution is written as: Multiplying both sides of ( 16) by    ϕ and integrating over Q T = Ω × [0, T], we get that: On the other hand, equation (17) gives: From equations ( 18), (21), and ( 22), and using the boundary conditions, we obtain that: Without loss of generality, we continue the proof for the case of zonal sensors (the same can be easily done for pointwise sensors).We have that: Using ( 19), ( 23) and ( 24), we deduce that (D  ,   ) 1≤≤  is gradient ω-strategic if, and only if, From Lemma 5 in [19], we get that the last expression is equivalent to, which is also equivalent to, Because E α (λ   α ) > 0 for all  ∈ [0, T] and for all  ∈ N * , then we have: and the result is proved.
The following corollary is an immediate consequence of Theorem 16 in the one-dimensional case, i.e., when  = 1.

The Regional Gradient Reconstruction Method
Now we present the steps of an approach that permits the recovery of the initial gradient for (3) in ω.Our approach is an extension of the Hilbert uniqueness method (HUM) for fractional systems.
For every φ0 in K, we introduce the system: which possesses one and only one mild solution in L 2 (0, T; D (A)) ∩ C(0, T; E), written as follows: We associate with K × K the form: where Let  be an element of K. Hence, and, since ( 25) is approximately G-observable in ω, we have χ  ω  = 0. We conclude that  = 0 in ω.It follows that  = 0 from the fact that  ∈ K.
Let || • || K be the norm on K associated with •, • K , and let us denote again by K its completion by the norm || • || K .The space K endowed with || • || K is now a Hilbert space.
We introduce the following auxiliary system: controlled by the solution of ( 25).
We now define an operator that associates to every possible candidate of the initial gradient in ω, the projection on K via the operator This way, the problem of regional gradient reconstruction is reduced to a solvability problem of the equation: which leads to the next result.
Theorem 22.If system (3) is approximately G-observable in ω, then equation (30) has a unique solution φ0 ∈ K, which corresponds to the initial gradient ∇ 0 in ω.
Proof.We shall prove that Λ is coercive, that is, there exists σ > 0 that verifies Λ,  K ≥ σ  K for all  ∈ K.We take φ0 to be in K, [48], we have that Therefore, and equation ( 30) possesses only one solution.

Applications
In this section, we take Ω =]0, 1[×]0, 1[.Let ω ⊂ Ω be the desired sub-region.We consider the following time-fractional system: Our goal is to illustrate the steps used to recover the initial gradient vector ∇ 0 =   1  0 ,   2  0 in the sub-region ω.We present the method for the two types of sensors introduced in Section 4. Recall that the eigenvalues and eigenfunctions of  2 and, φ ,  ( 1 ,  2 ) = 2 sin(π 1 ) sin( π 2 ).

Zonal Sensors
Let us take a zonal sensor Proposition 23.This sensor (D,  ) is gradient ω-strategic if, and only if, where, and, Proof.In this case,  = 1,   = 1, and  = 2. Hence, and One can see that: and, Hence, using Theorem 16, (D,  ) is gradient ω-strategic if, and only if, , that is, if, and only if, ).The proof is complete.
Let us now introduce the set: Using Lemma 20, for any ℎ = (ℎ 1 , ℎ 2 ) and  = ( 1 ,  2 ) in K, the expression: defines a scalar product whenever the system ( 32) is approximately G-observable in ω and, , is the associated norm.Keeping in mind formula (14), we can write the adjoint system as follows: It follows from Theorem 22 that the equation Λℎ = (χ  ω ) * χ  ω ∇I 1−α T − ψ 1 (0) possesses one and only one solution in K.
Proof.Similar to the proof of Proposition 23.

Numerical Simulations
In this section, we illustrate the adopted method for solving the gradient reconstruction problem by presenting two examples that show its efficiency.In order to solve equation (30), we calculate the components of the operator Λ for some orthonormal basis φ  ∈N * of E  , denoted by: Λ   := Λφ  , φ  (E)  .
Algorithm 1: Solution to the gradient reconstruction problem.
Step 3. Take φ0 to be the reconstructed initial gradient vector in ω.
After implementing the proposed algorithm (Algorithm 1), we obtain the reconstructed initial gradient φ0 .
As we can see in Figure (2), the two graphs of the initial gradient vector and the recovered one are neighbor to one another in the desired region ω with a reconstruction error: This shows that the numerical approach is successful.We remark that the proposed algorithm does not put into consideration the value of the initial gradient outside of ω. Figure 3 portrays the manner in which the error evolves in terms of the placement of the sensor.As it is seen, there are many positions where the error is large or even explodes to infinity.In this case, we say that the sensor is non-strategic in ω.Moreover, it is clear that the optimal location of the sensor, in the sense that it gives the minimum value of the reconstruction error, is  = 0.2.

Example with a Zonal Sensor
Let us now consider the following fractional system: and take the measurements with a zonal sensor (D,  ) with D = [0.9, 1.0],  = χ D , and the subregion ω = [0.35, 0.65].We choose  0 () = ((1 − )) 2 to be the initial state and the gradient to be recovered as () = 2(1 − ) (1 − 2), which are both supposed to be unknown.We see in Figure 4 that the plot of the initial gradient vector is nearly identical to the plot of the reconstructed initial gradient.In fact, the reconstruction error takes the value:  As it can be seen in the two examples, the case of a zonal sensor gives numerical results with better and smaller reconstruction errors in comparison with the case when we consider a pointwise sensor.This might be due to the fact that, in that case, the sensor has a bounded observation operator and a geometrical support with a non-vanishing Lebesgue measure, which means that the measurements are given in a much larger set compared with the case of a pointwise sensor, where the measurements are provided in a single point , meaning that the quantity of obtained measurements is much less in this case.Therefore, in the case of a bounded observation operator, one has more information on the system than the one with an unbounded operator.These remarks are based upon the observations made during the implementation of the proposed algorithm, but more theoretical studies are needed, regarding the theory of strategic sensors, to confirm and validate, theoretically, the observations of our numerical simulations.

Conclusion
We dealt with the problem of regional gradient observability of linear time-fractional systems given in terms of the Caputo derivative.We developed a method that allows us to obtain the initial gradient vector in the desired region ω.We also gave a complete characterization of the regional gradient observability by means of gradient strategic sensors.Even though we studied two particular cases of sensors, namely pointwise and zonal, similar results can also be obtained for other kinds of sensors, for instance filament ones.The numerical simulations presented in this paper are very satisfying regarding the error rate and the computation time.We implemented the considered examples using the software Matlab R2014b on a 2.5GHz core i5 computer with 8 GB of RAM.The strength of the HUM approach lies in fact that it can be simulated numerically, providing the regional initial gradient with a satisfying control of the error.Moreover, it can be adapted to real-world applications.One weakness that can be faced while applying this approach to an example happens when one considers a dynamic A that possesses an eigenvalue with infinite multiplicity.In such a case, one needs an infinite number of sensors to observe the system, which can never be achieved in reality.For future work, we plan to extend the results of this paper to the case of semilinear fractional systems.Regarding the numerical simulations, we have considered here some academic examples in order to illustrate the obtained theoretical results.We claim that our results can be applied and useful to real-world situations, a question that will be addressed elsewhere.

Figure 1 :
Figure 1: Profile of an active plate.

Figure 4 :
Figure 4: The initial gradient vector and the reconstructed one in Ω for the example of Section 7.2.
•, • O is the scalar product in O.The bilinear form •, • K satisfies the conjugate symmetry and positive properties, i.e., ,  =  ,  and  ,  K ≥ 0. If the system (25) is approximately G-observable in ω, then the bilinear form (27) becomes a scalar product on K. Proof.By Remark 19, we only need to show that •, • K is definite, that is,  ,  K