Sentinel method and distributed systems with missing data

: In this work, we study an approximate controllability problem with constraint on the control. This problem appears naturally of weak and pointwise sentinel with a small or singleton region observation. The main tool is a theorem of uniqueness of the solution of ill-posed Cauchy problem for the parabolic equation.


Notion of the sentinel
The study of dynamic of spatiotemporal systems has generated wide literature with applications in many fields as such ecology, immunology, desertification, population dynamics, pollution as well as many others. Interesting problem for such systems concerns incomplete data and state measurement on a certain region of its geometric domain. In the case of distributed systems defined on a geometric domain Ω, numerous papers were devoted to the state controllability in the whole domain Ω (see Lions [10,12] and the references therein). This work caters with regional analysis paradigm developed by Zerrik [31], El Jai [7] and others, by using the weakly sentinel notion introduced by Rezzoug and Ayadi [1,20] for pollution estimation where the measurement region O is either Ω or in the pointwises of Ω. Precisely, we consider a parabolic distributed parameter system defined on the geometric domain Ω and we assume that the following assumptions are given : -An open regular and bounded set Ω of R n , n ≥ 1,with boundary Γ = ∂Ω. -A second order differential linear operator A with compact resolvent and which generates a strongly continuous semi-group (S(t)) t≥0 on the state space X = L 2 (Ω).
-A * will denote the adjoint operator of A. Then the considered system which is described by the following state equations in Ω, where F ∈ L 2 (Q), g ∈ L 2 (Σ) and y0 ∈ L 2 (Ω).
Have unique weak solution. And we note that for all y and q in the Sobolev space H 1 (Ω). In systems theory, the sentinel is related to the possibility of finding the state of the adjoint system dynamics independently of the missing and pollution terms, and of the choose of control spaces. The regional (boundary) sentinel explores the notion of sentinel in the particular case where the support of the initial state of adjoint system dynamics is into the subregion (a part of boundary) ω.

Regional sentinel
In this section, we choose O in the interior of Ω and we assume that the considered system is described by the in Ω, where f0, y0 are given ; f , y are unknown functions and λ, τ are small unknown parameters. Let h0 be a function One considers a functional defined by the formula where ϕ ∈ L 2 ((0, T ) × O).
The functional S(λ, τ ) is said to be regional sentinel defined by h0 if the following properties are satisfied : × Ω) such that its spacial support is into O.
2) u = inf ϕ for all ϕ satisfying the property one. Now we focus on the regional sentinel construction : let y(t, x) be the unique solution of the following equations in Ω, on Σ. (4) The derivative of the system (2) with respect to the parameter λ near (λ = 0, τ = 0) is given by the following in Ω, and also the derivative of the system (2) with respect to the parameter τ near (λ = 0, τ = 0) is given by the in Ω, on Σ.
The adjoint system associated to (6) is defined by the following equations in Ω, on Σ, with h0 and ϕ in L 2 (]0, T [ × O). The system (7) is decomposed into two systems, free one and forced one. The free system is given by the following equations in Ω, the forced system is given by the following equations in Ω, then the solution of (7) is written as q = q0 + q1.
The dynamic system (9) is said to be regionally controllable on the region ω if, for all desired state, there exists a control such that the final state is equal to the considered desired state on ω.
We consider q0(0, .) ∈ L 2 (Ω) as the desired state and we take a region ω = Ω\O. Then the regional controllability consists in finding a control u in L 2 (]0, T [ ; L 2 (O)) which permits, in a finite time, to bring the state q1 of system (9) from the initial state q1(T, x) = 0, to the final desired state −q0(0, x) on this region.
If the higher multiplicity of the eigenvalue of A is equal to one, then the system (9) is controllable in L 2 (ω), [7,31].
If the system (9) is ω regional controllable, then there exists a unique control u ∈ L 2 (]0, T [ ; L 2 (O)) which satisfies the definition 2.1 of the sentinel.
From the equation (6) and the equation (7) we can deduce and hence, for any y having its support outside O, we have O q(0)ydx = 0, hence

Estimate of the pollution terms
Now, Let ym(t, x) be the measured state of the system on the observatory O during the interval ]0, T [, then the measured regional sentinel is given by formula Theorem 2.2.
If the system (9) is ω−regionally controllable then we have the following estimation Proof. We know that using the equations (11) and (12) we have where and using the equations (5) and (7), we deduce that

Pointwise sentinel
In this section, we choose O = {b} a point in Ω and we assume that the considered system is described by the equation (2). Let h0 be a function given in L 2 (0, T ),one considers a functional defined by the formula where ϕ ∈ L 2 (0, T ).
The functional S(λ, τ ) is said to be pointwise sentinel defined by h0 if the following properties are satisfied : Now we focus on the Pointwise sentinel construction: let y(t, x) be the some solution of (4),the derivative solution with respect to the parameter λ is given by (5) and also the derivative solution with respect to the parameter τ is given by (5) The adjoint system associated to (5) is defined by the following equations in Ω, with h0 and ϕ in L 2 (0, T ). The system (18) is decomposed into two systems, free one and forced one. The free system is given by the following equations in Ω, The forced system is given by the following equations in Ω, there is one function ϕ such that Multiplying the equation (6) by q and integrating by parts, we have Let ym(t, x) be a measured state of the system on the observatory {b} during the interval ]0, T [, then the measured sentinel is given by and we write where

Estimate of the pollution terms
In this section, the objective is to estimate the pollution terms independently of the missing terms.

Theorem 3.1.
Under the hypothesis of the theorem 2.1, the pollution term of the system (5) is estimated independently of the missing term by whereŷ is the solution of (4) and ym is the observed state in {b} during the time interval [0, T ] .

Weak sentinel 4.1. Formulation problem
Where (.) is the partial derivative with respect to time t.
The problem (24) admits a unique solution. For the sake of simplicity, we denote y(x, t; λ, τ ) = y(λ, τ ). One supposes that the data ξ is rather regular, and that the terms of pollution "that one wants to estimate" are rather regular. It will be always supposed that the solution y check at least y ∈ L 2 (Q) .

Remark 4.2.
One will always indicate by y0 the solution y (x, t; 0, 0) ; thus The problem considered here consists in trying to estimate λ ξ starting from observations, distributed or borders, without seeking to estimate the term lack τ y0.
One starts with a distributed observation, therefore a distributed sentinel. (definition, existence and uniqueness of the sentinel) Let h ∈ L 2 (U) and for any control function u ∈ L 2 (U), set

The weak sentinel method
the functional S is said to be weak sentinel if it satisfies the following conditions : for all > 0 there exists u ∈ L 2 (U) such as u ∈ L 2 (U), of minimal norm.
The function u = −h give place to (26) so that the problem (27, 28) admits a single solution, which is defined by h. The problem is thus : (1) to calculate this solution ; (2) to see whether the corresponding sentinel justifies its name, i.e. gives information on pollution λ ξ.

Adjoint state
The adjoint state is introduced q by in Ω.
Where (.) is the partial derivative with respect to time t, h, u ∈ L 2 (U) .

Remark 4.4.
System (29) is the adjoint parabolic problem. It appears under this form in J.L.Lions sentinels theory as the associated adjoint state.
We multiply (29) by yτ and we integrate by parts, we have so that (28) is equivalent to There is thus business with a problem of the type "approximate controllability with zero" .

The main result
The main result is the following Lemma 4.1.
Let v ∈ L 2 (U) .Then there is no ρ ∈ L 2 (Q) , ρ = 0 such that ρ satisfies Proof. If the problem (32) admits a solution, then it is given by Where uj are eigenfunctions of Differentiate the solution (34) once with respect to t and twice with respect to x and substitute these derivatives into the first equation of (32). We then obtain As vχO ∈ L 2 (Q) then Consequently Then the solution of the first order linear is given by Consequently, if the problem (32) admits a solution, it is necessarily in the form : We prove now that ρ / ∈ L 2 (Q) . Indeed, But, λj is the eigenvalue of problem (34), then λj −→ Which means that the series whose general term αj (t) is not normally convergent. So, problem (32) admits no solution.
For > 0, h ∈ L 2 (U) , there exists some control u and some state q such that (29) and (31) hold. Moreover, there exists a unique pair ( uχO, q) with u of minimal norm in L 2 (U) , i.e. such that (29,31) and (27) hold.
Proof. Let q be a solution of the system (29) and q0 a solution of the following system in Ω. (45) We put Then, z is the solution of the following problem in Ω. (47) We now introduce the set of states reachable at time 0 defined by It is clear that F (0) is a vector subspace of L 2 (Ω). According to the HAHN-BANACH theorem, it will be dense in L 2 (Ω) if and only if its orthogonal in L 2 (Ω) is reduced to zero. As {0} ⊂ F ⊥ (0) , it remains to show Where z is solution of (47). It is therefore natural to define the adjoint ρ of z, this is the solution of the following in Ω, Where ρ is solution of (50). Now we multiply the first equation of system (47) by ρ. After integration by parts in Q, it comes Since z and ρ are solutions of (47) and (50) respectively, (51) becomes Therefore, ρ satisfies (50) and (52) and by applying Lemma 4.1, we deduce that As a consequence, ρ 0 = 0 which shows that F ⊥ (0) = {0} .

Characterization of optimal control
In this section, we will characterize the optimal control using a result of Fenchel-Rockafellar duality.
The optimality system satisfied by ( u, q) is established. Let ρ 0 ∈ L 2 (Ω) and ρ the associated solution of in Ω, on Σ. (53) We now introduce the functional J defined by Consider the following unconstrained problem (P ) : Then, we have Proposition 4.1.
The functional J defined in (54) is coercive.
Proof. To prove that J is coercive, it suffices to show the following relation : Let ρ 0 j ⊂ L 2 (Ω) be a sequence of initial data for the adjoint system (53) with ρ 0 j L 2 (Ω) −→ ∞. We normalize them as follows So ρ 0 j L 2 (Ω) ≤ 1. On the other hand, let ρj be the solution of (53) with initial data ρ 0 j . Then, we have We now show that the last integral in equation (58) is bounded. Indeed, we know that ρj is the solution of the (59) Multiplying the first equation of system (59) by ρj then integrating by parts on Q, yields By the Poincaré inequality, (60) becomes, Now, by Cauchy Schwartz inequality, one finds From (61), (62), we conclude that Returning to relation (58), two cases can occur : 1.
T 0 O ρ 2 j dxdt = 0.In this case, since ρ 0 j j is bounded in L 2 (Ω) , we can extract a subsequence ρ 0 j j such that Where ψ is solution of system (53) with initial data ψ 0 . Moreover, by lower semi continuity of the norm, it comes Therefore, And as ψ is solution of (53), and in view of (67), we have Thus, Moreover, from inequality (61), we deduce that But, From (70) and (71), we conclude that So by Lemma 4.1, it comes As a consequence, But, Thus, Hence relation (56) is satisfied.
Proof. As J attains its minimum value at ρ 0 ∈ L 2 (Ω), then, for any ψ 0 ∈ L 2 (Ω) and any r ∈ R we have On the other hand, Substituting (78) in (77) and after simplifications, we find On the other hand, From (79) and (80), we obtain for any ψ 0 ∈ L 2 (Ω) and r ∈ R, Dividing by r > 0 and by passing to the limit r → 0, we obtain The same calculations with r < 0 give so if we take u = ρχO in (58) and we multiply the first equation of the system (58) by ψ solution of (53) and we get after integration by parts over Q, It comes from the last two relations: Ω q(0)ψ 0 dx ≤ ψ 0 L 2 (Ω) ; ∀ψ 0 ∈ L 2 (Ω) .
There are thus always furtive pollution for a sentinel.

Conclusion
In this work we present the weak and pointwise sentinel to estimate the pollution term in diffusion equation when the state governed by unknown datum and missing initial condition when the classical approach of sentinel method gives us information related to the missing data for this we try to avoided this problems by notion of control. This method can be also used in pointwise sentinel and weakly sentinel.

Funding
This work was supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT).

Competing Interests
All authors have no conflict of interest to report.