An Asymptotic Analysis of the Gradient Remediability Problem for Disturbed Distributed Linear Systems

: The goal of this work is demonstrating, through the gradient observation of a disturbed distributed parameter systems of type linear ( 𝐷𝐷𝑃𝐿 -systems), the possibility for reducing the effect of any disturbances (pollution, radiation, infection, etc. ) asymptotically, by a suitable choice of related actuators of these systems. Thus, a class of asymptotically gradient remediable system ( 𝐴𝐺𝑅 -system) was developed based on finite time gradient remediable system ( 𝐺𝑅 -system). Furthermore, definitions and some properties of this concept 𝐴𝐺𝑅 -system and asymptotically gradient controllable system ( 𝐴𝐺𝒞 -controllable) were stated and studied. More precisely, asymptotically gradient efficient actuators ensuring the weak asymptotically gradient compensation system ( 𝑊𝐴𝐺𝐶 -system) of known or unknown disturbances are examined. Consequently, under convenient hypothesis, the existence and the uniqueness of the control of type optimal, guaranteeing the asymptotically gradient compensation system ( 𝐴𝐺𝐶 -system), are shown and proven. Finally, an approach that leads to a Mathematical approximation algorithm is explored.


Introduction:
Driven by environmental, pollution 1 , radiation and infection problems 2-3 , the authors have studied the problem with regard to the gradient observation of a class of -systems considering the possibility of lessening or compensating asymptotically the effect of any disturbances. Thus, the study constitutes a development to the case of asymptotic type for the previous investigates to the remediability linear parabolic problem of different systems, introduced in the finite time case 4-7 and asymptotic case 4, 8, 9 . One can note that studying compensation problem with respect to the gradient observation and the so-called gradient remediability, is of considerable interest 10 . Thus, it was shown that there exists a system that is not remediable, however may be gradient remediable.
Gradient remediability concept in usual and regional case is considered and studied for systems [10][11][12] . Regarding the asymptotic case aspect 13 , the great importance of the asymptotic analysis in systems theory [14][15] , takes into consideration the problem of -systems and studies a prospective extension of the development methods, in addition to analyzing the results in finite time. Hereafter, through likeness the relationship among the remediability and controllability of the gradient case has been inspected and studied in a considerable time.
Also, the link among remediability and controllability in asymptotic gradient case has been studied and analyzed. This paper is structured as follows: Section 2, is devoted to the introduction of the gradient remediability concepts of type exact and weak under convenient hypothesis. Section 3 relates to the asymptotic form in various cases in connection with suitable actuators and sensors. Also, an asymptotically gradient efficient actuators enable the guaranteeing an asymptotic gradient compensation of weak type is presented.
In section 4, weakly and exactly a asymptotically gradient controllable system ( -system) are defined and characterized. Then, the link between -system and weakly and exactly asymptotically gradient remediable system ( -system) are studied and analyzed, and it is shown that system is dependent on the appropriate sensors with corresponding actuators. While, in section 5, the -problem through the energy of type minimum is examined. In the last section, the control of optimal type, is used to obtain a mathematical algorithm approach.

Remark 1
The finite time gradient compensation problem is equivalent to: For any ∈ 2 (0, ; ), does there exists a control ∈ 2 (0, ; ) such that Consequently, the finite time gradient remediability of ( ) + ( ) can be also formulated as follows: For any ∈ 2 (0, ; ), there exists a control ∈ 2 (0, ; ) satisfying Eq.1. The characterizations consequences on the -systems and in limited time have been established by Rekkab and Benhadid, and they have shown that the remediability concept of type gradient is a weaker than controllability of type gradient 10 .

Asymptotic Gradient Compensation Problem: Formalism statement:
An asymptotic analysis of the problem is given by considering the system: The asymptotic gradient remediability problem was studied to consist an investigation regarding the output operator , the existence of an input one confirming the gradient compensation asymptotically of any disturbance, that is : For any ∈ 2 (0, +∞; ), there exists ∈ 2 (0, +∞; ) such that ∇ ∞ + C∇ ∞ = 0 2 Note that the operators ∞ and ∞ are not generally well defined. They are, if and only if the following condition is verified 14 : ∃ ∈ 2 (0, +∞; ℝ + ) such that ‖ ( )‖ ≤ ( ); ∀ ≥ 0 3

Characterization:
For the following results, let * and * be the adjoint operators of and respectively and ( * ( )) ≥0 is considered for the semigroup of ( ( )) ≥0 of type adjoint. Let also ′ , ′ and ′ be the dual space of , and . Under hypothesis Eq.4, the following general characterization results are obtained:

Asymptotic Gradient Efficient Actuators and Sensors:
The notion of asymptotic gradient efficient actuator have been presented analogy to the concept of gradient efficient actuator in finite time given as follows: 10 The suite ( , ) 1≤ ≤ , is called asymptotic gradient efficient actuators ( actuators) if, ( ∞ ) + ( ∞ ) is -systems. The case of asymptotic relation is difficult and requires more conditions. Asymptotic Gradient Controllability: Assuming the system that is described by the following equation: is supposed generates a strongly continuous semi-group ( ( )) ≥0 such that ∃ ∈ 2 (0, +∞; ℝ + ) such that ‖∇ ( )‖ ≤ ( ); ∀ ≥ 0 8 Next, some sufficient conditions to characterize the -system are given in the following results.
Let ℰ ′ , ′ be the dual spaces of ℰ and respectively, then using Lemma 1, it is easy to show the following results the following proposition 5 characterizes the -systems, and systems.

(ii)
-systems if and only if : The following results in proposition 6 demonstrate that the asymptotic controllability concept of type gradient is strongest than the asymptotic remediability of type gradient in various situations.

Remark 3
The opposite of Proposition 6 is not correct; this case may be exemplified via the following.

Example 1
Reflect the subsequent one dimensional system of type diffusion.
For its resolution, one can use a modification of (H. U. M) 14 .

Proposition 7
If the condition Eq.7, is verified, then ‖. ‖ * is a norm on ℝ q if and only if ( ∞ ) + ( ∞ ) is system and the operator ∞ is invertible. and since the Corollary 4, the result is obtained.

Proof
By utilizing Proposition 7, the mapping Λ ∞ has inverse, now, ∈ 2 (0, +∞; ), then there exists a unique ∈ ℝ such that Λ ∞ = − ∞ and by putting = ( ∞ ) * , yields that The current part of this paper, presents important approximations augmented with an approximation approach for -system. First we give an approximation of as a solution of a finite dimension linear system = and then the optimal control , with a comparison between the corresponding observation noted , , and the normal case.
: Control of type optimal links to * the solution of (ℙ).

Conclusion:
In this paper, the problem of analysis has been presented. Certainly, it is based on suitable hypothesis and an appropriate choice of operators and spaces. Furthermore, -system and actuators have been presented firstly. Also the problem of -system has been examined under a suitable hypothesis with appropriate choice of spaces and operators. More precisely, the relationship between -system and system has been demonstrated in different important results. Indeed, in the asymptotic case, it has been proved that the controllability concept of gradient type remains stronger than the remediability concept of gradient type, that is to say, -system can be asymptotically gradient remediable but, it is not -system. Thus, through the choice of sensors and hypothesis of -system, the problem of system with minimum energy has been studied. Moreover, the issue of how to discover an optimal control has been examined in a way compensateing for the influence of the disturbances about the observation of gradient via the use of H U M modified.
Regarding the digital processing, some mathematical approximations are proposed, using a multi-step algorithm.
Later, the obtained outcomes have been introduced for class -systems and may be interesting to expand this work to regional or regional bounded case with other classes under the suitable different select of spaces, for example, the possibility to replace the observability concept in this paper by an asymptotic observer.