Regional gradient compensation with minimum energy.pdf 1. Introduction The control problem of distributed parameter systems arises in engineering applications and many different contexts, which are characterized by a spatiotemporal evolution. Systems analysis consists of a set of concepts as controllability, observability, remediability,^that allows a better understanding of those systems and consequently enables to conduct them in a better way. Moreover, the analysis itself has to deal not with the whole domain, but with its specific subdomain of interest. Thus, motivated by practical applications, El Jai and Zerrik have introduced and studied the so-called regional analysis [1-5]. Such analysis aims to analyze or control a system in which an objective function is defined only on a prescribed subregion. Therefore, the system dynamics is defined in the whole the domain Ω , whilst the objective is focused on a given subregion ω, where ω⊂ Ω . An extension of this study that is very important in practical applications is that of regional analysis of the gradient developed in [6-10]. This study is of great interest from a more practical and control point of view since there exist systems that cannot be controllable but gradient controllable or that cannot be observable but gradient observable or that cannot be detectable but gradient detectable and they provide a means to deal with some problem from the real world. With the same preoccupation, the regional gradient remediability and regionally efficient gradient actuators are introduced and characterized recently for linear distributed systems in [11]. In this work, we show how to find practically the optimal control (convenient regionally gradient efficient actuators) ensuring the gradient compensation regionally, based on a result of characterization obtained in our previous work [11]. In addition, it constitutes also an extension of our previous work [12] to the regional case. This paper is organized as follows. In the second section, we start by presenting the considered problem. After, we recall the definitions of exact and weak regional gradient 20 S. Rekkab, H. Aichaoui, S. Benhadid remediability, the notion of regional gradient efficient actuators, and a characterization which shows that the regional gradient remediability of any system may depend on the structure of the actuators and sensors. In section 3, under a condition on the structure of the actuators and the weak regional gradient remediability hypothesis, using an extension of Hilbert Uniqueness Method (HUM), we examine the problem of gradient remediability with minimum energy regionally. Then, we give the optimal control which compensates an arbitrary disturbance. In the last section, approximations, simulations, and numerical results are presented. 2. Considered problem, definitions, and characterization Let Ω be an open and bounded subset of IRn ((n = 1,2,3) with a regular boundary ∂Ω. Fix T > 0 and let denoted by Q = Ω× ]0,'T[, ∑ = ∂Ω× ]0,'T[ . Consider the system described by the parabolic equation dyy(x,t) = (x,t) + Bu(t) + f (x,t) - Q, У(x,0) = y0 (x) - Ω, .У(ξ, t) = 0 - ∑, (2.1) where A is a second order linear differential operator which generates a strongly continuous semi-group (5 (t ))t≥0 on the Hilbert space L2 (Ω). (S' * (t ))t ≥0 is considered for the adjoint semi-group of (S(t))t≥0 . B∈L(U,X),u∈L2 (0,T;U) where U is a Hilbert space representing the control space and X = H0 (Ω) the state space. The disturbance term f ∈L2 (0,T;X)is generally unknown. In system (2.1), the disturbance function f has a space support which can be, in practical applications, a part ω of the domain Ω (ω⊂ Ω). The system (2.1) admits a unique solution y ∈ C(0, T;H0 (Ω)) ∩ C' (0, T;L2 (Ω)) [13] given by yu f (t) = S (t)У° +∫ S(t - 5)Bu (5) ds + ∫S (t - s) f (s) ds . 00 For ω⊂ Ω an open subregion of Ω with a positive Lebesgue measure, we consider the operators Xω : (l2 (ω)) → (ω)) , and Xω : L2 (Ω)→ L (ω), •У → .У ω , •У → У ω , while their adjoints denoted by χ*, and jχω respectively, are defined by X*ω : (l2 (ω)) → (l (ω)) , and χ ω : L2 (ω)→ L2 (Ω), * У on ω, y →χωy = {0 on Ω\\ω, * У on ω, y →χωy = {0 on Ω\\ω. Regional gradient compensation with minimum energy 21 Consider also the operator V defined by V: H1 (Ω)→(L≈ (Ω))n, y →Vy = k∂y, ∂^y-.....∂y. 1 к ∂ X1 ∂ X2 ∂ xn ) while V*its adjoint operator. The system (2.1) is augmented by the regional output equation zuω,f(t)=CχωVyu,f(t), (2.2) where C ∈ L ((L2 (ω))n ,O), O is a Hilbert space (observation space). In the case of an gradient observation on [0,T] with q sensors, we take generally O= IRq . In the autonomous case, without disturbance ( f = 0) and without control (u = 0), the gradient observation in ω is given by z0ω,0(t)=CχωVS(t)y0, it is then normal. However, if f ≠ 0 and u ≠ 0, the regional gradient observation is disturbed. The problem consists to study the existence of an input operator B (actuators), with respect to the output operator C (sensors), ensuring the gradient compensation at finite time T , of any disturbance acting on the system, that is to say: For any f∈L2 (0,T; X), there exists u∈L2 (0,T;U), such that zuω,f(t)=CχωVS(T)y0, this is equivalent to CXω^Hu + C Xω^Ff = 0, where H and F are two operators defined by H : L2 (0,T;,U)→ X, F: L2 (0,T;X)→ X, T and T u→Hu=∫S(T-s)Bu(s)ds, f →Ff =∫S(T - s)f ( s)ds. 00 This leads to the following definitions. Definition 1. 1) We say that the system (2.1) augmented by the output equation (2.2) is exactly regionally f-remediable in ω, if there exists a control u∈L2 (0,T;U), such that CXω^Hu + C Xω-^Ff = 0. 2) We say that the system (2.1) augmented by the output equation (2.2) is weakly regionally f-remediable in ω on [0, T ] , if for every ε>0,there exists a control u∈L2 (0,T;U)such that ∣∣CXωVHu + CXωVFf∣∣^R, < 0. 22 S. Rekkab, H. Aichaoui, S. Benhadid 3) We say that the system (2.1) augmented by the output equation (2.2) is regionally exactly (resp. weakly) remediable in ω, if for every f ∈ L2 (0,T ; X )the system (2.1) -(2.2) is exactly (resp. weakly) f-remediable in ω . We suppose that the system (2.1) is excited by p zone actuators (Ωi,g^ )1≤i≤,g^ ∈ L2 (Ωi), Ωi ⊂ω , ∀i = 1,^,p , in this case the control space is U= IRp and the operator B is given by B:IRp →X, p u (t) = (u1 (t),u2 (t),-,up (t)) ∣→ ^u = ΣXΩi. (x)gi (x')ui (t). i=1 Its adjoint is given by β*z = g1, z) Ω1 g 2, Ω2 ,-∖ gp, Ω ) ≡ 1r' . Also suppose that the output of the system (2.1) is given by q sensors (Di , hi )1≤i≤q , hi ∈ L2 (Di ), being the spatial distribution, Di=supphi⊂ω, for i = 1,.^, q and D^ ∩ D- =φ for i ≠ j, then the operator C is defined by C:(L2 (ω))n →IRq , y(t) Cy(t) = f∑h1,Уі(tDe'^{h2,Уі(tD,,-∙,'^{hq,Уі(t^D J , V i=1 i=1 i=1 q / its adjoint is given by C* with forθ = (θ1,θ2, ^, θq )" ∈ IRq , C*θ = I ΣxDi (x )θιhι (x), (x )θιhι (x),-, ∑X D, (x)θιhι (x )| ∈ Jl O»))” . Vi=1 i=1 i=1/ We recall the following notion of the regionally gradient efficient actuator [11|. Definition 2. The actuators (Ωi, g^ )1≤i≤, g^ ∈ LJ (Ωi) are said to be regionally gradient efficient, if the system (2.1) - (2.2) so excited is weakly regional gradient remediable. Form≥1, let Mm be the matrix of order (p×rm) defined by Mm =^gi, wm^p2 ω ))ιj ,1 ≤ i ≤ P and 1 ≤ j ≤ rm and let Gm be the matrix of order (q×rm) defined by ∂wmΛ I , 1 ≤ i ≤ q and 1 ≤ j ≤ rm . l2 (Di) Ju- Corollary 1 [11]. If there exists m0 ≥ 1, such that rankGmT0 = q, V k=1 ’ dxk I, (2.3) Regional gradient compensation with minimum energy 23 then the actuators (Ωi,gi )1≤.≤,g^ ∈ L2 (Ωi) are regionally gradient efficient if and only if ∩ ker(MmGmT )={0}. m≥1 3. Regional gradient remediability with minimum energy Under a condition (2.3) and the weak regional gradient remediability hypothesis, we study in this section the problem of the exact regional gradient remediability with minimal energy. For f ∈L2 (0,T;X), we study the existence and the unicity of an optimal control u∈L2 (0,T;U) ensuring, at the time T , the exact regional gradient remediability of the disturbance f , such that CχωVHu+CχωVFf=0. That is the set defined by (3.1) D = {u ∈ L} (0, T;IRp ) /Cχω-,VHu + CχωVFf = θ} is non empty. We consider the function J (u ) = ICχωVHu + CχωVFf ∣∣2r, +∣∣U∣∣ L2(0,T;IRp). The considered problem becomes minJ (u) . u∈D For its resolution, we will use an extension of the Hilbert Uniqueness Methods (HUM). For θ∈IRq , let us note 1 ι∣θ I*=[∫l ∖b*s * (t - s )v*χωC*θ∣ IRp ds 12. The corresponding inner product is given by (θ, σ * = β B* S * (t - s) v*χωC*θ, B* s * (t - s) v*χζc*σ ^s 0 and the operator Λ : IRq → IRq defined by Λθ = Cχω'^HH*V*χ*ωC*θ = CχωV∫(T - s)BB*(T - s) V*χζC*θ ds. 0 Then, we have the following proposition: Proposition 1. If the condition (2.3) is satisfied, then || ∣∣*is a norm on IRq if and only if system (2.1) - (2.2) is weakly regional gradient remediable on [0,T] and the operator Λ is invertible. 24 S. Rekkab, H. Aichaoui, S. Benhadid Proof. We have ∣∣θ∣, =[∫∣B*S*(t-)V*χωC*θ∣2^P d^j2 = 0 ⇒∣∣B*S*(T-.)V*χωC*θL2(tirp) = 0 ⇒B*S*(T-.)v*χ*ωC*θ=0⇒θ∈ker(B*S*(T-.)v*χ*ωC*)=ker(B*F*v*χ*ωC*). But ker(B*F*V*χ*ωC*)=∩≥ ker(Mmfmω), m≥1 where, for m ≥1, fω :θ∈IRq → fmω(θ)=((C*θ,Wm^^V*χωC*θ,Wm^,^J^V*χ*ωC*θ,Wmm^^)t ∈)m . Indeed, let θ∈ IRq , we have B* F *v*χωc *θ=B* S * (T-.)V*χ*ωC *θ and we have ∀m ≥1 ( rm Mmfmω(θ)= ∑^{v*χ*ωC*θ; wm^L2(ω∕g1, Wm^L2(Ω1) ^∑{V*χωC*θ; wm^L2 (ω∕ g 2, wm^L2 (П,) ∑{ V*χζC *θ; wm^L2 (ωS gP , wm^L2 (Ω p If we assume that θ ∈ ∩ ker(Mmfmω), then m≥1 θ∈ker(Λ^mfωT, ∀m ≥l⇒ΣV*χωC*θ; Wm^}L2(Ω^g≈, wm^ = 0,∀i∈{l,2,^, p},∀m ≥1 ⇒ ∑ eλm(T-.) τ∑(V*χ*ωC*θ , wm^{g,, wm^ = 0, ∀i ∈ {1,2, , P} , ∀m ≥ 1 ⇒ B*F*V*χ*ωC*θ = 0⇒ θ∈ker(B*F*V*χ*ωC*), where ∩ ker(Mm fmω)⊂ ker(B*F*v*χ*ωC*), m≥1 Regional gradient compensation with minimum energy 25 that is ∩≥ ker (Mmfω) = ker (B F *V*χζC*) m≥1 and we have also ∩ ker(MmGmT ) = ∩ ker(Mmfmω). Indeed, let θ∈IRq , then m≥1 m≥1 θ∈∩ ker(MmGmT ) ⇔ (MmGmT )θ=0, ∀m≥1, mm mm m≥1 q rm n ∂ w ⇔ΣΣgi, wm^{hl ∑-∂Xm^ θl = 0, ∀m ≥ 1, ∀i =1,-, P , l=1j=1 rm k=1 ∂Xk L2(Dl ) ⇔ Σ{g,, wm^(v*χ*ωC*, wm^j2(ω) = 0,∀m ≥ 1,∀i = 1, ^ , P , j =1 L ⇔(Mmfmω)θ=0, ∀m≥1 ⇔θ∈∩ ker(Mmfmω). mm mm m≥1 Where ker(B*F*V*χ*ωC*) = ∩ ker(MmGmT ) , this gives θ∈ ∩ ker(MmGmT ) and m≥1 m ≥1 since the Corollary 1, we obtain the result. On the other hand, the operator Λ is symmetric, indeed, , ,h} ∫e^■■^^-^'u,, (s)ds (4.5) m'=1h=1k=1 i=1 i ∂xk L2(Dj)0 f Nrm' n∂w t + ]Σ Σ Σ{dWχmh, hA ∫ e-mF-sU^f (s), Wmh},2 (ω) ds. m'=1h=1k=1 ∂xk L2(Dj )0 28 S. Rekkab, H. Aichaoui, S. Benhadid 4.2. Numerical simulations We recall the problem considered above: (P) l^Find u ∈ lL (0,T∖U), such that ( )tc Xω^Hu + C Xω'^Ff = 0. Based on Proposition 2 and using the above results, we can develop an algorithm which allows us to determine a sequence of controls which converges to the solution u* of (P). The output is given by (4.4) - (4.5). Algorithm Step 1: Data: the domain Ω , the subregion ω , the final time T, the initial state y0, the disturbance function f , the sensors (D, h) , the efficient gradient actuators (σ, g), and the accuracy threshold ε . Step 2: Choose a low truncation order M = N . Step 3: Compute z0ω,0 : the output, when f = 0 and u = 0 (an autonomous case). Step 4: Compute z0ω, f : the output, when f ≠ 0 and u = 0 (a disturbed case). Step 5: Solve a finite dimension linear system Αθ = b , where these coefficients are given by (4.1) - (4.2). Step 6: Calculate u given by (4.3). u ≠ 0 (a disturbed and Step 7: Compute zuω, f : the output when f ≠ 0 and controlled case, that is to say a compensate case). Step 8: If ∣∣zω f - zω 01∣l2( ) ≤ ε, then stop. Otherwise, Step 9: M M +1 and N N +1 and return to step 3. u* of (P) . Step 10: the optimal control u corresponds to the solution Now, we give a numerical example, which illustrates the efficiency of the approach given in the above section. Illustrative example. We consider without loss of generally the following diffusion system dyy (χ, t ) = δ y (χ, t )+∑xω,.g, (χ x)u, (t)+f (χ, t) ω×]0, t [ y(x,0) = y0 (x) Ω , y (ξ, t ) = 0 ∂Ω×]0, T [ with Ω = ]0,1[ and a Dirichlet boundary condition. In this case, the functions wm (.) are defined by wm (x) ^'∕2sin(mπx);m ≥ 1. The associated eigenvalues are simple and given by λm = -m2π2 ; m ≥ 1. Let ω = ]0.15,0.25[ (ω ⊂ Ω) be the geometrical support or the disturbance. Then in the case of: - an initial state: У0 (.)= 0, Regional gradient compensation with minimum energy 29 - a sensor: (D,h) with D = ω and h(x) = 42x^ (q = 1), - an efficient actuator: (σ,g) with σ=ωand g(x)=2x3 (p=1), - a disturbance function: defined by f(x)=240cosx. For M=N=10 and T = 0.5, we obtain numerical results illustrating the theoretical results established in previous sections. Hence, in Fig. 1, we give the representations of the discrete observation zu, f corresponding to the control u= uθ and the disturbance f f and z0,0 , which represent the normal observation, that is u = 0 and f = 0. Fig. 1. Representation of zu,f (line 1), z0,f (line 2) and z0,0 (line 3) This figure shows that the disturbance f is compensated by the optimal control u=uθ at the time T that is, we have zuω,f(T)= z0ω,0(T). f The optimal control uθ f ensuring the regional gradient remediability of the disturbance f is represented in Fig. 2. 30 S. Rekkab, H. Aichaoui, S. Benhadid Table 1 shows that if we want to eliminate the effect of the disturbance in a short time T , the cost increases. Table 1 Evolution cost with respect to the finite time T T Cost 0.3 1.06∙105 0.4 1.05∙105 0.5 1.03∙105 1 9.46∙104 2 8.99∙104 3 8.89∙104 5 8.85∙104 10 8.84∙104 100 8.84∙104 Conclusions Under a condition on the sensors and the weak regional gradient remediability hypothesis, we have studied the problem of exact regional gradient remediability with minimal energy. That is to say, when a system is subjected to disturbances, we have shown how to find the optimal control, which compensate the effect of the disturbance that can be located in a given subregion of the space domain, with respect to the regional gradient observation and this using an extension of the Hilbert Uniqueness Method. Illustrative examples, numerical approximations, and results are also presented. These results are developed for a class of discrete linear distributed parabolic systems, but the considered approach can be extended to regional or bounded gradient remediability of other class of systems with a convenient choice of space.

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