# Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality

Authors: Mariela  Olguín; Domingo A. Tarzia
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energyg. The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive h (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter h goes to zero.


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## Body

## 1. Introduction

We consider a bounded domainΩ⊂Rn whose regular boundary ∂Ω=Γ1∪Γ2 consists of the union of two disjoint portions Γ1 and Γ2 with meas(Γ1)>0. We consider the following free boundary problem (S):(1)u≥0;u-Δu-g=0;-Δu-g≥0inΩ;(2)u=bonΓ1;-∂u∂n=qonΓ2,where the function g in (1) can be considered as the internal energy in Ω, b is the constant temperature on Γ1, and q is the heat flux on Γ2. The variational formulation of the above problem is given as follows: find u=ug∈K such that ∀v∈K(3)au,v-u≥g,v-uH-∫Γ2qv-uds,where(4)V=H1Ω,K=v∈V:v≥0inΩ,v=bonΓ1,V0=v∈V:v=0onΓ1,H=L2Ω,Q=L2Γ2,u,vQ=∫Γ2uvds∀u,v∈Q,u,vH=∫Ωuvdx∀u,v∈H,au,v=∫Ω∇u·∇vdx∀u,v∈V.We note that a is bilinear, continuous, and symmetric on V and a coercive form on V0 [1]; that is to say, there exists a constant λ>0 such that(5)av,v≥λvV2∀v∈V0.In [2], the following continuous distributed optimal control problem associated with (S) or the elliptic variational inequality (3) was considered as follows.Problem
(
P
). Find the continuous distributed optimal control gop∈H such that [3–5](6)Jgop=ming∈H⁡Jg,where the quadratic cost functional J:H→R0+ is defined by(7)Jg=12ugH2+M2gH2with M>0, a given constant, and ug is the corresponding solution of the elliptic variational inequality (3) associated with the control g.Several continuous optimal control problems are governed by elliptic variational inequalities, for example, the process of biological waste-water treatment; reorientation of a satellite by propellers; and economics: the problem of consumer regulation of a monopoly and so forth. There exists an abundant literature for optimal control problems governed by elliptic variational equalities or inequalities [6–12], for numerical analysis of variational inequalities or optimal control problems [13–16].The objective of this work is to make the numerical analysis of the optimal control problem(P) which is governed by the elliptic variational inequality (3) by proving the convergence of a discrete solution to the continuous optimal control problems.In Section2, we establish the discrete elliptic variational inequality (10) which is the discrete formulation of the continuous elliptic variational inequality (3), and we obtain that these discrete problems have unique solutions for all positive h. Moreover, on the adequate functional spaces these solutions are convergent when h→0+ to the solutions of the continuous elliptic variational inequality (3).In Section3, we define the discrete optimal control problem (31) corresponding to continuous optimal control problem (6). We prove the existence of a discrete solution for the optimal control problem (Ph) for each parameter h and we obtain the convergence of this family with its corresponding discrete state system to the continuous optimal control with the corresponding continuous state system of the problem (P).


## 2. Discretization of the Problem(S)

LetΩ⊂Rn be a bounded polygonal domain; b a positive constant; and τh a regular triangulation with Lagrange triangles of type 1, constituted by affine-equivalent finite elements of class C0 over Ω, h being the parameter of the finite element approximation which goes to zero [17, 18]. We take h equal to the longest side of the triangles T∈τh and we can approximate the sets V and K by(8)Vh=vh∈C0Ω¯:vh∈P1TonT,∀T∈τh,Vh0=vh∈Vh:vh=0onΓ1,Kh=vh∈Vh:vh≥0,vh=bonΓ1,where P1(T) is the set of the polynomials of degree less than or equal to 1 in the triangle T. Let Πh:V→Vh be the corresponding linear interpolation operator and c0>0 a constant (independent of the parameter h) such that, ∀v∈Hr(Ω), 1<r≤2 [17]:(9)v-ΠhvH≤c0hrvr,v-ΠhvV≤c0hr-1vr.The discrete variational inequality formulation (Sh) of system (S) is defined as follows: find uhg∈Kh such that ∀vh∈Kh(10)auhg,vh-uhg≥g,vh-uhgH-q,vh-uhgQ.Theorem 1.
Letg∈H, b>0, and q∈Q; then there exists unique solution of the problem (Sh) given by the elliptic variational inequality (10).Proof.
It follows from the application of Lax-Milgram Theorem [1].Lemma 2.
Letg1,g2∈H and uhg1, uhg2∈Kh be the solutions of (Sh) for g1 and g2, respectively; then one has that(a)
there exists a constantC independent of h such that(11)uhgV≤C,∀h>0;(b)
∀h>0(12)uhg2-uhg1V≤1λg2-g1H;(c)
ifgn⇀g in H weak, then uhgn→uhg in V strong for each fixed h>0.Proof.
(a) If we considervh=b∈Kh in the discrete elliptic variational inequality (10) we have(13)λuhg-bV2≤auhg-b,uhg-b≤g,uhg-bH+q,b-uhgQ≤gH+qQγ0uhg-bV,where γ0 is the trace operator and therefore (11) holds.
(b) Asuhg1 and uhg2 are, respectively, the solutions of discrete elliptic variational inequalities (10) for g1 and  g2, we have(14)auhgi,vh-uhgi≥gi,vh-uhgiH-q,vh-uhgiQ,∀vh∈Khfor i=1,2. By coerciveness of a we deduce(15)λuhg2-uhg1V2≤auhg2-uhg1,uhg2-uhg1≤g2-g1,uhg2-uhg1H≤g2-g1Huhg2-uhg1V∀h>0;thus (12) holds.
(c) Leth>0. From item (a) we have that uhgn≤C∀n; then there exist η∈V such that uhgn⇀η in V weak (in H strong). If we consider the discrete elliptic inequality (10) we have(16)auhgn,vh-uhgn≥gn,vh-uhgnH-q,vh-uhgnQand using the fact that a is a lower weak semicontinuous application then, when n goes to infinity, we obtain that(17)aη,vh-η≥g,vh-ηH-q,vh-ηQand from uniqueness of the solution of problem (Sh), we deduce that η=uhg∈Kh.
Now, it is easily to see that(18)auhgn-uhg,uhgn-uhg≤-g-gn,uhgn-uhgHand from the coerciveness of a we obtain(19)λuhgn-uhgV2≤g-gn,uhgn-uhgH.As uhgn→uhg in H and gn⇀g in H, by passing to the limit when n→∞ in the previous inequality, we obtain(20)limn→∞uhgn-ugV=0.Henceforth we will consider the following definitions [2]: given μ∈[0,1] and g1,g2∈H, we have the convex combinations of two data items(21)g3μ=μg1+1-μg2∈H,the convex combination of two discrete solutions(22)uh3μ=μuhg1+1-μuhg2∈Kh,and we define uh4(μ) as the associated state system which is the solution of the discrete elliptic variational inequality (10) for the control g3(μ).Then, we have the following properties.Lemma 3.
Given the controlsg1,g2∈H, one has that(a)
(23)uh3H2=μuhg1H2+1-μuhg2H2-μ1-μuhg2-uhg1H2,(b)
(24)g3μH2=μg1H2+1-μg2H2-μ1-μg2-g1H2.Proof.
(a) From definition (22) we get(25)uh3H2=μ2uhg1H2+1-μ2uhg2H2+2μ1-μuhg1,uhg2H,uhg2-uhg1H2=uhg2H2+uhg1H2-2uhg1,uhg2H,and then we conclude (23).
(b) It follows from a similar method to part (a).Theorem 4.
Ifug and uhg are the solutions of the elliptic variational inequalities (3) and (10), respectively, for the control g∈H, then uhg→ug in V strong when h→0+.Proof.
From Lemma2 we have that there exists a constant C>0 independent of h such that uhgV≤C∀h>0, and then we conclude that there exists η∈V so that uhg⇀η in V weak as h→0+ and η∈K. On the other hand, given v∈K there exist vh∗ such that vh∗∈Kh for each h and vh∗→v in V strong when h goes to zero. Now, by considering vh∗∈Kh in the discrete elliptic variational inequality (10) we get(26)auhg,uhg≤auhg,vh∗-g,vh∗-uhg+q,vh∗-uhgQand when we pass to the limit as h→0+ in (26) by using the fact that bilinear form a is lower weak semicontinuous in V we obtain (27)aη,η≤aη,v-g,v-η+q,v-ηQ;that it is to say, (28)aη,v-η≥g,v-η-q,v-ηQ∀v∈Kand, from the uniqueness of the solution of the discrete elliptic variational inequality (3), we obtain that η=ug.
Now, we will prove the strong convergence. If we considerv=uhg∈Kh⊂K in the elliptic variational inequality (3) and vh=Πh(ug)∈Kh in (10), from the coerciveness of a and by some mathematical computation, we obtain that(29)λuhg-ugV2≤auhg-ug,uhg-ug≤auhg,Πhug-ug-g,Πhug-ug+q,Πhug-ugQ;then by passing to the limit when h→0+ it results in limh→0+uhg-ugV=0.


## 3. Discretization of the Optimal Control Problem

Now, we consider the continuous optimal control problem which was established in (6). The associated discrete cost functional Jh:H→R0+ is defined by the following expression:(30)Jhg=12uhgH2+M2gH2and we establish the discrete optimal control problem (Ph) as follows: find goph∈H such that(31)Jhgoph=ming∈H⁡Jhg,where uhg is the associated state system solution of the problem (Sh) which was described for the discrete elliptic variational inequality (10) for a given control g∈H.Theorem 5.
Given the controlg∈H, one has(a)
(32)lim∥g∥H→∞⁡Jhg=∞;(b)
Jh(g)≥M/2gH2-CgH for some constant C independent of h;(c)
the functionalJh is a lower weakly semicontinuous application in H;(d)
there exists a solution of the discrete optimal control problem (31) for all h>0.Proof.
(a) From the definition ofJh(g) we obtain (a) and (b).
(c) Letgn⇀g in H weak; then by using the equality gnH2=gn-gH2-gH2+2(gn,g)H we obtain that gH≤liminfn→∞gnH2. Therefore, we have(33)liminfn→∞⁡Jhgn≥12uhgH2+M2gH2=Jhg.
(d) It follows from [4].Lemma 6.
If the continuous state system has the regularityug∈Hr(Ω)(1<r≤2) then one has the following estimations ∀g∈H:(a)
(34)uhg-ugV≤Chr-1/2,(b)
(35)Jhg-Jg≤Chr-1/2,where C’s are constants independent of h.Proof.
(a) Asug∈K, we have that Πh(ug)∈Kh⊂K. If we consider vh=Πh(ug) in (10), by using the inequalities (29), we obtain(36)λuhg-ugV2≤auhg-ug,uhg-ug≤auhg,Πhug-ug-g,Πhug-ug+q,Πhug-ugQ≤CΠhug-ugV≤Cugrhr-1≤Chr-1,and then (34) holds.
(b) From the definitions ofJ and Jh, it results in(37)Jhg-Jg=12uhgH2-ugH2=12uhg-ugH2+ug,uhg-ugand therefore(38)Jhg-Jg≤12uhg-ugH+ugHuhg-ugH≤Chr-1/2.Following the idea given in [2] we define an open problem: given the controls g1,g2∈H and ∀μ∈[0,1], ∀h>0(39)0≤uh4μ≤uh3μinΩ,(40)uh4μH≤uh3μH.Remark 7.
We have that (39)⇒(40).Remark 8.
The equivalent inequality (39) for the continuous optimal control problem (P) is true; that is [2], for all g1,g2∈H, and ∀μ∈[0,1],(41)0≤u4μ≤u3μinΩ,where u3(μ)=μug1+(1-μ)ug2∈K, ugi(i=1,2) is the unique solution of the elliptic variational inequality (3) when we consider gi instead of g and u4(μ) is the unique solution of the elliptic variational inequality (3) when we consider g3(μ) instead of g.Remark 9.
If (40) (or (39)) is true, then the functional Jh is H-elliptic and a strictly convex application because we have(42)μJhg1+1-μJhg2-Jhg3μ=μ1-μ2uhg2-uhg1H2+Mg2-g1H2+12uh3H2-uh4H2≥μ1-μ2uhg2-uhg1H2+Mg2-g1H2>0and therefore, the uniqueness for the discrete optimal control problem (Ph) holds in Theorem 5.Now, we will show the convergence result for optimal control problems governed by elliptic variational inequalities in order to generalize the result for optimal control problems governed by elliptic variational equalities [19]. We remark that there exist a few numbers of papers for the numerical analysis of optimal control problems governed by elliptic variational inequalities, for example [20–22].Theorem 10.
Letugop∈K be the continuous state system associated with the optimal control gop∈H which is the solution of the continuous distributed optimal control problem (6). If, for each h>0, one chooses an optimal control goph∈H which is the solution of the discrete distributed optimal control problem (31) and its corresponding discrete state system uhgoph∈Kh, one obtains that(43)uhgoph⟶ugoponVstrong,goph⟶goponHstrongwhenh⟶0+.Proof.
Leth>0 and let goph be a solution of (31), and let uhgoph be its associated discrete optimal state system which is the solution of the discrete elliptic variational inequality (10) for each h>0. From (30) we have that for all g∈H(44)Jhgoph=12uhgophH2+M2gophH2≤12uhgH2+M2gH2.Then, if we consider g=0 and uh0 its corresponding associated state system, it results in the following:(45)Jhgoph=12uhgophH2+M2gophH2≤12uh0H2.From Lemma 2 we have that uh0H≤C∀h; then we can obtain(46)uhgophH≤C∀h>0,gophH≤1Muh0H≤CM∀h>0.If we consider vh=b∈Kh in inequality (10) for goph, we obtain(47)auhgoph,b-uhgoph≥goph,b-uhgoph-q,b-uhgophQ;therefore(48)auhgoph-b,uhgoph-b≤goph,uhgoph-b-q,uhgoph-bQ,and from the coerciveness of the application a we have that uhgoph-bV≤C and in consequence uhgophV≤C.
Now we can say that there existη∈V and f∈H such that uhgoph⇀η in V weak (in H strong), and goph⇀f in H weak when h→0+. Then, η/Γ1=b and η≥0 in Ω; that is, η∈K.
Lettingv∈K, there exist vh∈Kh such that vh→v in V strong when h→0+. Then, if we consider the variational elliptic inequality (10) for g=goph we have(49)auhgoph,vh≥auhgoph,uhgoph+goph,vh-uhgoph-q,vh-uhgophQ.Taking into account that the application a is a lower weak semicontinuous application in V and by passing to the limit when h goes to zero in (49) we obtain that(50)aη,v-η≥f,v-η-q,v-ηQ,∀v∈Kand by the uniqueness of the solution of the problem given by the elliptic variational inequality (3), we deduce that η=uf.
Finally, the norm onH is a lower semicontinuous application in the weak topology; then we can prove that(51)Jf=12ufH2+M2fH2≤liminfh→0+⁡Jhgoph≤liminfh→0+⁡Jhg=12limh→0+⁡uhgH2+M2gH2=12ugH2+M2gH2=Jg,∀g∈Hand because of the uniqueness of the optimal problem (6), it results in f=gop and η=ugop.
Now, if we considerv=uhgoph∈Kh⊂K in the elliptic variational inequality (3) for the control gop and we define zh=uhgoph-ugop, we have that(52)azh,zh≤auhgoph,uhgoph-auhgoph,ugop-gop,uhgoph-ugop+q,uhgoph-ugopQ,and by considering v=Πh(ugop)∈Kh for g=goph in inequality (10) we obtain(53)auhgoph,uhgoph≤-goph,Πhugop-uhgoph+q,Πhugop-uhgophQ+auhgoph,Πhugopand then by the coerciveness of a we get(54)λzhV2≤q,Πhugop-ugopQ+auhgoph,Πhugop-ugop+goph-gop,uhgoph-ugop-gop,Πhugop-ugop.When we pass to the limit as h→0 in (54) and by using the strong convergence of uhgoph to ugop on H and the weak convergence of goph to gop on H, we have(55)limh→0+⁡ugop-uhgophV=0.
The strong convergence of the optimal controlsgoph to gop is obtained by using Theorem 5 and goph⇀gop weakly on H; that is,(56)Jgop=12ugopH2+M2gopH2≤liminfh→0+⁡Jhgoph≤liminfh→0+⁡Jhgop=liminfh→0+⁡12ugopH2+M2gopH2=Jgop;then limh→0gophH=gopH and therefore limh→0+goph-gopH=0.


## 4. Conclusions

We have proved the convergence of a discrete optimal control and its corresponding discrete state system governed by a discrete elliptic variational inequality to the continuous optimal control and its corresponding continuous state system which is also governed by a continuous elliptic variational inequality by using the finite element method with Lagrange’s triangles of type 1.



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