1. Introduction

The research in the field of electromagnetism is set to become a vital factor in biomedical technologies. Those studies included several areas like the usage of electromagnetic waves for probing organs and advanced MRI techniques, microwave biosensors, non-invasive electromagnetic diagnostic tools, therapeutic applications of electromagnetic waves, radar technologies for biosensing, the adoption of electromagnetic waves in medical sensing, cancer detection using ultra-wideband signal, the interaction of electromagnetic waves with biological tissues and living systems, theoretical modeling of electromagnetic propagation through human body and tissues and imaging applications of electromagnetic.

Actually, the principal goal of the study is to control such electromagnetic waves to be compatible with some biomedical needs like X-rays in the framework of medical screening and wireless power transfer of electromagnetic waves through the human body [1] where we want to make waves closer to a desired distribution.

In this chapter, we consider a linear wave equation with a potential term pxσ supposed dependent on space variable x and real parameter σ∈01, this term generally comprises the dielectric permittivity of the medium which has different properties and cannot be exactly presented, this is because of the difference or lack of knowledge of the physical properties of the material penetrated from the electromagnetic waves. The initial position and velocity are also supposed unknown.

In this study, we consider an optimal control problem for electromagnetic wave equation depending upon a parameter and with missing initial conditions. We use the method of no-regret control which was introduced firstly in statistics by Savage [2] and later by Lions [3, 4] where he used this concept in optimal control theory, and its related idea is “low-regret” control to apply it to control distributed systems of incomplete data which has the attention of many scholars [5, 6, 7, 8, 9, 10, 11, 12], motivated by various applications in ecology, and economics as well [13]. Also, we use the notion of average control because our system depends upon a parameter, Zuazua was the first who introduced this new concept in [14].

The rest of this chapter is arranged as follows. Section 2, lists the definition of the problem we are studying. Section 3, is devoted to the study of the averaged no-regret control and the averaged low-regret control for the electromagnetic wave equation. Ultimately, we prove the existence of a unique average low-regret control, and the characterization of the average optimal is given in Section 4. Finally, we make a conclusion in Section 5.

2. Statement of the problem

Consider a bounded open domain Ω with a smooth boundary ∂Ω. We set Σ=0T×∂Ω and Q=Ω×0T. We introduce the following linear electromagnetic wave equation depending on a parameter

∂2y∂t2−Δy+pxσy=0y=v0yx0=y0x;∂y∂tx0=y1xin Qon Σ0on Σ\Σ0in ΩE1

where p∈L∞Ω is the potential term supposed dependent on a real parameter σ∈01 presents the dielectric permittivity and permeability of the medium and such that 0<α1≤pxσ≤α2a.e.in Ω, v is a boundary control in L2Σ0, y0∈H01Ω,y1∈L2Ω are the initial position and velocity respectively, both supposed unknown. For all σ∈01, the wave Eq. (1) has a unique solution yvy0y1σ in C0TH1Ω∩C10TL2Ω [15].

Denote by g=y0y1∈H01Ω×L2Ω the initial data. We want to choose a control u independently of σ and g in a way such that the average state function y approaches a given observation yd∈L2Q. To achieve our goal, let’ associate to (1) the following quadratic cost functional

Jvg=∫01yvgσdσ−ydL2Q2+NvL2Σ02E2

where N∈R+∗.

In this work, we aim to characterize the solution u of the optimal control problem with missing data given by

infv∈L2Σ0Jvgsubject to1E3

independently of g and σ.

3. Averaged no-regret control and averaged low-regret control for the electromagnetic wave equation

A classical method to obtain the optimality system is then to solve the minmax problem

infv∈L2Σ0supg∈H01Ω×L2ΩJvg,E4

but Jvg is not upper bounded since supg∈H01Ω×L2ΩJvg=+∞. A natural idea of Lions [3] is to search for controls v such that

Jvg−J0g≤0,∀g∈H01Ω×L2ΩE5

Those controls v are called averaged no-regret controls.

As in [16, 17], we introduce the averaged no-regret control defined by.

Definition 1 [1] We say that v∈L2Σ0 is an averaged no-regret control for (1) if v is a solution of

infv∈L2Σ0supg∈H01Ω×L2Ω(Jvg−J(0g)).E6

Let us start by giving the following important lemma.

Lemma 1 For all v∈L2Σ0 and g∈G we have

Jvg−J0g=Jv0−J00E7

−2∫Ωy0x∫01∂ζ∂tx0dσdx+2∫Ωy1x∫01ζx0dσdxE8

where ζ is given by the following backward wave equation

∂2ζ∂t2−Δζ+pxσζ=∫01yv0σdσζ=0ζxT=0,∂ζ∂txT=0in Qon Σin ΩE9

which has a unique solution in C0TH01Ω∩C10TL2Ω [15].

Proof. It’s easy to check that for all vg∈L2Σ0×G

Jvg−J0g=Jv0−J00+2∫Q∫01yv0dσ∫01y0gdσdxdt.E10

Use (9) and apply Green formula to get

∫Q∫01yv0dσ∫01y0gdσdxdt=∫0T∫Ω∂2ζ∂t2−Δζ+pxσζ∫01y0gdσdxdtE11

=−2∫Ωy0x∫01∂ζ∂tv0dσdx+2∫Ωy1x∫01ζv0dσdx.E12

■

The no-regret control seems to be hard to characterize (see [11]), for this.

reason we relax the no-regret control problem by making some quadratic perturbation as follows.

Definition 2 [17] We say that uγ∈L2Σ0 is an averaged low-regret control for (1) if uγ is a solution of

infv∈L2Σ0supg∈H01Ω×L2Ω(Jvg−J(0g)−γy0H01Ω2−γy1L2Ω2),γ>0.E13

Using (9) the problem (13) can be written as

infv∈L2Σ0Jv0−J00+supg∈G2(∫Ωy1x∫01ζvσx0dσdx−∫Ωy0x∫01∂ζ∂tvσx0dσdx−γy0H01Ω2−γy1L2Ω2).,γ>0.E14

And thanks to Legendre transform (see [18, 19]), we have

supg∈G2−∫Ωy0x∫01∂ζ∂tv0dσdx+∫Ωy1x∫01ζv0dσdx−γy0H01Ω2−γy1L2Ω2E15

=1γ∫01∂ζvσ∂tx0dσH01Ω2+1γ∫01ζvσx0dσL2Ω2.E16

Then, the averaged low-regret control problem (9) is equivalent to the following classical optimal control problem

infv∈L2Σ0JγvE17

where

Jγv=Jv0−J00+1γ∫01∂ζvσ∂tx0dσH01Ω2+1γ∫01ζvσx0dσL2Ω2.E18

4. Characterizations

In the recent section, we aim to find a full characterization for the averaged no-regret control and averaged low-regret control via optimality systems.

Theorem 1.1 There exists a unique averaged low-regret control uγ solution to (17), (18).

Proof. We have for every v∈L2Σ0:Jγv≥−J00, this means that (17), (18) has a solution.

Let vnγ∈L2Σ0 be a minimizing sequence such that

limn→∞Jγvnγ=Jγuγ=dγ.E19

We know that

Jγvnγ=Jvnγ0−J00+1γ∫01∂ζvnγσ∂tx0dσH01Ω2+1γ∫01ζvnγσx0dσL2Ω2≤dγ+1.E20

This implies the following bounds

vnγL2Σ0≤Cγ,E21

∫01yvnγ0σdσL2Q≤Cγ,E22

∂2ζn∂t2−Δζn+pxσζnL2Q≤Cγ,E23

where Cγ is a positive constant independent of n. Moreover, by continuity w.r.t. data and (21) we get

yvnγ0σL2Q≤Cγ.E24

By similar way an by using (22) we obtain

ζvnγσL20TH01Ω≤Cγ.E25

Then, from (21) we deduce that there exists a subsequence still denoted vnγ such that vnγ⇀uγweakly in L2Σ0, and from (22) we get

yvnγ0σ⇀yγ weakly in L2Q.E26

Also, because of continuity w.r.t. data we have yvnγ0σ⇀yuγ0σ weakly in L2Q, by limit uniqueness yγ=yuγ0σ solution to

∂2yγ∂t2−Δyγ+pxσyγ=0yγ=uγ0yγx0=y0x;∂yγ∂tx0=y1xin Q,on Σ0,on Σ\Σ0,in Ω.E27

In other hand, use (24) and (22) to apply the convergence dominated theorem and, we have

∫01yvnγ0σdσ⇀∫01yuγ0σdσ weakly in L2Q.E28

From (25) we deduce the existence of a subsequence still be denoted by ζvnγσx0 such that

ζvnγσx0⇀ζuγσx0 weakly in H01Ω,E29

then

∂2ζn∂t2−Δζn+pxσζn→∂2ζγ∂t2−Δζγ+pxσζγinD′Q,E30

where DQ=C0∞Q, and (23) leads to

∂2ζn∂t2−Δζn+pxσζn⇀fweakly inL2Q.E31

Again, by limit uniqueness ∂2ζγ∂t2−Δζγ+pxσζγ=∫01yuγ0σdσ in L2Q. Finally, ζγ is a solution to

∂2ζγ∂t2−Δζγ+pxσζγ=∫01yuγ0σdσζγ=0ζγxT=0,∂ζγ∂txT=0 in Q,on Σ,in Ω.E32

The uniqueness of uγ follows from strict convexity and weak lower semi-continuity of the functional Jγv.■

After proving existence and uniqueness, we aim in the next theorem to give a full description to the average low-regret control for the electromagnetic wave equation.

Theorem 1.2 For all γ>0, the average low-regret control uγ is characterized by the following optimality system

∂2yγ∂t2−Δyγ+pxσyγ=0,∂2ζγ∂t2−Δζγ+pxσζγ=∫01yuγ0σdσ,∂2ργ∂t2−Δργ+pxσργ=0,∂2qγ∂t2−Δqγ+pxσqγ=∫01ργ+yuγ0σdσ−ydin Q,yγ=uγ0on Σ0onΣ\Σ0,ζγ=0,ργ=0,qγ=0 on Σ,yγ0x=0,∂yγ∂t0x=0,ζγxT=0,∂ζγ∂txT=0,ργx0=−1γ∫01∂ζuγ∂tx0dσ,∂ργ∂tx0=1γ∫01ζuγx0dσ,qγTx=0,∂qγ∂tTx=0in Ω,E33

with

uγ=1N∫01∂pγ∂ηdσinL2Σ0.E34

Proof. From the first order necessary optimality conditions, we have

Jγ′uγw=∫01yuγ0dσ−yd∫01yw0dσL2Q+NuγwL2Σ0+1γ∫01ζ′uγ0dσ∫01ζ′w0dσL2Ω+1γ∫01ζuγ0dσ∫01ζw0dσL2Ω=0E35

for all w∈L2Σ0.

Now, let us introduce ργ=ρuγ0 unique solution to

∂2ργ∂t2−Δργ+pxσργ=0ργ=0ργx0=−1γ∫01∂ζuγ∂tx0dσ;∂ργ∂tx0=1γ∫01ζuγx0dσin Q,on Σ,in Ω.E36

So that for every w∈L2Σ0, we obtain

Jγ′uγw=∫01yuγ0+ργdσ−yd∫01yw0dσ−y00L2Q+NuγwL2Σ0=0E37

We finally define another adjoint state qγ=quγ as the unique solution of

∂2qγ∂t2−Δqγ+pxσqγ=∫01ργ+yuγ0σdσ−ydqγ=0qγxT=0,∂qγ∂txT=0in Q,on Σ,in Ω.E38

Then (35) becomes

uγ=1N∫01∂qγ∂ηdσinL2Σ0.E39

■

The previous Theorem gives a low-regret control characterization. For the no-regret control, we need to prove the convergence of the sequence of averaged low-regret control to the averaged no-regret control. Then, we announce the following Proposition.

For some constant C independent of γ, we have

uγL2Σ0≤C,E40

∫01yuγ0σdσL2Q≤C,E41

yuγ0σdσL2Q≤C,E42

∫01∂ζuγσ∂tx0dσH−1Ω≤Cγ,E43

∫01ζuγσx0dσL2Ω≤Cγ,E44

ργL∞0TH01Ω≤C,E45

qγL∞0TH01Ω≤C.E46

Proof. Since uγ is a solution to (17) and (18), we get

Jγuγ≤Jγ0,E47

then

Juγ0+1γ∫01∂ζuγσ∂tx0dσL2Ω2+1γ∫01ζuγσx0dσL2Ω2≤J00,E48

this gives (40), (41), (42) and (43). The bound (43) follows by a way similar to (24).

From energy conservation property with (43) and (44).

Eργt=12Ω∂ργ∂t2+∇ργ2+qxσργ2dx=Eργ0≤C,E49

we find (45).

To get qγ estimates, just reverse the time variable by taking s=T−t to find (46).

Lemma 2 The averaged low-regret control uγ tends weakly to the averaged no-regret control u when γ→0.

Proof. From (40) we deduce the existence of a subsequence still be denoted uγ such that

uγ⇀uweakly inL2Σ0,E50

let us prove u is an averaged no-regret control. We have for all v∈L2Σ0

Juγg−J0g−γy0H01Ω2−γy1L2Ω2≤supg∈H01Ω×L2ΩJvg−J0g,E51

take γ→0 to find

Jug−J0g≤supg∈H01Ω×L2ΩJvg−J0g,E52

i.e. is an averaged no-regret control. ■.

Finally, we can present the following theorem giving a full characterization the average no-regret control.

Theorem 1.3 The average no-regret control u is characterized by the following optimality system

∂2y∂t2−Δy+pxσy=0,∂2ζ∂t2−Δζ+pxσζ=∫01yu0σdσ,∂2ρ∂t2−Δρ+pxσρ=0,∂2q∂t2−Δq+pxσq=∫01ρ+yu0σdσ−ydin Q,y=u0onΣ0onΣ\Σ0,ζ=0ρ=0;q=0on Σ,y0x=0,∂y∂t0x=0,ζxT=0,∂ζ∂txT=0,ρx0=λ1x,∂ρ∂tx0=λ2x,qTx=0,∂q∂tTx=0in Ω,E53

with

u=1N∫01∂p∂ηdσ in L2Σ0,E54

and

λ1x=limγ→0−1γ∫01∂ζuγ∂tx0dσ weakly in H01Ω,

λ2x=limγ→01γ∫01ζuγx0dσ weakly in L2Ω.

Proof. From (42) continuity w.r.t data, we can deduce that

yuγ0σ⇀yu0σ weakly in L2Ω,E55

solution to

∂2y∂t2−Δy+pxσy=0y=u0yx0=y0x;∂y∂tx0=y1xin Q,on Σ0,on Σ\Σ0,in Ω.E56

Again, by (41) and dominated convergence theorem

∫01yuγ0σdσ⇀∫01yu0σdσweakly inL2Σ0.E57

The rest of equations in (53) leads by a similar way, except the convergences of initial data ρx0, ∂ρ∂tx0 which will be as follows.

From (43) and (44) we deduce the convergences of

−1γ∂ζuγσ∂tx0⇀λ1x weakly in H01Ω,E58

and

1γζuγσx0⇀λ2x weakly in L2Ω.E59

5. Conclusion

As we have seen, the averaged no-regret control method allows us to find a control that will optimize the situation of the electromagnetic waves with missing initial conditions and depending upon a parameter. The method presented in the paper is quite general and covers a wide class of systems, hence, we could generalize the situation to more control positions (regional, punctual,…) and different kinds of missing data (source term, boundary conditions,…).

The results presented above can also be generalized to the case of other systems which has many biomedical applications. This problem is still under consideration and the results will appear in upcoming works.

Acknowledgments

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

Carmen Astete

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