Open access peer-reviewed chapter - ONLINE FIRST

Averaged No-Regret Control for an Electromagnetic Wave Equation Depending upon a Parameter with Incomplete Initial Conditions

By Abdelhak Hafdallah and Mouna Abdelli

Submitted: September 4th 2020Reviewed: December 10th 2020Published: March 11th 2021

DOI: 10.5772/intechopen.95447

Downloaded: 16

Abstract

This chapter concerns the optimal control problem for an electromagnetic wave equation with a potential term depending on a real parameter and with missing initial conditions. By using both the average control notion introduced recently by E. Zuazua to control parameter depending systems and the no-regret method introduced for the optimal control of systems with missing data. The relaxation of averaged no-regret control by the averaged low-regret control sequence transforms the problem into a standard optimal control problem. We prove that the problem of average optimal control admits a unique averaged no-regret control that we characterize by means of optimality systems.

Keywords

  • optimal control
  • averaged no-regret control
  • electromagnetic wave equation
  • parameter depending equation
  • systems with missing data

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 pxσsupposed dependent on space variable xand real parameter σ01, 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.

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2. Statement of the problem

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

2yt2Δy+pxσy=0y=v0yx0=y0x;ytx0=y1xin Qon Σ0on Σ\Σ0in ΩE1

where pLΩis the potential term supposed dependent on a real parameter σ01presents the dielectric permittivity and permeability of the medium and such that 0<α1pxσα2a.e.in Ω, vis a boundary control in L2Σ0, y0H01Ω,y1L2Ωare the initial position and velocity respectively, both supposed unknown. For all σ01, the wave Eq. (1) has a unique solution yvy0y1σin C0TH1ΩC10TL2Ω[15].

Denote by g=y0y1H01Ω×L2Ωthe initial data. We want to choose a control uindependently of σand gin a way such that the average state function yapproaches a given observation ydL2Q. To achieve our goal, let’ associate to (1) the following quadratic cost functional

Jvg=01yvgσydL2Q2+NvL2Σ02E2

where NR+.

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

infvL2Σ0Jvgsubject to1E3

independently of gand σ.

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

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

infvL2Σ0supgH01Ω×L2ΩJvg,E4

but Jvgis not upper bounded since supgH01Ω×L2ΩJvg=+. A natural idea of Lions [3] is to search for controls vsuch that

JvgJ0g0,gH01Ω×L2ΩE5

Those controls vare called averaged no-regret controls.

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

Definition 1 [1] We say that vL2Σ0is an averaged no-regret control for (1) if vis a solution of

infvL2Σ0supgH01Ω×L2Ω(JvgJ(0g)).E6

Let us start by giving the following important lemma.

Lemma 1 For all vL2Σ0and gGwe have

JvgJ0g=Jv0J00E7
2Ωy0x01ζtx0dσdx+2Ωy1x01ζx0dσdxE8

where ζis given by the following backward wave equation

2ζt2Δζ+pxσζ=01yv0σζ=0ζxT=0,ζtxT=0in Qon Σin ΩE9

which has a unique solution in C0TH01ΩC10TL2Ω[15].

Proof.It’s easy to check that for all vgL2Σ0×G

JvgJ0g=Jv0J00+2Q01yv001y0gdxdt.E10

Use (9) and apply Green formula to get

Q01yv001y0gdxdt=0TΩ2ζt2Δζ+pxσζ01y0gdxdtE11
=2Ωy0x01ζtv0dσdx+2Ωy1x01ζv0dσdx.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 uγL2Σ0is an averaged low-regret control for (1) if uγis a solution of

infvL2Σ0supgH01Ω×L2Ω(JvgJ(0g)γy0H01Ω2γy1L2Ω2),γ>0.E13

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

infvL2Σ0Jv0J00+supgG2(Ωy1x01ζvσx0dσdxΩy0x01ζtvσx0dσdxγy0H01Ω2γy1L2Ω2).,γ>0.E14

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

supgG2Ωy0x01ζtv0dσdx+Ωy1x01ζv0dσdxγy0H01Ω2γy1L2Ω2E15
=1γ01ζvσtx0H01Ω2+1γ01ζvσx0L2Ω2.E16

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

infvL2Σ0JγvE17

where

Jγv=Jv0J00+1γ01ζvσtx0H01Ω2+1γ01ζvσx0L2Ω2.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 uγsolution to (17), (18).

Proof.We have for every vL2Σ0:JγvJ00, this means that (17), (18) has a solution.

Let vnγL2Σ0be a minimizing sequence such that

limnJγvnγ=Jγuγ=dγ.E19

We know that

Jγvnγ=Jvnγ0J00+1γ01ζvnγσtx0H01Ω2+1γ01ζvnγσx0L2Ω2dγ+1.E20

This implies the following bounds

vnγL2Σ0Cγ,E21
01yvnγ0σL2QCγ,E22
2ζnt2Δζn+pxσζnL2QCγ,E23

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

yvnγ0σL2QCγ.E24

By similar way an by using (22) we obtain

ζvnγσL20TH01ΩCγ.E25

Then, from (21) we deduce that there exists a subsequence still denoted vnγsuch that vnγuγweakly in L2Σ0, and from (22) we get

yvnγ0σyγ weakly in L2Q.E26

Also, because of continuity w.r.t. data we have yvnγ0σyuγ0σweakly in L2Q, by limit uniqueness yγ=yuγ0σsolution to

2yγt2Δyγ+pxσyγ=0yγ=uγ0yγx0=y0x;yγtx0=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

01yvnγ0σ01yuγ0σ weakly in L2Q.E28

From (25) we deduce the existence of a subsequence still be denoted by ζvnγσx0such that

ζvnγσx0ζuγσx0 weakly in H01Ω,E29

then

2ζnt2Δζn+pxσζn2ζγt2Δζγ+pxσζγinDQ,E30

where DQ=C0Q, and (23) leads to

2ζnt2Δζn+pxσζnfweakly inL2Q.E31

Again, by limit uniqueness 2ζγt2Δζγ+pxσζγ=01yuγ0σin L2Q. Finally, ζγis a solution to

2ζγt2Δζγ+pxσζγ=01yuγ0σζγ=0ζγxT=0,ζγtxT=0 in Q,on Σ,in Ω.E32

The uniqueness of uγfollows 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 uγis characterized by the following optimality system

2yγt2Δyγ+pxσyγ=0,2ζγt2Δζγ+pxσζγ=01yuγ0σ,2ργt2Δργ+pxσργ=0,2qγt2Δqγ+pxσqγ=01ργ+yuγ0σydin Q,yγ=uγ0on Σ0onΣ\Σ0,ζγ=0,ργ=0,qγ=0 on Σ,yγ0x=0,yγt0x=0,ζγxT=0,ζγtxT=0,ργx0=1γ01ζuγtx0,ργtx0=1γ01ζuγx0,qγTx=0,qγtTx=0in Ω,E33

with

uγ=1N01pγηinL2Σ0.E34

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

Jγuγw=01yuγ0yd01yw0L2Q+NuγwL2Σ0+1γ01ζuγ001ζw0L2Ω+1γ01ζuγ001ζw0L2Ω=0E35

for all wL2Σ0.

Now, let us introduce ργ=ρuγ0unique solution to

2ργt2Δργ+pxσργ=0ργ=0ργx0=1γ01ζuγtx0;ργtx0=1γ01ζuγx0in Q,on Σ,in Ω.E36

So that for every wL2Σ0, we obtain

Jγuγw=01yuγ0+ργyd01yw0y00L2Q+NuγwL2Σ0=0E37

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

2qγt2Δqγ+pxσqγ=01ργ+yuγ0σydqγ=0qγxT=0,qγtxT=0in Q,on Σ,in Ω.E38

Then (35) becomes

uγ=1N01qγηinL2Σ0.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 Cindependent of γ, we have

uγL2Σ0C,E40
01yuγ0σL2QC,E41
yuγ0σL2QC,E42
01ζuγσtx0H1ΩCγ,E43
01ζuγσx0L2ΩCγ,E44
ργL0TH01ΩC,E45
qγL0TH01ΩC.E46

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

JγuγJγ0,E47

then

Juγ0+1γ01ζuγσtx0L2Ω2+1γ01ζuγσx0L2Ω2J00,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=12Ωργt2+ργ2+qxσργ2dx=Eργ0C,E49

we find (45).

To get qγestimates, just reverse the time variable by taking s=Ttto find (46).

Lemma 2 The averaged low-regret control uγtends weakly to the averaged no-regret control uwhen γ0.

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

uγuweakly inL2Σ0,E50

let us prove uis an averaged no-regret control. We have for all vL2Σ0

JuγgJ0gγy0H01Ω2γy1L2Ω2supgH01Ω×L2ΩJvgJ0g,E51

take γ0to find

JugJ0gsupgH01Ω×L2ΩJvgJ0g,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 uis characterized by the following optimality system

2yt2Δy+pxσy=0,2ζt2Δζ+pxσζ=01yu0σ,2ρt2Δρ+pxσρ=0,2qt2Δq+pxσq=01ρ+yu0σydin Q,y=u0onΣ0onΣ\Σ0,ζ=0ρ=0;q=0on Σ,y0x=0,yt0x=0,ζxT=0,ζtxT=0,ρx0=λ1x,ρtx0=λ2x,qTx=0,qtTx=0in Ω,E53

with

u=1N01pη in L2Σ0,E54

and

λ1x=limγ01γ01ζuγtx0weakly in H01Ω,

λ2x=limγ01γ01ζuγx0weakly in L2Ω.

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

yuγ0σyu0σ weakly in L2Ω,E55

solution to

2yt2Δy+pxσy=0y=u0yx0=y0x;ytx0=y1xin Q,on Σ0,on Σ\Σ0,in Ω.E56

Again, by (41) and dominated convergence theorem

01yuγ0σ01yu0σweakly inL2Σ0.E57

The rest of equations in (53) leads by a similar way, except the convergences of initial data ρx0, ρtx0which will be as follows.

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

1γζuγσtx0λ1x weakly in H01Ω,E58

and

1γζuγσx0λ2x 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.

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Acknowledgments

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

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Abdelhak Hafdallah and Mouna Abdelli (March 11th 2021). Averaged No-Regret Control for an Electromagnetic Wave Equation Depending upon a Parameter with Incomplete Initial Conditions [Online First], IntechOpen, DOI: 10.5772/intechopen.95447. Available from:

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