## 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

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

where

Denote by

where

In this work, we aim to characterize the solution

independently of

## 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

but

Those controls

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

Definition 1 [1] We say that

Let us start by giving the following important lemma.

Lemma 1 For all

where

which has a unique solution in

** Proof.**It’s easy to check that for all

Use (9) and apply Green formula to get

■

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

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

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

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

where

## 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

** Proof.**We have for every

Let

We know that

This implies the following bounds

where

By similar way an by using (22) we obtain

Then, from (21) we deduce that there exists a subsequence still denoted

Also, because of continuity w.r.t. data we have

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

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

then

where

Again, by limit uniqueness

The uniqueness of

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

with

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

for all

Now, let us introduce

So that for every

We finally define another adjoint state

Then (35) becomes

■

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

** Proof.**Since

then

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).

we find (45).

To get

Lemma 2 The averaged low-regret control

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

let us prove

take

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

with

and

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

solution to

Again, by (41) and dominated convergence theorem

The rest of equations in (53) leads by a similar way, except the convergences of initial data

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

and

## 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).