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

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.

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 $p\left(x,\sigma \right)$ supposed dependent on space variable $x$ and real parameter $\sigma \in \left(0,1\right)$, 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 $\mathrm{\Omega }$ with a smooth boundary $\mathrm{\partial \Omega }$. We set $\mathrm{\Sigma }=\left(0,T\right)\times \mathrm{\partial \Omega }$ and $Q=\mathrm{\Omega }\times \left(0,T\right)$. We introduce the following linear electromagnetic wave equation depending on a parameter

$$\left\{\begin{array}{l}\frac{{\partial }^{2}y}{\partial t^2}-\mathrm{\Delta }y+p\left(x,\sigma \right)y=0\\ y=\left\{\begin{array}{c}v\\ 0\end{array}\right.\\ y\left(x,0\right)=y_0\left(x\right);\frac{\partial y}{\partial t}\left(x,0\right)=y_1\left(x\right)\end{array}\right.\begin{array}{c}\text{in} Q\\ \mathrm{on} {\mathrm{\Sigma }}_0\\ \mathrm{on} \mathrm{\Sigma }\backslash {\mathrm{\Sigma }}_0\\ \text{in} \mathrm{\Omega }\end{array}$$E1

where $p\in L^{\infty }\left(\mathrm{\Omega }\right)$ is the potential term supposed dependent on a real parameter $\sigma \in \left(0,1\right)$ presents the dielectric permittivity and permeability of the medium and such that $0<{\alpha }_1\le p\left(x,\sigma \right)\le {\alpha }_2$$a.e.$in $\mathrm{\Omega }$, $v$ is a boundary control in $L^2\left({\mathrm{\Sigma }}_0\right)$, $y_0\in H_0^1\left(\mathrm{\Omega }\right),$$y_1\in L^2\left(\mathrm{\Omega }\right)$ are the initial position and velocity respectively, both supposed unknown. For all $\sigma \in \left(0,1\right)$, the wave Eq. (1) has a unique solution $y\left(v,y_0,y_1,\sigma \right)$ in $C\left(\left[0,T\right];H^1\left(\mathrm{\Omega }\right)\right)\cap C^1\left(\left[0,T\right];L^2\left(\mathrm{\Omega }\right)\right)$ [15].

Denote by $g=$$\left(y_0,y_1\right)\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$ the initial data. We want to choose a control $u$ independently of $\sigma$ and $g$ in a way such that the average state function $y$ approaches a given observation $y_d$$\in L^2\left(Q\right)$. To achieve our goal, let’ associate to (1) the following quadratic cost functional

$$J\left(v,g\right)={‖{\int }_0^{1}y\left(v,g,\sigma \right)\mathit{d\sigma }-y_d‖}_{L^2\left(Q\right)}^2+N{‖v‖}_{L^2\left({\mathrm{\Sigma }}_0\right)}^2$$E2

where $N\in {\mathbb{R}}_{+}^{\ast }$.

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

$$\underset{v\in L^2\left({\mathrm{\Sigma }}_0\right)}{inf}J\left(v,g\right)\text{subject\hspace{0.17em}to}\left(1\right)$$E3

independently of $g$ and $\sigma$.

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

$$\underset{v\in L^2\left({\mathrm{\Sigma }}_0\right)}{inf}\left(\underset{g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)}{sup}\left(J\left(v,g\right)\right)\right),$$E4

but $J\left(v,g\right)$ is not upper bounded since ${sup}_{g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)}\left(J\left(v,g\right)\right)=+\infty$. A natural idea of Lions [3] is to search for controls $v$ such that

$$J\left(v,g\right)-J\left(0,g\right)\le 0,\forall g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)$$E5

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 $v\in L^2\left({\mathrm{\Sigma }}_0\right)$ is an averaged no-regret control for (1) if $v$ is a solution of

$$\underset{v\in L^2\left({\mathrm{\Sigma }}_0\right)}{inf}\left(\underset{g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)}{sup}(J\left(v,g\right)-J(0,g\left)\right)\right).$$E6

Let us start by giving the following important lemma.

Lemma 1 For all $v\in L^2\left({\mathrm{\Sigma }}_0\right)$ and $g\in G$ we have

$$J\left(v,g\right)-J\left(0,g\right)=J\left(v,0\right)-J\left(0,0\right)$$E7

$$-2{\int }_{\mathrm{\Omega }}{y}_0\left(x\right){\int }_0^1\frac{\partial \zeta }{\partial t}\left(x,0\right)\mathit{\text{dσdx}}+2{\int }_{\mathrm{\Omega }}{y}_1\left(x\right){\int }_0^1\zeta \left(x,0\right)\mathit{\text{dσdx}}$$E8

where $\zeta$ is given by the following backward wave equation

$$\left\{\begin{array}{l}\frac{{\partial }^2\zeta }{\partial t^2}-\mathrm{\Delta }\zeta +p\left(x,\sigma \right)\zeta ={\int }_0^{1}y\left(v,0,\sigma \right)\mathit{d\sigma }\\ \zeta =0\\ \zeta \left(x,T\right)=0,\frac{\partial \zeta }{\partial t}\left(x,T\right)=0\end{array}\right.\begin{array}{c}\text{in} Q\\ \mathrm{on} \mathrm{\Sigma }\\ \text{in} \mathrm{\Omega }\end{array}$$E9

which has a unique solution in $C\left(\left[0,T\right];H_0^1\left(\mathrm{\Omega }\right)\right)\cap C^1\left(\left[0,T\right];L^2\left(\mathrm{\Omega }\right)\right)$ [15].

Proof. It’s easy to check that for all $\left(v,g\right)$$\in L^2\left({\mathrm{\Sigma }}_0\right)\times G$

$$J\left(v,g\right)-J\left(0,g\right)=J\left(v,0\right)-J\left(0,0\right)+2{\int }_Q\left({\int }_0^{1}y\left(v,0\right)\mathit{d\sigma }\right)\left({\int }_0^{1}y\left(0,g\right)\mathit{d\sigma }\right)\mathit{\text{dxdt}}.$$E10

Use (9) and apply Green formula to get

$${\int }_Q\left({\int }_0^{1}y\left(v,0\right)\mathit{d\sigma }\right)\left({\int }_0^{1}y\left(0,g\right)\mathit{d\sigma }\right)\mathit{\text{dxdt}}=\underset{0}{\overset{T}{\int }}\underset{\mathrm{\Omega }}{\int }\left(\frac{{\partial }^2\zeta }{\partial t^2}-\mathrm{\Delta }\zeta +p\left(x,\sigma \right)\zeta \right)\left({\int }_0^{1}y\left(0,g\right)\mathit{d\sigma }\right)\mathit{\text{dxdt}}$$E11

$$=-2{\int }_{\mathrm{\Omega }}{y}_0\left(x\right){\int }_0^1\frac{\partial \zeta }{\partial t}\left(v,0\right)\mathit{\text{dσdx}}+2{\int }_{\mathrm{\Omega }}{y}_1\left(x\right){\int }_0^1\zeta \left(v,0\right)\mathit{\text{dσdx}}.$$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 $u_{\gamma }\in L^2\left({\mathrm{\Sigma }}_0\right)$ is an averaged low-regret control for (1) if $u_{\gamma }$ is a solution of

$$\underset{v\in L^2\left({\mathrm{\Sigma }}_0\right)}{inf}\left(\underset{g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)}{sup}(J\left(v,g\right)-J(0,g)-\gamma {‖y_0‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2-\gamma {‖y_1‖}_{L^2\left(\mathrm{\Omega }\right)}^2)\right),\gamma >0.$$E13

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

$$\underset{v\in L^2\left({\mathrm{\Sigma }}_0\right)}{inf}\left(\begin{array}{c}J\left(v,0\right)-J\left(0,0\right)+\underset{g\in G}{sup}2({\int }_{\mathrm{\Omega }}{y}_1\left(x\right){\int }_0^1\zeta \left(v,\sigma \right)\left(x,0\right)\mathit{\text{dσdx}}-{\int }_{\mathrm{\Omega }}{y}_0\left(x\right){\int }_0^1\frac{\partial \zeta }{\partial t}\left(v,\sigma \right)\left(x,0\right)\mathit{\text{dσdx}}\\ -\gamma {‖y_0‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2-\gamma {‖y_1‖}_{L^2\left(\mathrm{\Omega }\right)}^2).\end{array}\right)\begin{array}{c},\gamma >0.\end{array}$$E14

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

$$\underset{g\in G}{sup}2\left(-{\int }_{\mathrm{\Omega }}{y}_0\left(x\right){\int }_0^1\frac{\partial \zeta }{\partial t}\left(v,0\right)\mathit{\text{dσdx}}+{\int }_{\mathrm{\Omega }}{y}_1\left(x\right){\int }_0^1\zeta \left(v,0\right)\mathit{\text{dσdx}}-\gamma {‖y_0‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2-\gamma {‖y_1‖}_{L^2\left(\mathrm{\Omega }\right)}^2\right)$$E15

$$=\frac{1}{\gamma }{‖{\int }_0^1\frac{\partial \zeta \left(v,\sigma \right)}{\partial t}\left(x,0\right)\mathit{d\sigma }‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2+\frac{1}{\gamma }{‖{\int }_0^1\zeta \left(v,\sigma \right)\left(x,0\right)\mathit{d\sigma }‖}_{L^2\left(\mathrm{\Omega }\right)}^2.$$E16

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

$$\underset{v\in L^2\left({\mathrm{\Sigma }}_0\right)}{inf}{J}_{\gamma }\left(v\right)$$E17

where

$$J_{\gamma }\left(v\right)=J\left(v,0\right)-J\left(0,0\right)+\frac{1}{\gamma }{‖{\int }_0^1\frac{\partial \zeta \left(v,\sigma \right)}{\partial t}\left(x,0\right)\mathit{d\sigma }‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2+\frac{1}{\gamma }{‖{\int }_0^1\zeta \left(v,\sigma \right)\left(x,0\right)\mathit{d\sigma }‖}_{L^2\left(\mathrm{\Omega }\right)}^2.$$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 $u_{\gamma }$ solution to (17), (18).

Proof. We have for every $v\in L^2\left({\mathrm{\Sigma }}_0\right):$$J_{\gamma }\left(v\right)\ge -J\left(0,0\right)$, this means that (17), (18) has a solution.

Let $\left(v_n^{\gamma }\right)\in L^2\left({\mathrm{\Sigma }}_0\right)$ be a minimizing sequence such that

$$\underset{n\to \infty }{lim}{J}_{\gamma }\left(v_n^{\gamma }\right)=J_{\gamma }\left(u_{\gamma }\right)=d_{\gamma }.$$E19

We know that

$$J_{\gamma }\left(v_n^{\gamma }\right)=J\left(v_n^{\gamma },0\right)-J\left(0,0\right)+\frac{1}{\gamma }{‖{\int }_0^1\frac{\partial \zeta \left(v_n^{\gamma },\sigma \right)}{\partial t}\left(x,0\right)\mathit{d\sigma }‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2+\frac{1}{\gamma }{‖{\int }_0^1\zeta \left(v_n^{\gamma },\sigma \right)\left(x,0\right)\mathit{d\sigma }‖}_{L^2\left(\mathrm{\Omega }\right)}^2\le d_{\gamma }+1.$$E20

This implies the following bounds

$${‖v_n^{\gamma }‖}_{L^2\left({\mathrm{\Sigma }}_0\right)}\le C_{\gamma },$$E21

$${‖{\int }_0^{1}y\left(v_n^{\gamma },0,\sigma \right)\mathit{d\sigma }‖}_{L^2\left(Q\right)}\le C_{\gamma },$$E22

$${‖\frac{{\partial }^2{\zeta }_n}{\partial t^2}-\mathrm{\Delta }{\zeta }_n+p\left(x,\sigma \right){\zeta }_n‖}_{L^2\left(Q\right)}\le C_{\gamma },$$E23

where $C_{\gamma }$ is a positive constant independent of $n$. Moreover, by continuity w.r.t. data and (21) we get

$${‖y\left(v_n^{\gamma },0,\sigma \right)‖}_{L^2\left(Q\right)}\le C_{\gamma }.$$E24

By similar way an by using (22) we obtain

$${‖\zeta \left(v_n^{\gamma },\sigma \right)‖}_{L^2\left(0,T;H_0^1\left(\mathrm{\Omega }\right)\right)}\le C_{\gamma }.$$E25

Then, from (21) we deduce that there exists a subsequence still denoted $\left(v_n^{\gamma }\right)$ such that $v_n^{\gamma }\rightharpoonup u_{\gamma }$weakly in $L^2\left({\mathrm{\Sigma }}_0\right)$, and from (22) we get

$$y\left(v_n^{\gamma },0,\sigma \right)\rightharpoonup y_{\gamma } \text{weakly in} L^2\left(Q\right).$$E26

Also, because of continuity w.r.t. data we have $y\left(v_n^{\gamma },0,\sigma \right)\rightharpoonup y\left(u_{\gamma },0,\sigma \right)$ weakly in $L^2\left(Q\right)$, by limit uniqueness $y_{\gamma }=y\left(u_{\gamma },0,\sigma \right)$ solution to

$$\left\{\begin{array}{l}\frac{{\partial }^2{y}_{\gamma }}{\partial t^2}-\mathrm{\Delta }{y}_{\gamma }+p\left(x,\sigma \right)y_{\gamma }=0\\ y_{\gamma }=\left\{\begin{array}{c}{u}_{\gamma }\\ 0\end{array}\right.\\ y_{\gamma }\left(x,0\right)=y_0\left(x\right);\frac{\partial y_{\gamma }}{\partial t}\left(x,0\right)=y_1\left(x\right)\end{array}\right.\begin{array}{c}\text{in} Q,\\ \mathrm{on} {\mathrm{\Sigma }}_0,\\ \mathrm{on} \mathrm{\Sigma }\backslash {\mathrm{\Sigma }}_0,\\ \text{in} \mathrm{\Omega }.\end{array}$$E27

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

$${\int }_0^{1}y\left(v_n^{\gamma },0,\sigma \right)\mathit{d\sigma }\rightharpoonup {\int }_0^{1}y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma } \text{weakly in} L^2\left(Q\right).$$E28

From (25) we deduce the existence of a subsequence still be denoted by $\zeta \left(v_n^{\gamma },\sigma \right)\left(x,0\right)$ such that

$$\zeta \left(v_n^{\gamma },\sigma \right)\left(x,0\right)\rightharpoonup \zeta \left(u_{\gamma },\sigma \right)\left(x,0\right) \text{weakly in} H_0^1\left(\mathrm{\Omega }\right),$$E29

then

$$\frac{{\partial }^2{\zeta }_n}{\partial t^2}-\mathrm{\Delta }{\zeta }_n+p\left(x,\sigma \right){\zeta }_n\to \frac{{\partial }^2{\zeta }_{\gamma }}{\partial t^2}-\mathrm{\Delta }{\zeta }_{\gamma }+p\left(x,\sigma \right){\zeta }_{\gamma }\text{in}{D}^{\prime }\left(Q\right),$$E30

where $D\left(Q\right)=C_0^{\infty }\left(Q\right)$, and (23) leads to

$$\frac{{\partial }^2{\zeta }_n}{\partial t^2}-\mathrm{\Delta }{\zeta }_n+p\left(x,\sigma \right){\zeta }_n\rightharpoonup f\text{weakly\hspace{0.17em}in}{L}^2\left(Q\right).$$E31

Again, by limit uniqueness $\frac{{\partial }^2{\zeta }_{\gamma }}{\partial t^2}-\mathrm{\Delta }{\zeta }_{\gamma }+p\left(x,\sigma \right){\zeta }_{\gamma }={\int }_0^{1}y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma }$ in $L^2\left(Q\right)$. Finally, ${\zeta }_{\gamma }$ is a solution to

$$\left\{\begin{array}{l}\frac{{\partial }^2{\zeta }_{\gamma }}{\partial t^2}-\mathrm{\Delta }{\zeta }_{\gamma }+p\left(x,\sigma \right){\zeta }_{\gamma }={\int }_0^{1}y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma }\\ {\zeta }_{\gamma }=0\\ {\zeta }_{\gamma }\left(x,T\right)=0,\frac{\partial {\zeta }_{\gamma }}{\partial t}\left(x,T\right)=0\end{array}\right.\begin{array}{c} \text{in} Q,\\ \mathrm{on} \mathrm{\Sigma },\\ \text{in} \mathrm{\Omega }.\end{array}$$E32

The uniqueness of $u_{\gamma }$ follows from strict convexity and weak lower semi-continuity of the functional $J_{\gamma }\left(v\right)$.■

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 $\gamma >0$, the average low-regret control $u_{\gamma }$ is characterized by the following optimality system

$$\left\{\begin{array}{ll}\frac{{\partial }^2{y}_{\gamma }}{\partial t^2}-\mathrm{\Delta }{y}_{\gamma }+p\left(x,\sigma \right)y_{\gamma }=0,& \\ \frac{{\partial }^2{\zeta }_{\gamma }}{\partial t^2}-\mathrm{\Delta }{\zeta }_{\gamma }+p\left(x,\sigma \right){\zeta }_{\gamma }={\int }_0^{1}y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma },& \\ \frac{{\partial }^2{\rho }_{\gamma }}{\partial t^2}-\mathrm{\Delta }{\rho }_{\gamma }+p\left(x,\sigma \right){\rho }_{\gamma }=0,& \\ \frac{{\partial }^2{q}_{\gamma }}{\partial t^2}-\mathrm{\Delta }{q}_{\gamma }+p\left(x,\sigma \right)q_{\gamma }={\int }_0^1\left({\rho }_{\gamma }+y\left(u_{\gamma },0,\sigma \right)\right)\mathit{d\sigma }-y_d& \text{in} Q,\\ y_{\gamma }=\left\{\begin{array}{c}{u}_{\gamma }\\ 0\end{array}\right.\begin{array}{c}\mathrm{on} {\mathrm{\Sigma }}_0\\ \mathrm{on\Sigma }\backslash {\mathrm{\Sigma }}_0\end{array},{\zeta }_{\gamma }=0,& \\ {\rho }_{\gamma }=0,q_{\gamma }=0& \mathrm{on} \mathrm{\Sigma },\\ y_{\gamma }\left(0,x\right)=0,\frac{\partial y_{\gamma }}{\partial t}\left(0,x\right)=0,& \\ {\zeta }_{\gamma }\left(x,T\right)=0,\frac{\partial {\zeta }_{\gamma }}{\partial t}\left(x,T\right)=0,& \\ {\rho }_{\gamma }\left(x,0\right)=-\frac{1}{\gamma }{\int }_0^1\frac{\partial \zeta \left(u_{\gamma }\right)}{\partial t}\left(x,0\right)\mathit{d\sigma },\frac{\partial {\rho }_{\gamma }}{\partial t}\left(x,0\right)=\frac{1}{\gamma }{\int }_0^1\zeta \left(u_{\gamma }\right)\left(x,0\right)\mathit{d\sigma },& \\ q_{\gamma }\left(T,x\right)=0,\frac{\partial q_{\gamma }}{\partial t}\left(T,x\right)=0& \text{in} \mathrm{\Omega },\end{array}\right.$$E33

with

$$u_{\gamma }=\frac{1}{N}{\int }_0^1\frac{\partial p_{\gamma }}{\partial \eta }\mathit{d\sigma }\text{in}{L}^2\left({\mathrm{\Sigma }}_0\right).$$E34

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

$$\begin{array}{c}{J}^{{\gamma }^{\prime }}\left(u_{\gamma }\right)\left(w\right)={\left({\int }_0^{1}y\left(u_{\gamma },0\right)\mathit{d\sigma }-y_d,{\int }_0^{1}y\left(w,0\right)\mathit{d\sigma }\right)}_{L^2\left(Q\right)}+N{\left(u_{\gamma },w\right)}_{L^2\left({\mathrm{\Sigma }}_0\right)}+\\ {\left(\frac{1}{\gamma }{\int }_0^1{\zeta }^{\prime }\left(u_{\gamma }\right)\left(0\right)\mathit{d\sigma },{\int }_0^1{\zeta }^{\prime }\left(w\right)\left(0\right)\mathit{d\sigma }\right)}_{L^2\left(\mathrm{\Omega }\right)}+{\left(\frac{1}{\gamma }{\int }_0^1\zeta \left(u_{\gamma }\right)\left(0\right)\mathit{d\sigma },{\int }_0^1\zeta \left(w\right)\left(0\right)\mathit{d\sigma }\right)}_{L^2\left(\mathrm{\Omega }\right)}=0\end{array}$$E35

for all $w\in L^2\left({\mathrm{\Sigma }}_0\right)$.

Now, let us introduce ${\rho }_{\gamma }$$=\rho \left(u_{\gamma },0\right)$ unique solution to

$$\left\{\begin{array}{l}\frac{{\partial }^2{\rho }_{\gamma }}{\partial t^2}-\mathrm{\Delta }{\rho }_{\gamma }+p\left(x,\sigma \right){\rho }_{\gamma }=0\\ {\rho }_{\gamma }=0\\ {\rho }_{\gamma }\left(x,0\right)=-\frac{1}{\gamma }{\int }_0^1\frac{\partial \zeta \left(u_{\gamma }\right)}{\partial t}\left(x,0\right)\mathit{d\sigma };\frac{\partial {\rho }_{\gamma }}{\partial t}\left(x,0\right)=\frac{1}{\gamma }{\int }_0^1\zeta \left(u_{\gamma }\right)\left(x,0\right)\mathit{d\sigma }\end{array}\right.\begin{array}{c}\text{in} Q,\\ \mathrm{on} \mathrm{\Sigma },\\ \text{in} \mathrm{\Omega }.\end{array}$$E36

So that for every $w\in L^2\left({\mathrm{\Sigma }}_0\right)$, we obtain

$$J^{{\gamma }^{\prime }}\left(u_{\gamma }\right)\left(w\right)={\left({\int }_0^{1}y\left(u_{\gamma },0\right)+{\rho }_{\gamma }\mathit{d\sigma }-y_d,{\int }_0^{1}y\left(w,0\right)\mathit{d\sigma }-y\left(0,0\right)\right)}_{L^2\left(Q\right)}+N{\left(u_{\gamma },w\right)}_{L^2\left({\mathrm{\Sigma }}_0\right)}=0$$E37

We finally define another adjoint state $q_{\gamma }=q\left(u_{\gamma }\right)$ as the unique solution of

$$\left\{\begin{array}{l}\frac{{\partial }^2{q}_{\gamma }}{\partial t^2}-\mathrm{\Delta }{q}_{\gamma }+p\left(x,\sigma \right)q_{\gamma }={\int }_0^1\left({\rho }_{\gamma }+y\left(u_{\gamma },0,\sigma \right)\right)\mathit{d\sigma }-y_d\\ q_{\gamma }=0\\ q_{\gamma }\left(x,T\right)=0,\frac{\partial q_{\gamma }}{\partial t}\left(x,T\right)=0\end{array}\right.\begin{array}{c}\text{in} Q,\\ \mathrm{on} \mathrm{\Sigma },\\ \text{in} \mathrm{\Omega }.\end{array}$$E38

Then (35) becomes

$$u_{\gamma }=\frac{1}{N}{\int }_0^1\frac{\partial q_{\gamma }}{\partial \eta }\mathit{d\sigma }\text{in}{L}^2\left({\mathrm{\Sigma }}_0\right).$$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 $\gamma$, we have

$${‖u_{\gamma }‖}_{L^2\left({\mathrm{\Sigma }}_0\right)}\le C,$$E40

$${‖{\int }_0^{1}y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma }‖}_{L^2\left(Q\right)}\le C,$$E41

$${‖y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma }‖}_{L^2\left(Q\right)}\le C,$$E42

$${‖{\int }_0^1\frac{\partial \zeta \left(u_{\gamma },\sigma \right)}{\partial t}\left(x,0\right)\mathit{d\sigma }‖}_{H^{-1}\left(\mathrm{\Omega }\right)}\le C\sqrt{\gamma },$$E43

$${‖{\int }_0^1\zeta \left(u_{\gamma },\sigma \right)\left(x,0\right)\mathit{d\sigma }‖}_{L^2\left(\mathrm{\Omega }\right)}\le C\sqrt{\gamma },$$E44

$${‖{\rho }_{\gamma }‖}_{L^{\infty }\left(0,T;H_0^1\left(\mathrm{\Omega }\right)\right)}\le C,$$E45

$${‖q_{\gamma }‖}_{L^{\infty }\left(0,T;H_0^1\left(\mathrm{\Omega }\right)\right)}\le C.$$E46

Proof. Since $u_{\gamma }$ is a solution to (17) and (18), we get

$$J_{\gamma }\left(u_{\gamma }\right)\le J_{\gamma }\left(0\right),$$E47

then

$$J\left(u_{\gamma },0\right)+\frac{1}{\gamma }{‖{\int }_0^1\frac{\partial \zeta \left(u_{\gamma },\sigma \right)}{\partial t}\left(x,0\right)\mathit{d\sigma }‖}_{L^2\left(\mathrm{\Omega }\right)}^2+\frac{1}{\gamma }{‖{\int }_0^1\zeta \left(u_{\gamma },\sigma \right)\left(x,0\right)\mathit{d\sigma }‖}_{L^2\left(\mathrm{\Omega }\right)}^2\le J\left(0,0\right),$$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).

$$E_{{\rho }_{\gamma }}\left(t\right)={\frac{1}{2}}_{\mathrm{\Omega }}\left[{\left|\frac{\partial {\rho }_{\gamma }}{\partial t}\right|}^2+{\left|\nabla {\rho }_{\gamma }\right|}^2+q\left(x,\sigma \right){\left|{\rho }_{\gamma }\right|}^2\right]\mathit{dx}=E_{{\rho }_{\gamma }}\left(0\right)\le C,$$E49

we find (45).

To get $q_{\gamma }$ estimates, just reverse the time variable by taking $s=T-t$ to find (46).

Lemma 2 The averaged low-regret control $u_{\gamma }$ tends weakly to the averaged no-regret control $u$ when $\gamma \to 0$.

Proof. From (40) we deduce the existence of a subsequence still be denoted $u_{\gamma }$ such that

$$u_{\gamma }\rightharpoonup u\text{weakly\hspace{0.17em}in}{L}^2\left({\mathrm{\Sigma }}_0\right),$$E50

let us prove $u$ is an averaged no-regret control. We have for all $v\in L^2\left({\mathrm{\Sigma }}_0\right)$

$$J\left(u_{\gamma },g\right)-J\left(0,g\right)-\gamma {‖y_0‖}_{H_0^1\left(\mathrm{\Omega }\right)}^2-\gamma {‖y_1‖}_{L^2\left(\mathrm{\Omega }\right)}^2\le \underset{g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)}{sup}\left(J\left(v,g\right)-J\left(0,g\right)\right),$$E51

take $\gamma \to 0$ to find

$$J\left(u,g\right)-J\left(0,g\right)\le \underset{g\in H_0^1\left(\mathrm{\Omega }\right)\times L^2\left(\mathrm{\Omega }\right)}{sup}\left(J\left(v,g\right)-J\left(0,g\right)\right),$$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

$$\left\{\begin{array}{ll}\frac{{\partial }^{2}y}{\partial t^2}-\mathrm{\Delta }y+p\left(x,\sigma \right)y=0,& \\ \frac{{\partial }^2\zeta }{\partial t^2}-\mathrm{\Delta }\zeta +p\left(x,\sigma \right)\zeta ={\int }_0^{1}y\left(u,0,\sigma \right)\mathit{d\sigma },& \\ \frac{{\partial }^2\rho }{\partial t^2}-\mathrm{\Delta }\rho +p\left(x,\sigma \right)\rho =0,& \\ \frac{{\partial }^{2}q}{\partial t^2}-\mathrm{\Delta }q+p\left(x,\sigma \right)q={\int }_0^1\left(\rho +y\left(u,0,\sigma \right)\right)\mathit{d\sigma }-y_d& \text{in} Q,\\ y=\left\{\begin{array}{c}u\\ 0\end{array}\right.\begin{array}{c}\mathrm{on}{\mathrm{\Sigma }}_0\\ \mathrm{on}\mathrm{\Sigma }\backslash {\mathrm{\Sigma }}_0\end{array},\zeta =0& \\ \rho =0;q=0& \mathrm{on} \mathrm{\Sigma },\\ y\left(0,x\right)=0,\frac{\partial y}{\partial t}\left(0,x\right)=0,& \\ \zeta \left(x,T\right)=0,\frac{\partial \zeta }{\partial t}\left(x,T\right)=0,& \\ \rho \left(x,0\right)={\lambda }_1\left(x\right),\frac{\partial \rho }{\partial t}\left(x,0\right)={\lambda }_2\left(x\right),& \\ q\left(T,x\right)=0,\frac{\partial q}{\partial t}\left(T,x\right)=0& \text{in} \mathrm{\Omega },\end{array}\right.$$E53

with

$$u=\frac{1}{N}{\int }_0^1\frac{\partial p}{\partial \eta }\mathit{d\sigma } \text{in} L^2\left({\mathrm{\Sigma }}_0\right),$$E54

and

${\lambda }_1\left(x\right)=\underset{\gamma \to 0}{lim}-\frac{1}{\gamma }{\int }_0^1\frac{\partial \zeta \left(u_{\gamma }\right)}{\partial t}\left(x,0\right)\mathit{d\sigma }$ weakly in $H_0^1\left(\mathrm{\Omega }\right)$,

${\lambda }_2\left(x\right)=\underset{\gamma \to 0}{lim}\frac{1}{\gamma }{\int }_0^1\zeta \left(u_{\gamma }\right)\left(x,0\right)\mathit{d\sigma }$ weakly in $L^2\left(\mathrm{\Omega }\right)$.

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

$$y\left(u_{\gamma },0,\sigma \right)\rightharpoonup y\left(u,0,\sigma \right) \text{weakly in} L^2\left(\mathrm{\Omega }\right),$$E55

solution to

$$\left\{\begin{array}{l}\frac{{\partial }^{2}y}{\partial t^2}-\mathrm{\Delta }y+p\left(x,\sigma \right)y=0\\ y=\left\{\begin{array}{c}u\\ 0\end{array}\right.\\ y\left(x,0\right)=y_0\left(x\right);\frac{\partial y}{\partial t}\left(x,0\right)=y_1\left(x\right)\end{array}\right.\begin{array}{c}\text{in} Q,\\ \mathrm{on} {\mathrm{\Sigma }}_0,\\ \mathrm{on} \mathrm{\Sigma }\backslash {\mathrm{\Sigma }}_0,\\ \text{in} \mathrm{\Omega }.\end{array}$$E56

Again, by (41) and dominated convergence theorem

$${\int }_0^{1}y\left(u_{\gamma },0,\sigma \right)\mathit{d\sigma }\rightharpoonup {\int }_0^{1}y\left(u,0,\sigma \right)\mathit{d\sigma }\text{weakly\hspace{0.17em}in}{L}^2\left({\mathrm{\Sigma }}_0\right).$$E57

The rest of equations in (53) leads by a similar way, except the convergences of initial data $\rho \left(x,0\right)$, $\frac{\partial \rho }{\partial t}\left(x,0\right)$ which will be as follows.

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

$$-\frac{1}{\gamma }\frac{\partial \zeta \left(u_{\gamma },\sigma \right)}{\partial t}\left(x,0\right)\rightharpoonup {\lambda }_1\left(x\right) \text{weakly in} H_0^1\left(\mathrm{\Omega }\right),$$E58

and

$$\frac{1}{\gamma }\zeta \left(u_{\gamma },\sigma \right)\left(x,0\right)\rightharpoonup {\lambda }_2\left(x\right) \text{weakly in} L^2\left(\mathrm{\Omega }\right).$$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).