# Domination and Compensation in Finite Dimension Dynamical Systems

Larbi Afifi, El Mostafa Magri, Abdelhaq El Jai

► To cite this version:

Larbi Afifi, El Mostafa Magri, Abdelhaq El Jai. Domination and Compensation in Finite Dimension Dynamical Systems. Australian Journal of Mathematical Analysis and Applications, 2010, 4 (49), p. 2443 - 2457. 10.4236/ica.2013.42026. hal-01296545

HAL Id: hal-01296545

https://univ-perp.hal.science/hal-01296545

Submitted on 4 Oct 2019

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Domination and Compensation in Finite Dimension Dynamical Systems

## Larbi AFIFI

Faculty of Sciences, University Hassan II Ain Chock
B.P.5366-Maârif, Casablanca, Morocco
l.afifi@fsac.ac.ma, larbi_afifi@yahoo.fr

## El Mostafa MAGRI

Faculty of Sciences, University Hassan II Ain Chock
B.P.5366-Maârif, Casablanca, Morocco
m.magri@fsac.ac.ma

## Abdelhaq EL JAI

MEPS - Théorie des Systèmes, Université de Perpignan
52 Avenue Paul Alduy, Perpignan Cedex
aej@univ-perp.fr

## Résumé

This work concerns the analysis of a class of linear dynamical systems. We study the possibility of comparing input operators, with respect to the output one, and we give characterization results. Various situations are examined, applications and illustrative examples are presented.

Under convenient hypothesis, we also show how to find the optimal control ensuring the compensation of a disturbance in the finite time or asymptotic cases. The relationship with the notions of controllability, stability and stabilizability is examined. Here also, applications and examples illustrating different results and situations are given.

**Keywords** : Dynamical systems, observation, control, domination, remediability, disturbance
# 1 Introduction and problem statement

In this work, we consider a class of finite dimension dynamical systems described by a linear state equation as follows:

$$ (S) \begin{cases} \dot{z}(t) = Az(t) + B_1u_1(t) + B_2u_2(t); \quad 0 < t < T \\ z(0) = z_0 \in \mathbb{R}^n \end{cases} \tag{1} $$

where $A \in M_n(\mathbb{R})$, $B_1 \in M_{n,p}(\mathbb{R})$, $B_2 \in M_{n,m}(\mathbb{R})$, $u_1 \in L^2(0, T; \mathbb{R}^p)$ and $u_2 \in L^2(0, T; \mathbb{R}^m)$.

The system (S) is augmented by the output equation:

$$ y(t) = Cz(t); \quad 0 < t < T \tag{2} $$

where $C \in M_{q,n}(\mathbb{R})$. The state of system (S) at time $t$, is given by:

$$ z(t) = e^{At}z_0 + H_1(t)u_1 + H_2(t)u_2 \tag{3} $$

where for $t \in [0, T]$, $H_1(t)$ and $H_2(t)$ are the operators defined by:

$$ \begin{gathered} H_1(t): \quad L^2(0, t; \mathbb{R}^p) \longrightarrow \mathbb{R}^n \\ u_1 \longrightarrow \int_0^t e^{A(t-s)} B_1 u_1(s) ds \end{gathered} \tag{4} $$

and

$$ \begin{gathered} H_2(t): \quad L^2(0, t; \mathbb{R}^m) \longrightarrow \mathbb{R}^n \\ u_2 \longrightarrow \int_0^t e^{A(t-s)} B_2 u_2(s) ds \end{gathered} \tag{5} $$

For $i = 1, 2$, we note $H_i = H_i(T)$. Then:

$$ y(T) = Ce^{AT}z_0 + CH_1u_1 + CH_2u_2 \tag{6} $$

The system (S) is excited by two input terms $B_1u_1$ and $B_2u_2$ where one, the second for example, is considered as an intentional or accidental disturbance. The other term $B_1u_1$ is introduced in order to compensate [1,2] at the final time $T$, the effect of the disturbance by bringing back the observation to the normal situation which is $Ce^{AT}z_0$. That is to say: for any $u_2 \in L^2(0, T; \mathbb{R}^m)$, there exists a control $u_1 \in L^2(0, T; \mathbb{R}^p)$ such that:

$$ Ce^{A(T)}z_0 + \int_0^T Ce^{A(T-s)}B_1u_1(s)ds + \int_0^T Ce^{A(T-s)}B_2u_2(s)ds = Ce^{A(T)}z_0 $$
or equivalently

$$CH_1u_1 + CH_2u_2 = 0$$

This leads to the notion of domination which consists to study the possibility of comparing the input operators $B_1$ and $B_2$, with respect to the output one $C$.

The domination notion is introduced and studied separately for controlled and observed distributed systems [3]. In this paper, we examine the problem of domination in connection with the compensation one.

First, we consider the finite time case. We define and we characterize the notion of domination. Sufficient conditions, applications and illustrative examples are also given. The minimum energy problem [7,8] is examined using Hilbert Uniqueness Method, such a problem can be studied as a general optimal control one. The obtained results are extended to the asymptotic case and various situations are examined. The relationship with the notions of stability and stabilizability [4,5,6] are equally studied.

# 2 Finite time C-domination

## 2.1 Definitions and characterizations

We define hereafter the notion of C-domination.

**Definition 2.1**

We say that $B_1$ dominates $B_2$ on $[0, T]$ with respect to $C$ (or $B_1$ C-dominates $B_2$ on $[0, T]$), if for any $u_2 \in L^2(0, T; \mathbb{R}^m)$, there exists a control $u_1 \in L^2(0, T; \mathbb{R}^p)$ such that :

$$CH_1u_1 + CH_2u_2 = 0$$

In this case, and for $C$ and $T$ fixed, one can note $B_2 \le B_1$

We have the following characterization result.

**Proposition 2.2** The following properties are equivalent :

i) $B_1$ dominates $B_2$ on $[0, T]$ with respect to $C$.

ii) $Im(CH_2) \subset Im(CH_1)$.

iii) $Ker(H_1^*C^*) \subset Ker(H_2^*C^*)$.
iv) $\exists \gamma > 0$ such that for any $\theta \in \mathbb{R}^q$, we have

$$ \| B_2^* e^{A^*(T-)} C^* \theta \|_{L^2(0,T;\mathbb{R}^m)} \leq \gamma \| B_1^* e^{A^*(T-)} C^* \theta \|_{L^2(0,T;\mathbb{R}^p)} \quad (7) $$

**Proof:** Derive from the definition, the fact that

$$ \mathrm{Ker}(H_i^* C^*) = \mathrm{Ker}(B_i^* e^{A^*(T-)} C^*), \quad \text{for } i = 1,2 $$

and also the following well known result.

**Lemma 2.3**

Let $X$, $Y$ and $Z$ be Banach reflexive spaces, $P \in \mathcal{L}(X, Z)$ and $Q \in \mathcal{L}(Y, Z)$. We have

$$ \mathrm{Im}(P) \subset \mathrm{Im}(Q) $$

if and only if

$$ \exists \gamma > 0 \text{ such that for any } z^* \in Z', \text{ we have } \|P^*z^*\|_{X'} \leq \gamma \|Q^*z^*\|_{Y'} $$

Concerning the relationship with the controllability notion, we have the following result.

**Proposition 2.4**

i) If the system

$$ (S_1) \left\{ \begin{array}{l} \dot{z}(t) = Az(t) + B_1u_1(t); \quad 0 < t < T \\ z(0) = z_0 \in \mathbb{R}^n \end{array} \right. \qquad (8) $$

is controllable on $[0, T]$, then $B_1$ dominates any operator $B_2$ on $[0, T]$, with respect<fnref n="1" /> to $C$.

ii) The converse is not true.

**Proof:**

i) Obviously, $(S_1)$ is controllable on $[0, T] \iff \mathrm{Im}H_1 = \mathbb{R}^n$, then

$$ \mathrm{Im}(CH_1) = \mathrm{Im}(C) $$

and hence

<footnote n="1">¹In fact, this is true for any output operator $C$.</footnote>
$$ Im(CH_2) \subset Im(CH_1) $$

Consequently $B_1$ dominates $B_2$ on $[0, T]$, with respect to $C$.

ii) **Counter example**: We consider the case where $n = 2, p = q = 1$ and

$$ A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}; \quad B_2 = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}; \quad B_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}; \quad C = \begin{pmatrix} 1 & 0 \end{pmatrix} $$

we have

$$ B_2^* e^{A^*(T-s)} C^* \theta = e^{(T-s)} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \theta = \begin{pmatrix} e^{(T-s)} \theta \\ 0 \end{pmatrix} $$

and

$$ B_1^* e^{A^*(T-s)} C^* \theta = \begin{pmatrix} 1 & 0 \end{pmatrix} \begin{pmatrix} e^{(T-s)} \theta \\ 0 \end{pmatrix} = e^{(T-s)} \theta $$

then

$$ \| B_2^* e^{A^*(T-s)} C^* \theta \|_{L^2(0,T;\mathbb{R}^2)} = \| B_1^* e^{A^*(T-s)} C^* \theta \|_{L^2(0,T;\mathbb{R})} $$

The inequality (7) is then true for $\gamma = 1$. Hence $B_1$ C-dominates $B_2$, but

$$ \mathrm{rank} \left( B_1 \ AB_1 \right) = \mathrm{rank} \left( \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \right) = 1 < 2 $$

Consequently ($S_1$) is not controllable on $[0, T]$. $\square$

<figure><img src="image_2.png" /></figure>

We give hereafter a sufficient condition ensuring such a domination.

**Proposition 2.5 If**

$$ \mathrm{rank} \left( CB_1 \ CAB_1 \ \dots \ CA^{n-1}B_1 \right) = q \quad (9) $$

then $B_1$ C-dominates any operator $B_2$ on $[0, T]$.

**Proof**: Using Caylay-Hamilton theorem, we have

$$ \begin{align*} \mathrm{rank} \left( CB_1 \ CAB_1 \ \dots \ CA^{n-1}B_1 \right) &= q \\ &\iff \left( \begin{array}{c} B_1^* C^* \\ B_1^* A^* C^* \\ \vdots \\ B_1^* (A^*)^{n-1} C^* \end{array} \right)_{(np,q)} & y=0; \forall y \in \mathbb{R}^q \implies y=0 \\ &\iff \mathrm{Ker}(H_1)^* C^* = \{0\} \end{align*} $$
Hence, if $\mathrm{Ker} [(H_1)^*C^*] = \{0\}$, then $\mathrm{Ker} [(H_1)^*C^*] \subset \mathrm{Ker} [(H_2)^*C^*]$ and then, $B_1$ dominates $B_2$ on $[0, T]$ with respect to $C$. $\square$

**Remark 2.6**

i) One can have

$$ \mathrm{rank} ( CB_1 \ CAB_1 \ ... \ CA^{n-1}B_1 ) = q $$

even if the system $(S_1)$ is not controllable on $[0, T]$.

ii) $B_1$ may dominate another operator $B_2$, with respect to $C$ on $[0, T]$, without having

$$ \mathrm{rank} ( CB_1 \ CAB_1 \ ... \ CA^{n-1}B_1 ) = q $$

This is illustrated in the following example.

**Example 2.7**

i) We consider the case where $n = 2, p = q = 1$ and

$$ A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}; \quad B_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}; \quad C = \begin{pmatrix} 1 & 0 \end{pmatrix} $$

The controllability matrix is given by

$$ ( B_1 \ AB_1 ) = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} $$

its rank is then $1 < 2$. Consequently, the corresponding system is not controllable on $[0, T]$. On the other hand

$$ ( CB_1 \ CAB_1 ) = \begin{pmatrix} 1 & 1 \end{pmatrix} $$

Its rank is $1 = q$, then $B_1$ dominates any operator $B_2$ on $[0, T]$ with respect to $C$.

ii) Now, for $m = n = 2, p = 1, q = 2$ and

$$ A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}; \quad B_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}; \quad B_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}; \quad C = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}; \quad \theta = \begin{pmatrix} \theta_1 \\ \theta_2 \end{pmatrix} $$

we have
$$
\begin{align*}
e^{A^*(T-.s)} C^* \theta &= e^{(T-s)} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} \theta_1 \\ \theta_2 \end{pmatrix} \\
&= e^{(T-s)} \begin{pmatrix} \theta_1 + \theta_2 \\ \theta_1 + \theta_2 \end{pmatrix}
\end{align*}
$$

and $B_1^* e^{A^*(T-s)} C^* \theta = e^{(T-s)} 2(\theta_1 + \theta_2)$, then

$$
\| B_1^* e^{A^*(T-\cdot)} C^* \theta \|_{L^2(0,T;\mathbb{R})}^2 = 4 \int_0^T e^{2(T-s)} (\theta_1 + \theta_2)^2 ds
$$

On the other hand

$$
\begin{align*}
\| B_2^* e^{A^*(T-\cdot)} C^* \theta \|_{L^2(0,T;\mathbb{R}^2)}^2 &= \int_0^T e^{2(T-s)} \left\| \begin{pmatrix} \theta_1 + \theta_2 \\ \theta_1 + \theta_2 \end{pmatrix} \right\|^2 ds \\
&= 2 \int_0^T e^{2(T-s)} (\theta_1 + \theta_2)^2 ds
\end{align*}
$$

hence

$$
\| B_2^* e^{A^*(T-\cdot)} C^* \theta \|_{L^2(0,T;\mathbb{R}^2)} \le \| B_1^* e^{A^*(T-\cdot)} C^* \theta \|_{L^2(0,T;\mathbb{R})}
$$

Consequently, $B_1$ C-dominates $B_2$ on $[0, T]$, even if

$$
\mathrm{rank} ( CB_1 \ CAB_1 ) = \mathrm{rank} \left( \begin{array}{cc} 2 & 2 \\ 2 & 2 \end{array} \right) = 1 \neq 2
$$

<figure><img src="image_3.png" /></figure>

In the following result, we give a necessary and sufficient condition for such a domination.

**Proposition 2.8**

$B_1$ C-dominates $B_2$ on $[0, T]$, if and only if

$$
\mathrm{Im} ( CB_2 \ CAB_2 \ \dots \ CA^{n-1} B_2 ) \subset \mathrm{Im} ( CB_1 \ CAB_1 \ \dots \ CA^{n-1} B_1 )
$$

**Proof** : Using proposition 2.2, $B_1$ C-dominates $B_2$ on $[0, T]$, if and only if

$$
\mathrm{Ker}(H_1^* C^*) \subset \mathrm{Ker}(H_2^* C^*)
$$

Using Caylay-Hamilton theorem, we deduce that for $i = 1, 2$
$$y \in \mathrm{Ker}[(H_i)^*C^*] \iff \left( \begin{array}{c} B_i^*C^* \\ B_i^*A^*C^* \\ \vdots \\ B_i^*(A^*)^{n-1}C^* \end{array} \right)_{(np,q)} \quad y=0$$

hence

$$\mathrm{Ker} \left( \begin{array}{c} B_i^*C^* \\ B_i^*A^*C^* \\ \vdots \\ B_i^*(A^*)^{n-1}C^* \end{array} \right) = \mathrm{Ker}(H_i^*C^*)$$

Consequently, $B_1$ C-dominates $B_2$ on $[0, T]$, if and only if

$$\mathrm{Im} ( CB_2 \ CAB_2 \ ... \ CA^{n-1}B_2 ) \subset \mathrm{Im} ( CB_1 \ CAB_1 \ ... \ CA^{n-1}B_1 )$$

□

**Remark 2.9**

1. This necessary and sufficient condition is not depending on the time parameter $T$.

2. In the particular case where $B_2$ is invertible, $B_1$ C-dominates $B_2$ on $[0, T]$ if and only if

$$\mathrm{rank} ( CB_1 \ CAB_1 \ ... \ CA^{n-1}B_1 ) = \mathrm{rank} ( C )$$

## 2.2 Minimum energy problem

In this part, we assume that $B_1$ C-dominates $B_2$ on $[0, T]$, then for any $v \in L^2(0, T; \mathbb{R}^m)$, there exists a control $u \in L^2(0, T; \mathbb{R}^p)$ such that

$$CH_1u + CH_2v = 0 \qquad (10)$$

For $v \in L^2(0, T; \mathbb{R}^m)$, we examine the existence and the uniqueness of the optimal control $u \in L^2(0, T; \mathbb{R}^p)$ satisfying (10), i.e. ensuring the compensation of the opposing term $B_2v$.

For this, we use an extension of the Hilbert Uniqueness Method. Indeed, for $\theta \in \mathbb{R}^q$, let us note:
$$ \| \theta \|_{*} = \left( \int_{0}^{T} \|(H_{1})^{*} C^{*} \theta \|_{\mathbb{R}^{p}}^{2} ds \right)^{\frac{1}{2}} = \left( \int_{0}^{T} \| B_{1}^{*} e^{A^{*}(T-s)} C^{*} \theta \|_{\mathbb{R}^{p}}^{2} ds \right)^{\frac{1}{2}} $$

$\| \theta \|_{*}$ is a semi-norm on $\mathbb{R}^{q}$.

We assume that $\| \cdot \|_{*}$ is a norm on $\mathbb{R}^{q}$. If $Ker [(H_{1})^{*}C^{*}] = \{0\}$, this is equivalent to the asymptotic remediability [1,2,3] of the system (1)+(2) on $[0, T]$. The corresponding inner product is given by :

$$ \langle \theta, \sigma \rangle_{*} = \int_{0}^{T} \langle B_{1}^{*} e^{A^{*}(T-s)} C^{*} \theta, B_{1}^{*} e^{A^{*}(T-s)} C^{*} \sigma \rangle ds $$

and the operator $\Lambda_{C}: \mathbb{R}^{q} \to \mathbb{R}^{q}$ defined by

$$ \Lambda_{C}\theta = C H_{1}(H_{1})^{*} C^{*} \theta = \int_{0}^{T} C e^{(T-s)} B_{1} B_{1}^{*} e^{A^{*}(T-s)} C^{*} \theta ds $$

is symmetric and positive definite, and then invertible. We give hereafter the expression of the optimal control ensuring the compensation of the effect of the term $B_{2}v$, at the final time $T$.

**Proposition 2.10**

For $v \in L^{2}(0, T; \mathbb{R}^{m})$, there exists a unique $\theta_{v} \in \mathbb{R}^{q}$ such that

$$ \Lambda_{C}\theta_{v} = -CH_{2}v $$

and the control

$$ u_{\theta_{v}}(\cdot) = B_{1}^{*} e^{A^{*}(T-\cdot)} C^{*} \theta_{v} $$

verifies

$$ CH_{1}u_{\theta_{v}} + CH_{2}v = 0 $$

Moreover, it is optimal and

$$ \|u_{\theta_{v}}\|_{L^{2}(0,T;\mathbb{R}^{p})} = \|\theta_{v}\|_{*} $$

Let us note that one can also consider a general optimal control problem with a cost function defined on $L^{2}(0, T; \mathbb{R}^{p})$ by
$$
\begin{align}
J(u) ={}& \langle P(CH_1 u + CH_2 v), CH_1 u + CH_2 v \rangle \nonumber \\
& + \int_0^T \langle Q(CH_1(t)tu + CH_2(t)v), CH_1(t)u + CH_2(t)v \rangle dt \tag{11} \\
& + \int_0^T \langle Ru(t), u(t) \rangle dt \nonumber
\end{align}
$$

where P, Q and R are symmetric matrixes with Q positive, P and R are
positive definite.

In the next section, we present an extension to the asymptotic case.

**3 Extension to the asymptotic case**

In this part, we consider a class of linear dynamical systems described by the
following state equation

$$
(S_1^\infty) \left\{
\begin{array}{l}
\dot{z}(t) = Az(t) + B_1u_1(t) + B_2u_2(t); \quad t > 0 \\
z(0) = z_0 \in \mathbb{R}^n
\end{array}
\right.
\qquad (12)
$$

with $A \in M_n(\mathbb{R})$, $B_1 \in M_{n,p}(\mathbb{R})$, $B_2 \in M_{m,n}(\mathbb{R})$, $u_1 \in L^2(0, +\infty; \mathbb{R}^p)$ and $u_2 \in L^2(0, +\infty; \mathbb{R}^m)$

The system (12) is augmented by the output equation :

$$
y(t) = Cz(t); \quad t > 0 \tag{13}
$$

with $C \in M_{q,n}(R)$. We have

$$
z(t) = e^{At}z_0 + H_1(t)u_1 + H_2(t)u_2
$$

Let

$$
z = \begin{pmatrix} z_+ \\ z_- \end{pmatrix}
$$

where $z_+$ and $z_-$ are respectively the projections of the state $z$ on the
unstable and the stable subspaces :

$$
E_+ = \bigoplus_{R(e)(\lambda) \ge 0} \mathrm{Ker}(A - \lambda I_n)^{m(\lambda)} \quad (14)
$$

and
$$E_{-} = \bigoplus_{R(e)(\lambda)<0} \mathrm{Ker}(A - \lambda I_{n})^{m(\lambda)} \quad (15)$$

where $m(\lambda)$ is the multiplicity of the eigenvalue $\lambda$. $E_+$ and $E_-$ are invariant with respect to the operator $A$. We have

$$\left\{ \begin{array}{l} (S^{+}) \dot{z}^{+}(t) = A_{+}z^{+}(t) + PBu_{1}(t) + PBu_{2}(t) \\ (S^{-}) \dot{z}^{-}(t) = A_{-}z^{-}(t) + (I - P)Bu_{1}(t) + (I - P)Bu_{2}(t) \end{array} \right. \quad (16)$$

$P$ is the projection operator on the unstable part and $A_+$, respectively $A_-$, is the matrix induced by $A$ on $E_+$, respectively $E_-$.

In the case where we observe only the stable part, i.e. if

$$E_{+} \subset \mathrm{Ker}(C)$$

the following operators $K^{\infty}(C)$ and $L^{\infty}(C)$ respectively given by

$$K^{\infty}(C) : L^{2}(0, +\infty; \mathbb{R}^{p}) \longrightarrow \mathbb{R}^{q}$$

$$u \quad \longrightarrow \int_{0}^{\infty} Ce^{At}B_{1}u(t)dt$$

and

$$L^{\infty}(C) : L^{2}(0, +\infty; \mathbb{R}^{m}) \longrightarrow \mathbb{R}^{q}$$

$$v \quad \longrightarrow \int_{0}^{\infty} Ce^{At}B_{2}v(t)dt$$

are well defined. We have the same result if the considered system is exponentially stable, i.e. the matrix $A$ is such that $Re(\lambda_i) < 0$ for $i = 1, n$; where $\lambda_1, ..., \lambda_n$ are the eigenvalues of $A$. But as it will be shown later, this is not necessary.

We assume that operators $K^{\infty}(C)$ and $L^{\infty}(C)$ are well defined. By the same, we say that $B_1 C$-dominates $B_2$ asymptotically, if for every $u_2 \in L^2(0, +\infty; \mathbb{R}^m)$, there exists $u_1 \in L^2(0, +\infty; \mathbb{R}^p)$ such that:

$$K^{\infty}(C)u_{1} + L^{\infty}(C)u_{2} = 0$$

With a similar approach, it is easy to show the following characterization result of the asymptotic domination.
**Proposition 3.1** *The following properties are equivalent :*

i) $B_1$ C-dominates $B_2$ asymptotically.

ii) $Im [L^\infty(C)] \subset Im [K^\infty(C)]$

iii) $Ker [K^\infty(C)] \subset Ker [L^\infty(C)]$

iv) $\exists \gamma > 0$ such that $\forall \theta \in \mathbb{R}^q$, we have

$$ \| B_2^* e^{A^*} \cdot C^* \theta \|_{L^2(0, +\infty; \mathbb{R}^m)} \leq \gamma \| B_1^* e^{A^*} \cdot C^* \theta \|_{L^2(0, +\infty; \mathbb{R}^p)} $$

v)

$$ Im ( CB_2 \ CAB_2 \ ... \ CA^{n-1}B_2 ) \subset Im ( CB_1 \ CAB_1 \ ... \ CA^{n-1}B_1 ) $$

Let us note that if the eigenvalues $\lambda_1, ..., \lambda_n$ of A are such that : $Re(\lambda_i) < 0$ for $i = 1, n$, the proposition 2.4 remain true in the asymptotic case. One can also consider the asymptotic optimal control problem. The approach and the results are similar.

We give hereafter illustrative examples showing particularly that the notions of stability (or even the stabilizability) and also the controllability are not necessary for considering the asymptotic domination.

**Example 3.2**

Let us consider the case of an unstable system with $m = n = 2, p = q = 1$ and

$$ A = \begin{pmatrix} 2 & 0 \\ 1 & -1 \end{pmatrix}; \quad B_2 = \begin{pmatrix} 1 & 0 \\ 1 & 2 \end{pmatrix}; \quad B_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}; \quad C = \begin{pmatrix} 1 & -3 \end{pmatrix} $$

Obviously, the system is not stable. We have

$$ e^{tA} = Pe^{t\Delta}P^{-1} $$

where

$$ \Delta = \begin{pmatrix} 2 & 0 \\ 0 & -1 \end{pmatrix}, \quad P = \begin{pmatrix} 3 & 0 \\ 1 & 1 \end{pmatrix} $$

then

$$ Ce^{tA} = ( e^{-t} \ -3e^{-t} ) $$
In this case $K^\infty(C)$ and $L^\infty(C)$ are well defined and

$$ ( CB_1 CAB_1 ) = ( 1 -1 ) $$

and $rank(C) = 1$. Consequently, $B_1$ C-dominates $B_2$ asymptotically.

□

**Example 3.3** We consider an unstable system with $m = n = 2$, $p = q = 1$ and $B_2$ is invertible.

i) For

$$ A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}; \quad B_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}; \quad C = \begin{pmatrix} 1 & 1 \end{pmatrix}; \quad F = \begin{pmatrix} a & b \end{pmatrix} $$

we have

$$ A + B_1 F = \begin{pmatrix} 1 + a & b \\ 0 & 2 \end{pmatrix} $$

$(A, B_1)$ is not stabilizable because for any $F = \begin{pmatrix} a & b \end{pmatrix}$, $A + B_1 F$ is not stable. However, $B_1$ C-dominates $B_2$ asymptotically.

ii) Let

$$ A = \begin{pmatrix} -1 & 0 \\ 1 & 2 \end{pmatrix}; \quad B_1 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}; \quad C = \begin{pmatrix} 1 & 1 \end{pmatrix} $$

For $F = \begin{pmatrix} a & b \end{pmatrix}$, the matrix $A + B_1 F = \begin{pmatrix} -1 & 0 \\ a+1 & b+2 \end{pmatrix}$ is stable for $b < -2$, then $(A, B_1)$ is stabilizable.

On the other hand, we have $AB_1 = \begin{pmatrix} 0 \\ 2 \end{pmatrix}$, then $rank(B_1 AB_1) = 1 \neq 2$ and consequently $(A, B_1)$ is not controllable (in fact, the asymptotic controllability is not well defined in the considered case). But $B_1$ C-dominates $B_2$ asymptotically, because

$$ rank(CB_1 CAB_1) = rank(C) \qquad (17) $$

Let us remark that in the general case, if $C$ is an invertible matrix (for example if $C$ is the identity matrix), then the relation (9) (the relation (17) in the considered example) is equivalent to the controllability rank condition. But this hypothesis is strong and is not very useful.
<figure><img src="image_4.png" /></figure>

Finally let us note that concerning the asymptotic minimum energy problem, and with an extension of H.U.M., it is easy to show the existence and the uniqueness of the optimal control and also how to find it.

With convenient hypothesis, one can also consider the asymptotic version of the cost function given by (11).

**Acknowledgment** : The authors wish to thank the Academy Hassan II of Sciences and Technics for its support to the Systems Theory Network.

Références

[1] L. Afifi, A. Chafiai and A. El Jai (2001)
Spatial Compensation of boundary disturbances by boundary actuators the
International Journal of Applied Mathematics and Computer Science,
Vol. 11, N°4.

[2] L. Afifi, A. El Jai and E. M. Magri (2008)
Compensation problem in finite dimension linear dynamical systems. In-
ternational Journal of Applied Mathematical Sciences. Vol.2, N.45, pp
2219 - 2228.

[3] L. Afifi, M. Bahadi, A. El Jai et A. El Mizane (2009) *Asymptotic Analysis, Approximations and Simulations of the Compensation Problem in Hyperbolic systems*. Applied Mathematical Sciences, volume 3, N°15, 737-765.

[4] L. Afifi, E. M. Magri and A. El Jai (2010)
*Weak and exact domination in distributed systems*. To appear in the In-
ternational Journal of Applied Mathematics and Computer Science. Vol.
20, No. 3.

[5] R.F. Curtain and A.J. Pritchard (1978)
*Infinite Dimensional Linear Systems Theory*. Lecture Notes in Control
and Information Sciences, vol. 8, Berlin.

[6] Curtain, R.F. and Zwart, H.J. (1995). *An Introduction to Infinite-Dimensional Linear Systems Theory*. Texts in Applied Mathematics, Springer-Verlag, New York.

[7] J.-L. Lions (1980)
*On the foundation of optimal control of distributed systems*. Lecture notes
in control and inf. sc. 22, Springer.

[8] J.-L. Lions (1988)
*Contrôlabilité exacte, perturbations et stabilisation des systèmes dis-
tribués*. Paris, Masson.
Received : July, 2010