ABSTRACT

We consider and we study a general concept of domination for controlled and observed distributed systems. We give characterization results and the main properties of this notion for controlled systems, with respect to an output operator. We also examine the case of actuators and sensors. Various other situations are considered and applications are given. Then, we extend this study by comparing observed systems with respect to a control operator. Finally, we study the relationship between the notion of domination and the compensation one, in the exact and weak cases.

1. Introduction

This work concerns the systems analysis and more precisely a general concept of domination. This notion consists to study the possibility of comparison or classification of systems. It was introduced firstly in [1] for controlled and observed lumped systems and then in [2] for a class of distributed parameter systems. The developed approach concerns separately the input and output operators. Various results are given and illustrated by applications and examples. A duality between the two cases is established. An extension of [2] to the regional case is given in [3]. The regional aspect of this problem is motivated by the fact that a system may dominates another one in a region, but not on the whole geometrical support of the system.

Let us note that in the case of the dual notions of observability and controllability, the literature is very rich. However, the purpose is different and generally, the main problem is how to reconstruct the state of the considered system or to reach a desired state, i.e. to study if a system is (or not) observable or controllable.

In this paper, we consider and we study a more general domination problem in the case of a class of controlled and observed systems [4-6]. The developed approach depends on the different parameters of the considered systems, such their dynamics, their input and output operators. Indeed, we consider without loss of generality, a class of linear distributed systems as follows

(1)

where generates a strongly continuous semi-group

(s.c.s.g.) on the state.,

and are respectively the state and the control spaces, assumed to be Hilbert spaces. The system (1) is augmented with the following output equation

(2)

with, is the observation space, a Hilbert space. The operator is the dynamics of the system, the operators and are respectively the input and output operators. The state of the system at time is given by

(3)

where

(4)

and the observation by

(5)

The first problem consists to study a possible comparison of controlled systems as system (1), with respect to an output operator. We give the main properties and characterization results. The case of sensors and actuators is also examined. Illustrative examples and applications are presented and various other situations are examined.

Then, an analogous study concerning the domination of observed systems, with respect to an input operator, is given. Finally, we study the relationship between the notion of domination and the compensation problem [7,8].

2. Domination for Controlled Systems

2.1. Problem Statement and Definitions

We consider the following linear distributed systems

(6)

(7)

where, for; is a linear operator generating a s.c.s.g. on the state space.,

; is a control space. The systems and are respectively augmented with the output equations

The state of at the final time is given by

(8)

where

(9)

The corresponding observation at time is given by

(10)

The purpose is to study a possible comparison of systems and (or the input operators and if) with respect to the output operator.

It is based on the dynamics and, the control operators, and the observation operator. Without loss of generality, one can assume that . We introduce hereafter the corresponding notion of domination.

Definition 1. We say that

1) dominates (or the pair dominates) exactly on with respect to the operator, if

2) dominates (or the pair dominates) weakly on, with respect to the operator, if

In this situation, we note respectively

Let us give following properties and remarks :

1) Obviously, the exact domination with respect to an output operator, implies the weak one with respect to. The converse is not true, this is shown in [2] for and).

2) If the system is controllable exactly (respectively weakly), or equivalently

then dominates exactly (respectively weakly) any system, with respect to any output operator.

3) In the case where, dominates exactly (respectively weakly), we say simply that dominates exactly (respectively weakly). Then, we note

Hence, one can consider a single system with two inputs as follows

(11)

augmented with an output equation

In this case, the domination of control operators and with respect to the observation operator is similar. The definitions and results remain practically the same.

4) The exact or weak domination of systems (or operators) is a transitive and reflexive relation, but it is not antisymmetric. Thus, for example in the case where, for any non-zero operator and, we have1, even if for.

5) Concerning the relationship with the notion of remediability [7,8], we consider without loss of generality, a class of linear distributed systems described by the following state equation

(12)

where is a known or unknown disturbance. The system (12) is augmented with the following output equation

(13)

The state of the system at time is given by

where

If the system (12), augmented with (13), is exactly (respectively weakly) remediable on, or equivalently (respectively

), then dominates any operator exactly (respectively weakly) with respect to the operator.

6) For and, one retrieve the particular notion of domination as in [2].

We give hereafter characterization results concerning the exact and weak domination.

2.2. Characterizations

The following result gives a characterization of the exact domination with respect to the output operator.

Proposition 2. The following properties are equivalent 1) The system dominates exactly with respect to the operator.

2) For any, there exists such that

(14)

3) There exists such that for any, we have

(15)

Proof.

The equivalence between i) and ii) derives from the definition.

The equivalence between ii) and iii) is a consequence of the fact that if and are Banach spaces; and then

if and only if, there exists such that for any, we have

where, and are respectively the dual spaces of, and.

Concerning the weak case, we have the following characterization result.

Proposition 3.

The system dominates weakly, with respect to, if and only if

(16)

Proof.

Derives from the definition and the fact that is equivalent to

It is well known that the choice of the input operator play an important role in the controllability of a system [4-6,9-11]. Here also, the domination for controlled systems, with respect to an output operator, depends on the dynamics and particularly on the choice of the control operators. However, even if (with the same actuator), the pair may dominates. This is illustrated in the the following example.

Example 4. We consider the system described by the one dimension equation

The operator generates the s.c.s.g. defined by

where, with, is a complete system of eigenfunctions of associated to the eigenvalues .

For, we have

(17)

Hence, if Equation (17) becomes

Let and .

The corresponding semi-groups, noted and, are respectively defined by

and

Then for with

1) If then for any, we have

consequently, the pair dominates the pair exactly, and hence weakly.

2) If then for any,

Hence, the pair dominates the pair exactly (and weakly).

In the next section, we examine the case of a finite number of actuators, and then the case where the observation is given by sensors.

2.3. Case of Actuators and Sensors

This section is focused on the notions of actuators and sensors [4,8,10], i.e. on input and output operators. In what follows, we assume that and, without loss of generality, we consider the analytic case where and generate respectively the s.c.s.g. and defined by

(18)

and

(19)

where is a complete orthonormal basis of eigenfunctions of, associated to the real eigenvalues such that; is the multiplicity of.

is a complete orthonormal basis of eigenfunctions of, associated to the real eigenvalues such that; is the multiplicity of.

2.3.1. Case of Actuators

In the case where is excited by zone actuators , we have and

(20)

where and;

. We have

(21)

By the same, if is excited by zone actuators, we have and

(22)

with, ,

and

(23)

As it will be seen in the next section, this leads to characterization results depending on and the corresponding controllability matrix, and then on the observability one in the case where the observation is given by a finite number of sensors. First, let us show the following preliminary result.

Proposition 5. We have

and

where and are the corresponding controllability matrices defined by

and

Proof. We have

Therefore, if and only if

By analyticity, this is equivalent to

or

where

The proof of the second equality of the proposition is similar.

The following result deriving from proposition 2, gives characterizations of exact and weak domination in the case of actuators.

Proposition 6.

1) dominates exactly with respect to the operator if and only if there exists such that for any, we have

2) dominates weakly with respect to the operator, if and only if for any, we have

Let us note that if, the domination concerns the operators and, and then the corresponding actuators. This leads to the following definition.

Definition 7. If dominates exactly (respectively weakly) with respect to the operator, we say that dominate exactly (respectively weakly) with respect to.

In the usual case, the observation is given by sensors. This is examined in following section.

2.3.2. Case of Sensors

Now, if the output is given by sensors, we have

and

We have the following proposition.

Proposition 8. dominates weakly with respect to the sensors, if and only if

(24)

where and are the corresponding observability matrices defined by

and

Proof. dominates weakly with respect to the sensors, if and only if, for any,

implies that

or equivalently, for any,

we then have the result.

Let us give the following remarks.

1) If, we have, for.

2) One actuator may dominates actuators, with respect to an output operator (sensors).

3) In the case of one actuator and one sensor, i.e. for and we have

and

Then

(25)

4) In the case of a finite number of sensors, the exact and weak domination are equivalent.

3. Application to Diffusion Systems

To illustrate previous results and other specific situations, we consider without loss of generality, a class of diffusion systems described by the following parabolic equation.

(26)

where is a bounded subset of with a sufficiently regular boundary; and for is augmented with the output equation

(27)

We examine respectively, hereafter the case of one and two space dimension.

3.1. One Dimension Case

In this section, we consider the systems and described by the following one dimension equations, with and.

(28)

(29)

admits a complete orthonormal system of eigenfunctions associated to the eigenvalues

with

Each system is augmented with the output equation corresponding to a sensor,

(30)

According to proposition 8, dominates with respect to the sensor, if and only if,

(31)

Let such that We suppose that and are respectively excited by the actuators and, i.e. and.

Then

• dominates with respect to the sensor and

• dominates with respect to the sensor

Let us also note that in the one dimension case, any operators and are comparable. this is not always possible in the two-dimension case which will be examined in the next section.

3.2. Two Dimension Case

Now, we consider the case where and the systems described by the following equations

Here, we have andfor admits a complete orthonormal system of eigenfunctions associated to the eigenvalues defined by

(32)

and are respectively augmented with the output equations

and

Let us first note that:, then is a double eigenvalue, corresponding to the eigenfunctions and

By the same, , then is also a double eigenvalue, corresponding to the eigenfunctions and

The examples given hereafter show the following situations :

• An actuator may dominates another one with respect to a sensor.

• None of the systems does not dominates the other.

Example 9. In the case where,

, and we have

(33)

where denotes the y-axis. Therefore dominates with respect to the corresponding output operator

On the other hand, for, and we have

(34)

where denotes the x-axis. Then dominates with respect to the corresponding output operator

Example 10. Now, for, , and we have

(35)

Then none of the operators and does not dominates the other.

4. Domination of Output Operators

In this section, we introduce and we study the notion of domination for observed systems (output operators) with respect to an input one. We consider first a dual problem where the control concerns the initial state, and then a general controlled system.

4.1. A Dual Problem

In this section, we examine a dual problem concerning the output operators and observed systems. We consider the system

(36)

The initial state depends on an input operator and is of the form We assume that is a linear operator with a domain dense in, a separable Hilbert space, and generates a strongly continuous semi-group on the state., is a Hilbert space. The system is augmented with the following output equations

(37)

(38)

For; the observations are given by

We have, with

Its adjoint operator is defined by

Noting ; and considering the dual systems

and

we obtain the following characterization result.

Proposition 11. (respectively

) if and only if, the controlled system

dominates exactly (respectively weakly).

From this general result, one can deduce analogous results and similar properties to those given in previous sections.

4.2. Domination of Output Operators

We consider the following linear distributed system

(39)

where generates a s.c.s.g. on the state space; and is the control space and the system (S) is augmented with the output equations

where is an Hilbert space. The observation with respect to operator at the final time is given by

(40)

We introduce hereafter the appropriate notion of domination for the considered case.

Definition 12. We say that 1) dominates exactly with respect to the system (S) (or the pair) on if .

2) dominates weakly with respect to the system (S) (or the pair) on if

.

Here also, we can deduce similar characterization results in the weak and exact cases. On the other hand, one can consider a natural question on a possible transitivity of such a domination. As it will be seen, this may be possible under convenient hypothesis. In order to examine this question, we consider without loss of generality, the linear distributed systems with the same dynamics .

(41)

(42)

where generates a s.c.s.g. on the state space;, ,

,; and are two control spaces. The systems and are augmented with the output equations

where, for; is a Hilbert space. The observations with respect to operator at the final time are respectively given by

(43)

(44)

By the same, the observations with respect to operator at time are given by

(45)

(46)

We have the following result deriving from the definitions.

Proposition 13. If the following conditions are satisfied 1) dominates exactly (respectively weakly) with respect to operator2) dominates exactly (respectively weakly) with respect to operator3) dominates exactly (respectively weakly) with respect to operatorthen dominates exactly (respectively weakly) with respect to operator.

We examine hereafter, the relationship between the notions of domination and compensation.

4.3. Domination and Compensation

In this section, we study the relationship between the notions of domination and compensation [7,8]. We consider without loss of generality, the following systems.

(47)

(48)

where generates a s.c.s.g. on the state space;, ,

, and; and

are two control spaces. and are respectively augmented with the output equations

The states of these systems at the final time are respectively given by

(49)

where the operators; and are defined by

(50)

(51)

The corresponding observations are given by

(52)

(53)

and. First let us recall the notion of compensation.

Definition 14. The system augmented with output equation (or) is 1) exactly remediable on if for any, there exists such that, or equivalently

(54)

2) weakly remediable on if for any and any there exists

such that, or equivalently

(55)

Here, the question is not to examine if a system is (or not) remediable (for this one can see [7,8]), but to study the nature of the relation between the notions of domination and compensation, respectively in the exact and weak cases. We have the following result.

Proposition 15. If the following conditions are verified 1) is exactly (respectively weakly) remediable.

2) dominates exactly (respectively weakly) with respect to the operator.

3) (respectively

).

then is exactly (respectively weakly) remediable.

We have the similar result concerning the output domination and the remediability notion.

Proposition 16. If the following conditions are satisfied 1) is exactly (respectively weakly) remediable.

2) dominates exactly (respectively weakly) with respect to the operator.

then is exactly (respectively weakly) remediable.

Let us note that this section is a generalization of the previous one where has the form. The results can be applied easily to a diffusion system and to other systems and situations.

REFERENCES

NOTES

1 denotes the operator corresponding to, i.e. defined by