Domination in Controlled and Observed Distributed Parameter Systems

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.


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][5][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 A generates a strongly continuous semi-group where    0 t S t  on the state Z .,

 
, B U Z   (s.c.s.g.)   2 0, ; ; u L T U  Z and U are respectively the state and the control spaces, assumed to be Hilbert spaces.The system (1) is augmented with the following output equation is the observation space, a Hilbert space.The operator A 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 where and the observation by    A parison 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].

Problem Statement and Definitions
We consider the following linear distributed systems where, for ; A is a linear operator generating a s.c.s.g. on the state space

S    
  for 1, 2; The state of at the final time is given by where 2 0 : 0, ; The corresponding observation at time is given by The purpose is to study a possible comparison of systems and  2 (or the input operators 1

B and if
A A  C ) with respect to the output operator .

It is based on the dynamics 1
A and 2 A , the control operators 1 , 2 and the observation operator .Without loss of generality, one can assume that 1,0 . We introduce hereafter the corresponding notion of domination.
In this situation, we note respectively , , and , , 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 is controllable exactly (respectively weakly), or equivalently   , with respect to any output operator .
exactly (respectively weakly), we say simply that 1 dominates exactly (respectively weakly).Then, we note Hence, one can consider a single system with two inputs as follows augmented with an output equation In this case, the domination of control operators 1 and 2 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 A A  , for any non-zero operator and ) 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   2 2 1 1 0, : 0, : . .where is a known or unknown disturbance.The system (12) is augmented with the following output equation The state of the system at time is given by , one retrieve the particular notion of domination as in [2].
We give hereafter characterization results concerning the exact and weak domination.

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 , there exists 3) There exists   Y such that for any    , we have 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 Z are Banach spaces; Proof.
Derives from the definition and the fact that It is well known that the choice of the input operator play an important role in the controllability of a system [4][5][6][9][10][11].Here also, the domination for controlled systems, with respect to an output operator , depends on the dynamics i A and particularly on the choice of the control operators i .However, even if , (with the same actuator), the pair A B .This is illustrated in the the following example.
Example 4. We consider the system described by the one dimension equation The corresponding semi-groups, noted and , are respectively defined by , dominates the pair 1 A B exactly, and hence 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.

Case of Actuators and Sensors
This section is focused on the notions of actuators and sensors [4,8,10] A , associated to the real ei-

 
S p 1 is excited by zone actuators In the case where 2 is excited by zone actuators By the same, if , we have and 1 , , 0, ; 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 where n M and n are the corresponding controllability matrices defined by Therefore, if and only if ker 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  2 exactly with respect to the operator if and only if there exists   such that for any dominates  2 weakly with respect to the operator , if and only if for any    , we have , the domination concerns the operators 1 and 2 , and then the corresponding actuators.This leads to the following definition.

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

Case of Sensors
Now, if the output is given by We have the following proposition., ker ker where n and n are the corresponding observability matrices defined by dominates  2 weakly with respect to the sensors   1 , or equivalently, for any , we then have the result.Let us give the following remarks.

1) If 1 2
A A , we have , for . n n 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  1, and m  we have  , , , , , , In the case of a finite number of sensors, the exact and weak domination are equivalent.

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 equa-where  is a bounded subset of with a sufficiently regular boundary We examine respectively, hereafter the case of one and two space dimension.


In this section, we consider the systems 1 and  S 2 described by the following one dimension equations, with ,0 0 in 0, , in 0, 0, , 0 in 0, ,0 0 in 0, admits a complete orthonormal system of eigenfunctions associated to the eigenvalues with respect to the sensor   , f  , if and only if, is augmented with the output , , , 0 , , , 0 Let us also note that in the one dimension case, any operators 1 and 2 are comparable.this is not always possible in the two-dimension case which will be examined in the next section.

   
Now, we consider the case where 0,1 0,1    and the systems described by the following equations , ,0 0 in z x y t z x y t g x y u t g x y u t t S z x y t T z x y , ,0 0 in and are respectively augmented with the output equations , 0 ,

E y t f z t f z t 
Let us first note that: , then is a double eigenvalue, corresponding to the eigenfunctions 10,10


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
where , 0 g g with respect to the corresponding out-put operator On the other hand, for

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.

A Dual Problem
In this section, we examine a dual problem concerning the output operators and observed systems.We consider the system The is a Hilbert space.The system is augmented with the following output equations For ; the observations are given by 1, 2 i    We have Its adjoint operator is defined by and considering the dual systems we obtain the following characterization result.
) if and only if, the controlled system   1 S  dominates  exactly (respectively weakly).
From this general result, one can deduce analogous results and similar properties to those given in previous sections.

Domination of Output Operators
We consider the following linear distributed system where A generates a s.c.s.g. on the state 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 We introduce hereafter the appropriate notion of domination for the considered case.
Definition 12.We say that 1) 1 dominates C exactly with respect to the system (S) (or the pair  

2)
dominates C weakly with respect to the system (S) (or the pair   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 A generates a s.c.s.g. on the state ; is a Hilbert space.The observations with respect to operator at the final time are respectively given by By the same, the observations with respect to operator at time T are given by We have the following result deriving from the definitions.
Proposition  The corresponding observations are given by 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 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.
1 then     .The results can be applied easily to a diffusion system and to other systems and situations.

B Definition 7 . 1 exactly
If 1 dominates  2 exactly (res- pectively weakly) with respect to the operator , (respectively weakly) with respect to .

2 ; 1
and U are two control spaces.The systems  1 and  are augmented with the output equations S