Regional Controllability of Semi-Linear Distributed Parabolic Systems: Theory and Simulation ()
1. Introduction
Many scientific and engineering problems can be modeled by partial differential equations, integral equations, or coupled ordinary and partial differential equations that can be described as differential equations in infinite-dimensional spaces using semi groups. Nonlinear integrodifferential equations, with and without delays, serve as an abstract formulation for many partial integrodifferential equations which arise in problems connected with heat flow in materials with memory, viscoelasticity, and other physical phenomena. In particular, Sobolev-type equations occur in thermodynamics in the flow of fluid through fissured rocks, in the shear of second-order fluids, and in soil mechanics. So, the study of controllability results for such systems in infinite-dimensional spaces is important.
For the motivation of abstract systems and the controllability of linear systems, one can refer to the books by Curtain and Pritchard [1], and by Curtain and Zwart [2]. For an earlier survey on the controllability of nonlinear systems using fixed-point theorems, including nonlinear delays systems, see [3]. The approximate controllability of nonlinear systems when the semigroup generated by A is compact has been studied also by many authors. The results of Zhou [4] and Naito [5] give sufficient conditions on B with infinite-dimensional range or necessary and sufficient conditions based on more strict assumptions on B. Li and Yong [6] studied the same problem assuming the approximate controllability of the associated linear system under arbitrary perturbation in 
Bian [7] investigated the approximate controllability for a class of semi-linear systems, [8] used the Banach fixed-point theorem to obtain a local exact controllability in the case of nonlinearities with small Lipschitz constants. Zhang [9] studied the local exact controllability of semi-linear evolutions systems. Naito [5] and Seidmann [10] used the Schauder fixed-point theorem to prove the invariance of the reachable set under nonlinear perturbations. Klamka [11-13] studied sufficient conditions for constrained exact controllability in a prescribed time interval for semi-linear dynamical systems in which the nonlinear term is continuously Frechet differentiable are formulated and proved assuming that the controls take values in a convex and closed cone with vertex at zero. The method used covers a wide class of semi-linear abstract dynamical systems and is specially useful for semi-linear ones with delays. Balachandran and Sakthivel [14] studied the controllability of semilinear integrodifferential systems in Banach spaces by using the Schaefer fixed-point theorem. Fabre et al. [15] prove approximate controllability in 
for 1 ≤ p < 
by means of a control which can be internal or on the boundary and when the nonlinearity is globally Lipschitz. Other related abstract results were given by Zuazua [16], Lasiecka et al. [17] and Kassara et al. [18].
The study of various analytic concepts related to controllability and stability of such systems is, in general, delicate and considering only linear model can not be sufficient in particular when some properties of system needs to be satisfied only in some part of the system evolution domain. From practical point of view, it is very natural to consider the analysis of such systems only in some subregion of its evolution system domain. This is the aim of regional analysis.
The regional analysis of distributed parameter system has recieved an intensive study in the last three decades.
The term “regional analysis” has been used to refer to control problems in which the target of our interest is not fully specified as a state, but refers only to a smaller region 
of the system domain. This concept has been widely developed and interesting results have been obtained, in particular, the possibility to reach a state only on an internal subregion 
of 
(El Jai et al. [19]) or on a part of the boundary 
of (Zerrik et al. [20]). The principal reason for introducing this concept is that, first it makes sense for the usual controllability concept closer to real world problem and, second, it can be applied to systems which are not controllable on the whole domain. Here we are interested on regional controllability of semi-linear parabolic systems. More precisely the question concerns the possibility of regional controllability for semi-linear system in the case where the desired state is given only on an internal subregion 
of 
or on a part of the boundary 
of
.
The interest of this work focused on the development of an approach that leads to numerical implementation for the computation of the control which steers the system from an initial state to a given regional internal and boundary state. A typical motivating example is the case of a biological reactor, where the problem is to regulate the concentration of a susbstratum at the bottom of the reactor [21].
In Section 2, first we present some preliminary material and state internal regional controllability problem of semi-linear systems. Next, we concentrate on the determination of a control achieving regional internal controllability, and we develop a numerical approach that leads to a useful algorithm and successfully tested through a diffusion process. Section 3 is focused on the regional boundary target control problem, and an approach is developed that leads to a numerical algorithm for the computation of a control which achieves regional boundary controllability. Numerical illustrations show the efficiency of the approach and lead to conjectures.
2. Regional Internal Controllability
2.1. Statement of the Problem
Let 
be a regular bounded open set of IRn, 
with boundary
. For a given time T > 0, let
, and 
We consider a semi-linear parabolic system excited by controls which can be applied via various types of actuators given by the following equation
(2.1)
where A is a second-order linear differential operator, which generates a strongly continuous semi-group 
on Hilbert space 
and N a locally lipschitz continuous nonlinear operator.
, 
and 
where
where p represents the number of actuators. We denote by U the completion of the space 
endowed with the standard norm of
. Denote by 
the solution of (2.1) when it is excited by a control u, suppose that
. Let us recall that an actuator is conventionally defined by a couple 
where 
is the geometric support of the actuator and f is the spatial distribution of the action on the support D, see [22]. In this case, 
In the case of pointwise actuator (internal or boundary), 
and
, where 
is the Dirac mass concentrated at 
in this case, the actuator is denoted by 
and
.
Let 
be the solution of (2.1) excited a control u and assume that 
see [23].
For
, open, nonempty and of positive Lebesgue measure, we consider the operator restriction
and 
denotes the adjoint operator.
Definition 1
• System (2.1) is said to be 
-exactly regionally controllable if for all 
there exists a control 
such that 
• System (2.1) is said to be 
-approximately regionally controllable if for all 
and for all
, there exists a control 
such that
The notion of regional controllability considered as a particular case of output controllability was introduced and developed for linear system in (El Jai et al. [19], Zerrik et al. [20]).
It is clear that:
• If system (2.1) is regionally controllable on 
then it is regionally controllable on any 
• In the linear case, one can find states which are approximately regionally controllable on 
but not controllable on the whole domain
, see [19,22].
To study the controllability of the system (2.1), we consider its corresponding linear system 
(2.2)
The problem of regional controllability on 
for (2.1) can be stated as follows:
Problem
(2.3)
More precisely, it is asked to find a control which steers system (2.1), at time T, to a desired state defined in subregion
.
2.2. Hilbert Uniqueness Approach
The aim of this section is to give an extension of regional controllability and Hilbert uniqueness method introduced in the linear case by (El Jai et al. [19]) and [24] which allows the characterization of a control 
solution of (2.3). The system (2.2) is approximately controllable in 
and system (2.1) is excited by a zone actuator
. System (2.1) may be rewritten in the form
(2.4)
and the operator 
verify
(2.5)
Let 
(2.6)
Which has a unique solution 
see [25].
For a given
, we consider the system (2.6) and define the mapping
(2.7)
Which is a norm on G; since the system (2.2) is approximately controllable in
.
Consider the system
(2.8)
and the associated linear system
(2.9)
The system (2.8) may be decomposed in the following three systems
(2.10)
and
(2.11)
where 
is the solution of (2.6) and
(2.12)
We denote the completion of the set G with respect to the norm (2.7) again by G.
Let 
be defined by 
where
.
Now, with the nonlinear operator
The problem of regional controllability (2.3) turns up to solve the equation 
which is equivalent to
where 
is the operator defined by the formula
. Then we have
(2.13)
The linear system (2.9) is approximately regionally controllable in
, then 
is one to one see [19].
Apply 
the equation (2.13), we have
Now, we define the nonlinear operator 
by
(2.14)
Then the problem (2.3) of system (2.7) turns up to search a fixed point of
, then we have
Proposition 1
Assume that (2.5) holds. If the linear system (2.9) is approximately regionally controllable in
, then the control 
steers the system (2.8) to 
in 
at 
where 
is the solution of the system (2.6) and 
is a fixed point of the operator 
given by (2.14).
Sketch of the proof: The proof may be easily achieved with the following two steps:
Step 1: We prove that K is a compact operator and then deduce that 
is also compact.
Step 2: Applying the Schauder fixed-point theorem, we see that the operator 
has a fixed point. For more details we refer the reader to [26].
Remark 1
The above approach is a generalization of the Hilbert uniqueness method given in the linear case 
and the operator 
coincides with the isomorphism
.
Algorithm 1
Summing up, in the zone case, the regional controllability is obtained via the following simplified algorithm
• Step 1: we take the following initial conditions
, 
, f, D and
.
• Step 2: Using the pseudo-code.
■ Resolution of (2.6) and obtaining 
■ Resolution of (2.10) and obtaining 
■ Resolution of (2.11) and obtaining 
■ Resolution of (2.12) and obtaining 
■ Calculation of 
and obtaining
.
■ Resolution of 
and obtaining
.
■ Until 
• Step 3: The control
.
2.3. Simulations
The goal of this section is to test the efficiency of the previous algorithm. The obtained results are related to the considered subregion, the desired state and the actuator structure. Let 
and consider the one-dimensional diffusion system described by
(2.15)
2.3.1. Zone Actuator
In this case 
where 
The subregion under consideration is
.
Let 
be the desired regional state in
. Using the previous algorithm1, the simulation gives the Figure 1.
The regional desired state 
is reached with error
and transfer cost
.
2.3.2. Pointwise Actuator
In this case 
where 
We consider the subregion
.
Let 
be the desired regional state on
. The simulation gives the Figure 2.
The regional desired state 
is reached with error
and transfer cost
.
2.3.3. Relation between the Subregion and Location of the Pointwise Actuator
The following simulation results show the evolution of the desired state error with respect to the actuator location. Figure 3 shows that:
• For a given subregion 
and a desired state, there is an optimal actuator location (optimal in the sense that it leads to a solution which is very close to the desired state).
• When the actuator is located sufficiently far from the subregion
, the estimated state error is constant for any location.
• The worst locations correspond to non strategic actuators in
, as developed in the linear case see [19].
Figure 4 shows that, for a given subregion and a desired state, there is an optimal actuator location in the sense that it leads to a smaller transfer cost.
The results are similar for other types of actuators.
3. Regional Boundary Controllability
The aim of this section is to give an extension of the concepts of regional internal controllability [26] to the case where is a part of the boundary of the domain. The developed 
method is original and leads to a numerical algorithm illustrated by simulations.
3.1. Considered System and Problem Statement
Let 
be a bounded open domain in IRn (n = 1, 2, 3) with a regular boundary
. For
, we write
Figure 1. Desired state (continuous line) and final state (dashed line) on the region ω.
Figure 2. Desired state (continuous line) and final state (dashed line) on the region ω.
Figure 3. The evolution of the estimated state error with respect to the actuator locations.
Figure 4. The evolution of the transfer cost with respect to the actuator locations.
, 
and consider the following semi-linear parabolic system
(3.1)
where
• A is a second-order linear differential operator, which generates a strongly continuous semi-group 
on Hilbert space
.
• N a locally lipschitz continuous nonlinear operator.
• 
• 
where 
with 
be the solution of (3.1) excited by a control u.
We denote by U the completion of the space 
endowed with the standard norm of
.
Assume that 
The controls may be applied via various types of actuators see [22].
The associated linear system is
(3.2)
For Γ being a regular subset of 
which has positive Lebesgue measure, consider the restriction operator
where 
denotes its adjoint operator.
Let us 
whilst 
is considered for the adjoint operator.
We introduce the definition.
Definition 2
The system (3.1) is said to be 
-exactly (resp. 
- approximately) regionally controllable if for all
(resp. for all
) there exists a control
such that 
(resp. 
).
This definition generalizes the standard ones of exact and approximate controllability on the whole domain
.
Remark 2
1) The notion of regional controllability considered as a particular case of output controllability was introduced and developed for linear system in [20].
2) A system which is 
-exactly (resp. 
-approximately) regionally controllable is 
-exactly (resp. 
- approximately) regionally controllable for all
.
3) The above definitions do not allow for pointwise or boundary controls since, for such.
4) systems 
and the solution 
However, the extension can be carried out in a similar manner if one takes regular controls such that 
[27].
In the sequel, we explore the possibility of finding a control which ensues the transfer of system (3.1) to desired 
on the boundary subregion 
consider the problem
(3.3)
3.2. Theoretical Approach
Firstly, the following result provides a link between regional internal controllability see [26] and regional boundary controllability for semi-linear systems.
Consider the linear and continuous extension operator
such that 
for all 
For
, we denote by 
the extension of 
to 
and we define
Let
integer small, we set 
and 
, where 
is the open ball of radius r and center z, see [28]. Then, we have the following result.
Proposition 2
If the system (3.1) is 
-exactly (resp. 
-approximately) regionally controllable, then it is 
-exactly (resp. 
-approximately) regionally controllable.
Proof
Let 
then by trace theorem, there exists 
with a bounded support such that
Since the system (3.1) is 
-exactly controllable, then there exists a control 
such that
Thus 
and then 
Consequently, the system (3.1) is 
-exactly controllable.
Now, if the system (3.1) is 
-approximately controllable, for all
, there exists 
such that
and by continuity of the trace mapping
, we have
therefore
Consequently, the system (3.1) is 
-approximately controllable.
Secondly, we develop an approach devoted to characterize a control 
solution of problem (3.3), when the system (3.1) is 
-approximately controllable. The approach we shall use is based on an extension of regional controllability techniques for linear systems developed in (El Jai et al. [19]) and Hilbert uniqueness method see [24].
The system (3.2) is excited by a control applied by means of a zone actuator 
where 
is the actuator support and 
defines the spatial distribution of the control on D, then the system (3.2) may be written in the form
(3.4)
The operator 
verify
(3.5)
Let G be the set
For
, we denote by 
the extension of 
to 
Consider the system
(3.6)
where 
is the Laplace operator. The system (3.6) has a unique solution z in
. Let 
the restriction of z in 
The problem of reaching 
on 
may then be solved by reaching 
on 
Then the problem (3.3) is formulated as follows:
(3.7)
For
, the system
(3.8)
has a unique solution 
[28].
In G, we define the mapping
(3.9)
which is a norm on G; since the system is 
-approximately controllable see [19].
Consider the system
(3.10)
and its associated linear system is
(3.11)
The system (3.10) may be decomposed into the following three systems
(3.12)
and
(3.13)
where 
is the solution of (3.8) and
(3.14)
We denote the completion of the set G with respect to the norm (3.9) again by G.
Consider the operator 
defined by
where 
is the dual of 
and 
Let us now define the nonlinear operator
The problem of regional controllability (3.3) turns up to solve the equation
which is equivalent to
where 
is the operator defined by 
, which gives
(3.15)
Since the linear system (3.11) is 
-approximately regionally controllable, then 
is one to one see (El Jai et al. [19]).
Apply 
the equation (3.15), we have
Then a solution of problem (3.3) of system (3.10) turns up to search a fixed point of nonlinear operator 
define by
(3.16)
Then we have:
Proposition 3
If the linear system (3.11) is 
-approximately regionally controllable, then the control
drives the system (3.10) to 
in 
at
, where 
is the solution of the system (3.9) and 
is a fixed point of the operator 
given by (3.16).
Proof
Step 1: We prove that 
is a compact operator.
Let the ball 
in X, we have
and we set
Where 
is solution of the system (3.14).
We have
(3.17)
see [23] and there exists 
such that
Since 
is a strongly continuous semi-group on
, then there exists 
such that
and from (3.17), we have
Since 
is solution of the system (3.12), then 
and we have
(3.18)
Since 
is solution of the system (3.13), then we have
and
then
(3.19)
thus
By Gronwall’s lemma, we obtain
(3.20)
then
Hence, 
is uniformly bounded.
Let show that 
is relatively compact, indeed: for 
and
, we have
where
and
For all
, there exists 
such that 
which gives
from (3.18), (3.19) and (3.20), we have
and
Thus
where
and
For 
and
, we obtain 
Then, 
is relatively compact.
Finally, by the Arzelà-Ascoli theorem see [29,30], 
is a compact operator, then 
is also compact.
Step 2: From (3.16) and (3.20), we have
where
and
The constant 
verify
It is being used the fact that 
is small.
Let 
such that
, then we have
such that
.
Hence, by applying Schauder’s fixed point theorem see [30], the operator 
at least one fixed point, and the proof is completed.
Algorithm 2
With the same hypothesis as in the last section, we have the following algorithm
• Step 1: we choose the initial conditions, subregion
, 
, 
, and the function
, D and
.
• Step 2: using the pseudo-code.
■ Resolution of (3.8) and obtaining 
■ Resolution of (3.12), (3.13) and (3.14)
■ Calculation of 
and obtaining
.
■ Resolution of 
and obtaining
.
■ Until 
• Step 3: The control
.
3.3. Numerical Example
In this subsection, we present a numerical example which illustrate the previous algorithm. It shows that there exists a link between the subregion area and the reached state error, the results are related to the choice of the subregion and the desired state to be reached.
Consider the two-dimensional diffusion system
(3.21)
3.3.1. Zone Actuator
We consider
• The actuator is located in
.
• 
,
: Intern subregion target.
• 
: Boundary subregion target.
• 
: The desired state to be reached in 
• 
: The extension of desired state 
on 
• Using the previous algorithm 2 in the case zone actuator we have Figures 5-8.
Using the previous algorithm, the regional desired state 
is obtained with error
and cost 
Figure 5. Desired state on the region ωr.
Figure 7. Desired and final state on the region ωr.
Figure 8. Trace of desired and final state on the region Γ.
3.3.2. Relation between the Subregion Area and Reached State Error
The reached state error depends on the area of the subregion where the desired state has to be given. This error grows with the subregion area. It means that the larger the region is, the greater the error is (see Table 1).
The results are similar for other types of actuator.
3.3.3. Pointwise Actuator
In this case, we have
• The actuator is located in 
with b1 = 0.162, 
• 
, 
Intern subregion target.
• 
: Boundary subregion target.
• 
: The desired state to be reached in 
• 
: The extension of desired state 
on 
• Using the previous algorithm 2 in the case point wise actuator we have Figures 9-12.
Table 1. The relation between the subregion area and reached state error.
Figure 9. Desired state on the region ωr.
Figure 10. Final state on the region ωr.
Figure 11. Desired and final state on the region ωr.
Figure 12. Trace of desired and final state on the region Γ.
4. Conclusions
The work is provide an interesting tool to achieve regional internal and boundary target for a semi-linear parabolic system excited by actuator. The problems of regional controllability are solved using linear regional controllability techniques and by applying HUM method and fixed point theorems. The obtained result leads to an algorithm which was implemented numerically. Examples of various situations and simulations are given.
Various open questions are still under consideration. For example, this is the case of the problem where we test this algorithm for real applications. This case is presently being studied and the results will appear in a separate paper.
The problem of regional controllability problem for semi-linear parabolic systems with time delays is of great interest and the work is under consideration and will be the subject of the feature paper.