An Output Stabilization Problem of Distributed Linear Systems Approaches and Simulations ()
1. Introduction
One of the most important notions in systems theory is the concept of stability. An equilibrium state is said to be stable if the system remains close to this state for small disturbances; and for an unstable system the question is how to stabilize it by a feedback control.
For finite dimensional systems, the problem of stabilization was considered in many works and various results have been developed [1]. In the infinite dimensional case, the problem has been treated in Balakrishnan [2], Curtain and Zwart [3], Pritchard and Zabczyk [4], Kato [5], Triggiani [6]. Many approaches have been considered to characterize different kinds of stabilization for linear distributed systems: Lyapunov and Riccati equation for exponential stabilization, and dissipative type criterion for the case of strong stabilization [3-5,7]. The problem has been also treated by means of specific state space decomposition [6]. The above results concern the state, but in many real problem the stabilization is considered for the state gradient of the considered system, which means to find a feedback control such that the gradient
, when ![](https://www.scirp.org/html/6-7900146\e05a5173-0b9d-4246-a223-e722aef7a0d7.jpg)
For example the problem of thermal insulation where the purpose is to keep a constant temperature of the system with regards to the outside environment assumed to be with fluctuating temperature. Thus one has to regulate the system temperature in order to vanish the exchange thermal flux. This is the case inside a car where one has to change the level of the internal air conditioning with respect to the external temperature.
As we cannot always have external measurements, we use a sensor to measure the flux, which is a transducer producing a signal that is proportional to the local heat flux.
The purpose of this paper is the study of gradient stabilization. It is organized as follows: In the second section we define and characterize gradient stability. In the third section, we characterize gradient stabilizability, by finding a control that stabilizes the gradient of a linear distributed system and we give characterizations of such a control. In the fourth section we search the minimal cost control that stabilizes the system gradient. In the last section we give an algorithmic approach for control implementation and simulation examples.
2. Gradient Stability
This section is devoted to some preliminaries concerning definition and characterization of gradient stability for linear distributed systems.
2.1. Notations and Definitions
Let
be an open regular subset of
and let us consider the state-space system
(1)
where
is a linear operator generating a strongly continuous semigroup
,
, on the state space H which is continuously embedded in
.
H is endowed with its a complex inner product
. and the corresponding norm
.
We define the operator
by:
(2)
is endowed with its usual complex inner product
and the corresponding norm
where:
(3)
With
and
where
The mild solution of (1) is given by
.
Let
denote the adjoint operator of
, and we define the operator
which a bounded operator applying H into itself.
Definition 2.1
The system (1) is said to be
• Gradient weakly stable (g.w.s) if
, the corresponding solution
of (1) satisfies
![](https://www.scirp.org/html/6-7900146\31e1a519-3c96-4de9-aa34-ac247b9b4f6a.jpg)
• Gradient strongly stable (g.s.s) if for any initial condition
the corresponding solution
of (1) satisfies:
![](https://www.scirp.org/html/6-7900146\e2db2aae-1f03-4747-928d-0e199696b4ef.jpg)
• Gradient exponentially stable (g.e.s) if there exist M,
such that:
![](https://www.scirp.org/html/6-7900146\3f91abb9-2be5-4fdb-8310-45834569fc2b.jpg)
Remark 2.2
From the above definitions we have:
1) g.e.s
g.s.s
g.w.s.
2) If the system (1) is stable then it also gradient stable.
3) We can find systems gradient stable but not stable. This is illustrated in the following example.
Exemple 2.3
Let
, on
we consider the following system
(4)
Where
and
is the Laplace operator.
The eigenpairs
of A are given by:
![](https://www.scirp.org/html/6-7900146\cae40649-b858-4022-a7f3-f9802e73af4f.jpg)
A generates a strongly continuous semigroup
given by
![](https://www.scirp.org/html/6-7900146\b5cdb84b-7673-40ff-b285-992295af502c.jpg)
then (4) isn’t stable but
![](https://www.scirp.org/html/6-7900146\760d82a1-fad9-4273-96d1-7bb5d0479620.jpg)
Therefore the system (4) is g.e.s.
2.2. Characterizations
The following result links gradient stability of the system (1) to the spectrum properties of the operator A.
Let us consider the sets
![](https://www.scirp.org/html/6-7900146\53a1f2ab-bc2e-45a9-ab98-b8a38476a49d.jpg)
and
![](https://www.scirp.org/html/6-7900146\bfccdb90-df5e-46f2-845c-486433753ab8.jpg)
where
and
are the points spectrum and the kernel of the operator A.
Proposition 2.4
1) If the system (1) is g.w.s then ![](https://www.scirp.org/html/6-7900146\58c88bda-6a8d-4d9d-b716-a6763a26e645.jpg)
2) Assume that the state space H has an orthonormal basis
of eigenfunctions of A, if
and, for some
,
for all
then the system (1) is g.e.s.
Proof
1) Assume that there exists
such that
and there exists
such that
.
For
, the solution of (1) is
, so
![](https://www.scirp.org/html/6-7900146\6838c1ad-6056-4327-bdfa-2e76252d596d.jpg)
hence the system (1) is not g.w.s.
2) For
we have
![](https://www.scirp.org/html/6-7900146\ab5cdb5c-01e8-4bee-ba9e-d88a5fe9dac8.jpg)
where
is the multiplicity of the eigenvalue
gives:
for some ![](https://www.scirp.org/html/6-7900146\549c1071-8f08-400d-a816-5944c0fd75e0.jpg)
So we have the g.e.s of the system (1).
As example we consider (4). We have:
and
,
, then the system (4) is g.e.s.
For the gradient exponential stability, we need the following lemma.
Lemma 2.5
Assume that there exists a function
such that:
(5)
Then the operators
are uniformly bounded.
Proof
Let us show that
. Otherwise there exists a sequence
,
and
such that
is increasing without bound.
Now we have:
![](https://www.scirp.org/html/6-7900146\d6fbb0d4-fb7e-4890-9937-e21df6a459e5.jpg)
and the right-hand side goes to zero when
.
By Fatou’s lemma
when
, almost everywhere
.
Hence for some
we can find a subsequence
such that ![](https://www.scirp.org/html/6-7900146\c78fe2d0-0bea-4ebc-80f0-635c92d66a8a.jpg)
But with (5) we have
![](https://www.scirp.org/html/6-7900146\874a9390-37ac-4c14-bc64-2651fbbc1e1d.jpg)
when
which is a contradiction.
The conclusion follows from the uniform boundedness principle.
Proposition 2.6
Assume that (5) is satisfied and
(6)
Then the system (1) is g.e.s if and only if
![](https://www.scirp.org/html/6-7900146\a5cecb97-7578-494c-872e-88eaa8c130d9.jpg)
Proof
![](https://www.scirp.org/html/6-7900146\628e7584-0619-4a23-8de8-e837f950db9d.jpg)
where
, then
,
for some
hence
![](https://www.scirp.org/html/6-7900146\89070d92-d51a-402f-8e67-efdafd798aec.jpg)
Now we show that
.
Let t1 > 0, and
there exists ![](https://www.scirp.org/html/6-7900146\478c1ce9-0c31-40f6-8470-ff87a1bd75c0.jpg)
such that
for each
then
![](https://www.scirp.org/html/6-7900146\1af20d17-e95c-408f-a8e2-42762abe14c0.jpg)
With (4) we have
![](https://www.scirp.org/html/6-7900146\f8c041fc-d9ac-49b3-a996-ecb6477c9915.jpg)
Therefore
![](https://www.scirp.org/html/6-7900146\af24ee88-6361-4a67-9246-883abdcbb20c.jpg)
then
.
Hence for all
there exists M' such that
![](https://www.scirp.org/html/6-7900146\6ece8611-58f0-4033-ad79-aa67bdab1a92.jpg)
![](https://www.scirp.org/html/6-7900146\65f2b5a4-774c-417a-84b6-35e9200eef9c.jpg)
![](https://www.scirp.org/html/6-7900146\77b8b14a-ff71-4c58-bebd-811dad31f492.jpg)
So the system (1) is g.e.s.
The converse is immediate.
Example 2.7
The system (2) satisfies the conditions (5) and (6). Indeed:
Let
, and
.
We have
, which implies
![](https://www.scirp.org/html/6-7900146\dae5be9f-a3aa-43df-ba38-fad4252a6c8d.jpg)
we can show that ![](https://www.scirp.org/html/6-7900146\ea31741f-ef01-48eb-85d1-d262eea22026.jpg)
We have
.
Therefore the system (4) is g.e.s.
Corollaire 2.8
Under conditions (5) and (6) and assume, in addition, that there exists a self-adjoint positive operator
such that:
(7)
where
is a self-adjoint operator satisfying
(8)
then (1) is g.e.s.
Proof
We define the function
, ![](https://www.scirp.org/html/6-7900146\cb3f3a10-d4e1-4743-8774-759b8975dbff.jpg)
For
we have
and
![](https://www.scirp.org/html/6-7900146\e774377f-6698-4682-b256-2b8f10023114.jpg)
Thus
By (8), we obtain
.
Since
is dense in H we can extended this inequality to all
and the proposition 3.3 gives the conclusion.
For the gradient strong stability we have the following result.
Proposition 2.9
Assume that the equation
![](https://www.scirp.org/html/6-7900146\6ae46954-59c2-4ee7-a596-d1cee810fab6.jpg)
has a self-adjoint positive solution
where
is a self-adjoint operator satisfying (8). Moreover if the following condition holds
(9)
then (1) is g.s.s.
Proof
Let us consider the function:
![](https://www.scirp.org/html/6-7900146\098449a0-2cf6-4ccd-92f8-36162d3ceaac.jpg)
For
, we have
and
![](https://www.scirp.org/html/6-7900146\9d59922f-df61-4238-8279-c58c6ca2dccd.jpg)
we obtain
By (8),
and from (9), we have
![](https://www.scirp.org/html/6-7900146\07291f3b-f40d-4693-9fdd-6085f5455b00.jpg)
Then
![](https://www.scirp.org/html/6-7900146\dc674d65-7889-47f1-83b7-8ef0da9c53b1.jpg)
We deduce
(10)
From the density of
in H, and the continuity of
, (10) is satisfied for all
. This means that the gradient of (1) is strongly stable.
3. Gradient Stabilizability
Let us consider the system
(11)
with the same assumptions on A, and B is a bounded linear operator mapping U, the space of controls (assumed to be Hilbert space), into H.
Definition 3.1
The system (11) is said to be gradient weakly (respectively strongly, exponentially) stabilizable if there exists a bounded operator
such that the system
(12)
is g.w.s (respectively g.s.s, g.e.s).
Remark 3.2
1) If a system is stabilizable, then it is also gradient stabilizable.
2) Gradient stabilizability is cheaper than state stabilizability. Indeed if we consider the cost functional
![](https://www.scirp.org/html/6-7900146\0699ae8a-13da-4159-b936-8ee60531fd4f.jpg)
and the spaces
![](https://www.scirp.org/html/6-7900146\eb14cca4-a6a5-47d6-b380-1cc11b739851.jpg)
and
![](https://www.scirp.org/html/6-7900146\dd47630a-d7d9-40ac-9290-338c87e7d365.jpg)
Then we have
and therefore
![](https://www.scirp.org/html/6-7900146\6ad19404-964d-473b-9db3-4b5304bd852d.jpg)
3) The gradient stabilization may be seen as a special case of output stabilization with output operator
.
In the following we give the feedback control which stabilizes the gradient of the system (11), by two approaches.
The first is an extension of state space decomposition [6] and the second one is based on algebraic Riccati equation.
3.1. Decomposition Method
Let
be a fixed real and consider the subsets
and
of the spectrum
of A defined by
![](https://www.scirp.org/html/6-7900146\2074ad99-8bc7-4b4e-a551-f935b313c08c.jpg)
and
![](https://www.scirp.org/html/6-7900146\498555e3-9b65-4029-bc96-9579647ff68b.jpg)
Assume that
is bounded and is separated from the set
in such a way that a rectifiable simple closed curve can be drawn so as to enclose an open set containing
in its interior and
in its exterior. This is the case, for example, where A is selfadjoint with compact resolvent, there are at most finitely many nonnegative eigenvalues of A and each with finite dimensional eigenspace.
Then the state space H can be decomposed [5] according to:
(13)
with
,
, and
is the projection operator given by ![](https://www.scirp.org/html/6-7900146\628873cb-69e5-40e8-8bba-310c86d6acfe.jpg)
where C is a closed curve surrounding
.
The system (11) may be decomposed into the two subsystems
(14)
and
(15)
where
and
are the restrictions of A to
and
, and are such that
,
, and
is a bounded operator on Hu.
The solutions of (14) and (15) are given by
(16)
And
(17)
where
and
denote the restriction of
to
and
, which are strongly continuous semigroups generated by
and
.
For the system state, it is known (see [6]) that if the operator
satisfies the spectrum growth assumption
(18)
then stabilizing (11) comes back to stabilize (14).
The following proposition gives an extension of this result to the gradient case.
Proposition 3.3
Let the state space satisfy the decomposition (13) and
satisfy the following inequality
(19)
1) If the system (14) is gradient exponentially (respectively strongly) stabilizable by a feedback control
, with
, then the system (11) is gradient exponentially (respectively strongly) stabilizable using the control
.
2) If the system (14) is gradient exponentially (resp strongly) stabilizable by the feedback control:
with
then the system (11) is gradient exponentially (respectively strongly) stabilizable using the feedback operator
.
Proof
We give the proof for the exponential case. In view of the above decomposition, we have:
.
Hence if
satisfies (19) then for some
and
, we have:
,
.
It follows that the system (15) is gradient exponentially stabilizable taking v(t) = 0.
Let
be such that
, with
and there exists
, M2 > 0 such that ![](https://www.scirp.org/html/6-7900146\e28a1235-4390-4e52-8522-a90ddc7e409a.jpg)
Then with the feedback control
we have
, with ![](https://www.scirp.org/html/6-7900146\557ee227-17fa-4140-a472-b99390e8c97f.jpg)
From (17) and (18) we have
![](https://www.scirp.org/html/6-7900146\a4658b6e-ae6f-421c-8615-8deb6b083143.jpg)
with
.
Thus the system (11) excited by
satisfies
![](https://www.scirp.org/html/6-7900146\1b2af70e-633d-4914-ac79-25046176cf6c.jpg)
which shows that the system (11) is gradient exponentially stabilizable.
2) The case of strong stabilizability follows from similar above techniques.
Corollary 3.4
Let A satisfy the spectrum decomposition assumption (13) and suppose that (19) is satisfied. If in addition 1)
is a finite dimensional space 2) The system (14) is controllable on
then the system (11) is gradient exponentially stabilizable.
Proof
The system (14) is of finite dimension and is controllable on the space
then it is stabilizable on the same space, hence it is gradient stabilizable, the conclusion is obtained with the proposition 3.3.
3.2. Riccati Method
Let us consider the system (11) with the same assumptions on A and B. We denote by
,
the strongly continuous semigroup generated by
, where K is the feedback operator
.
Let
be a self-adjoint operator such that (8) is satisfied and suppose that the steady state Riccati equation
(20)
has a self-adjoint positive solution
, and let
.
Proposition 3.5
1) If
satisfies the conditions (5) and (6), then the system (11) is gradient exponentially stabilizable by the control ![](https://www.scirp.org/html/6-7900146\11379889-24a2-425a-8f20-a1b08c61dfb0.jpg)
2) If
then the system (11) is gradient strongly stabilizable.
3) Suppose that the system (11) is gradient exponentially stabilizable. If in addition the feedback operator K satisfies
for some c > 0 then the state of the system (12) remains bounded.
Proof
The first and second points are deduced from the second section.
For the thirst point: Let
we have
(21)
and from (21) we obtain
![](https://www.scirp.org/html/6-7900146\b34d0e71-9186-4745-8c1c-d9762f68037d.jpg)
Since the system (11) is gradient exponentially stabilizable then
, so there exists ![](https://www.scirp.org/html/6-7900146\50688203-560f-495b-864c-9321e1f9b127.jpg)
such that
for all
and by the density of
in H we have the conclusion.
3.3. Gradient Stabilization Control Problem
Here we explore the control that stabilizes the gradient of the system (11) as a solution of the minimization problem
(22)
where
with
![](https://www.scirp.org/html/6-7900146\e3bcf776-bda2-40d3-91a9-86d994d2cc96.jpg)
and R is a linear bounded operator mapping H into itself and satisfying (8).
We recall the classical result known for state stabilization if
for each initial state
then there exists a unique control
that minimizes (22) and given by
where P is a positive solution of the steady state Riccati Equation (20).
If in addition the operator R is coercive then the state of system (11) is exponentially stabilizable (see [7]).
In the following we give an extension of the above result to the gradient case.
We suppose that
for each initial state
, and R satisfies (8).
Proposition 3.6
The control given by
minimizes
where P assumed to be a self-adjoint, positive operator, and satisfies the steady state Riccati equation (20), if in addition the semigroup
satisfies the conditions (5) and (6) then the same control stabilizes the gradient of system (11)
Proof
The proof follows from [7], and the proposition 3.5.
4. Numerical Algorithm and Simulations
In this section we present an algorithm which allows the calculation of the solution of problem (22) stabilizing the gradient of the system (11). By the previous result this control may be obtained by solving the algebraic Riccati Equation (20). Let
where
is a hilbertian basis of H.
is a subspace of H endowed with the restriction of the inner product of H. The projection operator
is defined by
![](https://www.scirp.org/html/6-7900146\ac8a1b13-b1e1-4be1-b31b-562c00a5690d.jpg)
The projection of (20) on space
, is given formally by:
(23)
where
,
and
are respectively the projections of A, P and R on
, and
the projection of B which is mapping U the space of control into
.
We have
, that is
converges to P strongly in H, (see [8]).
We can write the projection of (11) like
(24)
the solution of this system is given explicitly by:
(25)
To calculate the matrix exponential we use the Padé approximation with scaling and squaring (see [9]).
If we denote
, we have
(26)
Let consider a time sequence
,
where
small enough.
With these notations, the gradient stabilization control may be obtained the algorithm steps (Table 1).
Remark 4.1
The dimension of the projection space n is choosing to be good approximation of the considered system and appropriate for numerical constraint.
Example 4.2
Let
, on
![](https://www.scirp.org/html/6-7900146\205c6bc3-aa70-4758-b977-f9e1a05f7f30.jpg)
which is an Hilbert space we consider the following system
(27)
where
, ![](https://www.scirp.org/html/6-7900146\98de17b8-e436-4766-9e97-5acd824115ef.jpg)
,
is the restriction operator on
, and we consider the problem (22) with
.
A generates a strongly continuous semigroup ![](https://www.scirp.org/html/6-7900146\8fb8ba27-ef8f-449c-82c6-16f34841239a.jpg)
given by:
, where
and
with
.
The state and the gradient of system (27) are unstable since
.
Let consider the subspace
![](https://www.scirp.org/html/6-7900146\330f6aa2-be45-467d-b273-66f553774d54.jpg)
Applying the algorithm taking the truncation at n = 5, we obtain figures 1 which illustrates the evolution of the system gradient and shows how the gradient evolves close to zero when the time t increases.
The gradient is stabilized with error equals 9.9836 × 10–7 and cost equals 2.6982 × 10–4. This shows the efficiency of the developed algorithm.
In table 2 we give the cost of gradient stabilization of system (27) for different supports control “D”.
The Table 2 shows that there is relation between area of control support and the cost of gradient stabilization, more precisely more this area decreases more cost in-
![](https://www.scirp.org/html/6-7900146\3d514fe5-9fc3-4709-b01b-1d0aed0a8bf6.jpg)
Table 2. Support control-cost stabilization.
Figure 1. The gradient evolution for the Neumann boundary condition case.
creases.
Example 4.3
Let
on
we consider the system (27) with Dirichlet boundary conditions:
(28)
where
,
,
, and we consider the problem (22) with
.
The eigenpairs of A are given by
,
, with
.
The state and the gradient of system (28) are unstable since
.
We consider the subspace
with
.
Applying the algorithm with truncation (n = 5), the figure 2 shows the gradient evolution at times t = 3, 5, and 13.
In table 3 we present the cost of gradient stabilization
Figure 2. The gradient evolution for the Direchlet boundary condition case.
![](https://www.scirp.org/html/6-7900146\557d5a2c-b331-4a41-bb66-96ebe69bae78.jpg)
Table 3. Support control-cost stabilization.
of system (28) for different zone control support “D”.
Also in this example, we remark that more the area of control support increases more the cost of gradient stabilization decreases.
4. Conclusions
In this paper the question of gradient stabilization is explored. According to the conditions, satisfied by the dynamic of system, and those satisfied by the state space, two methods are applied to characterize the controls of gradient stabilization namely, decomposition approach and Riccati method.
The obtained results are successfully illustrated by numerical simulations. Questions are still open, this is the case of regional gradient stabilization. It is under consideration and will be appear in separate paper.
5. Acknowledgements
The work presented here was carried out within the support of the Academy Hassan II of Sciences and Technology.