2416 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 9, SEPTEMBER 2013
Boundary Stabilization of EquilibriumSolutions to Parabolic Equations
Viorel Barbu
Abstract—In this note, a stabilizing feedback boundary controller for theequilibrium solutions to parabolic equations with Dirichlet boundary con-trol is designed. The feedback controller is expressed in terms of the eigen-functions corresponding to unstable eigenvalues of the lin-earized equation. For , this stabilizing procedure is applicable for
only, while for it is of conditional nature requiring theindependence of on the part of the boundary where the controlis applied.
Index Terms— Dirichlet boundary, Neumann boundary.
I. INTRODUCTION
The existence of a stabilizing linear boundary feedback controllerfor the linear parabolic equations with Dirichlet or Neumann boundaryconditions was established long time ago by R. Triggiani in his pio-neering work [14]. In a few words, the idea was to decompose thecontrol system in a finite dimensional unstable part corresponding tounstable eigenvalues and an infinite dimensional stable system. Af-terwards, one shows via Kalman’s theory that the finite dimensionalcontrol system is exactly controllable, which implies by the standardtechniques that there is a stabilizing controller which leads to the de-sign of a feedback controller via Riccati equation. Such an approach,which was used later on to Navier–Stokes equations (see [5], [6]), pro-vides a robust linear feedback controller, but has the disadvantage thatthe feedback synthesis requires a big amount of computation involvinghigh or infinite dimensional Riccati equations. The approach developedin [14] has shown that the boundary stabilization is possible via a feed-back controller with finite dimensional structure and provides a con-ceptual procedure to get it, though not in an explicit form. Here, wedevelop a new technique for the construction of a stabilizable feedbackboundary controller for parabolic equations which was previously usedin the author works [1], [2] for the Navier–Stokes equations. The stabi-lizing controller designed here cannot be derived from [14], though itis constructed on similar spectral principles: the separation of unstableand stable modes. It should be said that a similar direct constructionwas used recently by Fursikov and Gorshkov in [9] (see also [8]) forboundary stabilization of Navier–Stokes equations via start control sta-bilization. However, there is no overlap with the present work. Themain advantage of the approach we propose here is that it providesa simple feedback stabilizer expressed in terms of unstable system ofeigenfunctions for the linear system which is easily imple-mentable. For the heat equation on a finite rod, a simple andefficient method for boundary stabilization was developed in the lastdecade: the backstepping approach. The main references here are theworks of Boškovic et al., [4], Balogh and Krstic [3], Liu [11], andSmyshlyaev and Krstic [13]. The stabilization method we present here
Manuscript received August 24, 2012; revised October 31, 2012 and January17, 2013; accepted February 28, 2013. Date of publication March 21, 2013; dateof current version August 15, 2013. This work was supported by a grant of theRomanian National Authority for Scientific Research ID PN-CT-1 ERC/02.07.2012. Recommended by Associate Editor C. Prieur.The author is with the Octav Mayer Institute of Mathematics (Roma-
nian Academy) and Al.I. Cuza University, Iaşi 700056, Romania (e-mail:[email protected]).Digital Object Identifier 10.1109/TAC.2013.2254013
is conceptually different from the backstepping approach and it worksin all dimensions without any restriction (for ) on the instabilitylevel of the equation, provided the system is linearlyindependent on the part of the boundary where the Dirichlet control isapplied. However, for , our method is applicable only for low andmoderate levels of instability (see Example 1), while the last achieve-ment on the backstepping method ([3], [11], [13]) is applicable for anylevel of instability of heat equation.
II. PROBLEM STATEMENT
Our aim here is to stabilize the equilibrium solutions to the para-bolic equation
(2.1)
where is a bounded and open domain of with a smooth boundarybeing connected parts of .
The Dirichlet controller is applied on while is insulated.Here, is the normal derivative and is any solutionto the equation
Translating into zero via substitution we can rewrite(2.1) as
(2.2)
and the stabilization problem reduces to design a feedback controllersuch that the solution to the corresponding closed loop
system satisfies, for some
(2.3)
for all in a -neighborhood of the origin.The first step toward this goal is the stabilization of the linearized
system associated with (2.2), that is
(2.4)
where . The stabilizing feedback con-troller for (2.4) will be used afterwards to stabilize locallysystem (2.2), and implicitly the equilibrium solution .Notation: Denote by the space of Lebesgue square inte-
grable functions on with the norm and thescalar product . Similarly, is the space ofLebesgue square integrable functions on with the scalar product
. By with the norm we denote thestandard Sobolev spaces on . By we denote the space ofall continuous functions on .
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III. STABILIZATION OF THE LINEAR SYSTEM (2.4)
Everywhere in the following, we shall assume that(i) .In particular, this implies that is continuous on .We consider the linear self-adjoint operator in
The operator has a countable set of real eigenvalues with cor-responding eigenfunctions , that is, , .Each eigenvalue is repeated here according to its multiplicity andlet be such that for ; . (Sincethe resolvent of is compact and , it isclear that is finite).Denote by the normal derivative of to .By the unique continuation property of eigenfunctions , we know
that, for all , on . In the following, we shall assumethat(ii) The system is linearly independent on
.It should be mentioned that (ii) is a standard hypothesis in boundary
stabilization theory of parabolic-like equations (see e.g., [1], [2], [14]).We note that (ii) always holds if and, for , only in thiscase. For , there are, however, significant situations where (ii)holds (see Example 2 below).Consider the feedback controller
(3.1)
where are parameters to be made precise later on, and
(3.2)
(3.3)
(3.4)
We note that, by assumption (ii), the Gram matrixis nonsingular. Hence,
and are well defined.Theorem 3.1: Let and be positive and sufficiently large such that
(3.5)
Then, the feedback controller (3.1) stabilizes exponentially system(2.4). More precisely, the solution to the closed loop system
(3.6)
satisfies, for , the estimate
(3.7)
It should be remarked that Theorem 3.1 provides a simple algorithmfor the stabilization of the linear system (2.4) as well as for the non-linear system (2.1) to be discussed later on. Determine first the un-stable eigenvalues and the corresponding eigenfunctionsof the operator and construct afterwards the feedback controller (3.1),where are given (3.2)–(3.5) and satisfy condition (3.5). Ofcourse, in specific situations, the eigenvalues and cannot be com-puted exactly and so, instead of (3.1), we must consider an approxi-mating feedback controller of the form (3.1) corresponding to theapproximations and of and , respectively. However, theapproximating controller is still stabilizing in problem (2.4) by therobustness of the stabilizer controller. (See [1], pp. 64 and 159, for adiscussion on these topics).Example 1: We consider the stabilization problem treated in [3], [4],
[13]. Namely
(3.8)
with Dirichlet actuation in , where is a constant parameter.The eigenvalues of the operator with the do-
main arewith eigenfunctions . Then,
for , we have and so Theorem 3.1 is applicablewith the feedback controller (3.1) of the form
(3.9)
where is given by (3.2), and . By Theorem 3.1, for, this feedback controller stabilizes
(3.8) with the exponent decay . In [4], astabilizing feedback controller was constructed by the backsteppingmethod for . Later on, this condition was removed in [3],[13] by a sharpening of the method. It should be said, however, thatthe feedback controller (3.9) is simpler than that constructed via thebackstepping method for .Example 2: Consider the boundary stabilization of the heat equation
in
(3.10)
where and . The eigen-values of theoperator with the domain
are given
with the corresponding eigenfunctions
Then, is determined by the condition and, since
2418 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 9, SEPTEMBER 2013
it turns out that assumption (ii) holds on (as a matter of fact oneach ). Hence, Theorem 3.1 is applicable in thepresent situation and so there is a feedback controller of the form(3.1)–(3.3), which stabilizes system (3.10).Remark 3.2: The previous equation might suggest that, for ,
assumption (ii) is always satisfied, but the following example, com-municated us by one of the reviewers, shows that, in general, this isnot true if , where . Take, for instance, (3.10)with , and boundary conditions: on ,
on , where ,. Then, is an
eigenfunction for and for(both unstable eigenvalues). However, as easily seen, ,
are linearly dependent.Remark 3.3: Numerical tests for the computation of the stabilizing
controller (3.1) were performed in [12] for (3.10) on, and , 7. Also, the case ,, was numerically tested in [12].
Proof of Theorem 3.1: Consider the map defined by
(3.11)
For sufficiently large, the Dirichlet map is well defined and(see, e.g., [10]). Moreover, we have
(3.12)
Since commutes with the operator , we may rewrite (2.4) interms of and as
Equivalently
(3.13)
where . Moreover, for later purpose, it is convenient toexpress the feedback controller (3.1) in terms of as
(3.14)
Indeed, taking in (3.14), we obtain that
(3.15)
where is the adjoint of .Now, multiplying (3.11), where , by and recalling that
, we obtain via Green’s formula that
and so, by (3.2) and (3.4), we have
This yields
(3.16)
and, substituting into (3.15), we obtain (3.1), as claimed.Substituting (3.14) into (3.1), we obtain
(3.17)
Let , the algebric projection ofon and set , . Then, we may decomposesystem (3.17) as follows:
(3.18)
where , and .If we represent as , we see by (3.16) that
or, equivalently
(3.19)
and, by condition (3.5), we have that
(3.20)
On the other hand, taking into account that the spectrum of ,, we have
because the -semigroup generated by is analytic in .Taking into account the second equation in (3.18) and (3.20), we obtainthat
(3.21)
where . Keeping in mind that , by(3.21) and (3.14) we see that (3.7) holds.Remark 3.4: The above design of a stabilizable feedback controller
applies as well to equation (2.1) with homogeneous Dirichlet conditionon and Dirichlet actuation on , that is, on ; on
or to the Neumann boundary control ;, but we omit the details.
We also note that Theorem 3.1 and the above stabilization construc-tion extend word by word to the controlled parabolic linear equation
where , , .
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IV. STABILIZATION OF SYSTEM (2.1)
We strengthen assumption (i) to
(4.1)
where for , 2, for .If is an equilibrium solution to (2.1), we consider the
feedback controller
(4.2)
where are given by (3.2)–(3.4).Theorem 4.1: Let . Then, under assumptions (i), (ii),
(4.1) and (3.5), the feedback controller (4.2) stabilizes exponentiallythe solutions to system (2.1). More precisely, the solution to theclosed loop system
(4.3)
satisfies for and sufficiently small, the estimate
(4.4)
where .Proof: By substitution , we reduce the problem to
the stability of the null solution to system (2.2) with the boundary con-troller (3.1), that is
(4.5)
where
Arguing as in the previous case, we write (4.5) as
(4.6)
and setting , we obtain that
(4.7)
We set as above
This yields
and
We get
(4.8)
(4.9)
By virtue of (3.21), both operators and are exponentially stableon , respectively , and therefore so is the operator
, on the space . We set
and rewrite (4.8), (4.9) as
Equivalently
(4.10)
We are going to show that, for sufficiently small, (4.10) hasa unique solution . To this end, we proceedas in [5], [6]. Namely, consider the map
defined by
and show that, for sufficiently small, it maps the ball
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into itself and is a contraction on for sufficiently smalland suitable chosen. Indeed, by assumption (4.1), we have
Taking into account the Sobolev embedding theorem (we recall that), this yields
Hence
while
(4.11)
Then, arguing as in the proof of Theorem 5.1 in [6] (see, also, [5]),we see that maps into itself for suitable chosen and
sufficiently small. Moreover, by (4.11), it follows that is acontraction on . Hence, (4.10) has, for sufficientlysmall, a unique solution . By a standardargument (see, e.g., Proposition 5.9 in [6]), this implies also that
, and the latter extends to the solutionto (4.5). Then (4.4) follows.Example 3: Consider the classical Fitzhugh–Nagumo equation
which describes the dynamics of electrical potential across cell mem-brane ([7])
(4.12)
where . This equation has the unstable equilibrium solution.
The linearized operator
has the eigenvalues
with the eigenfunctions , .Then, for , we have and so, byTheorem 4.1, the controller
(4.13)
where , stabilizes exponentially theequilibrium solution for .
ACKNOWLEDGMENT
The author is indebted to the referees for their pertinent commentsand suggestions which helped to improve this technical note.
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