1688 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58,...

1688 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013

Rapid Stabilization for a Korteweg-de Vries EquationFrom the Left Dirichlet Boundary Condition

Eduardo Cerpa and Jean-Michel Coron

Abstract—This paper deals with the stabilization problem forthe Korteweg-de Vries equation posed on a bounded interval. Thecontrol acts on the left Dirichlet boundary condition. At the rightend-point, Dirichlet and Neumann homogeneous boundary condi-tions are considered. The proposed feedback law forces the expo-nential decay of the system under a smallness condition on the ini-tial data. Moreover, the decay rate can be tuned to be as large asdesired. The feedback control law is designed by using the back-stepping method.

Index Terms—Backstepping, Korteweg-de Vries equation, stabi-lization by feedback.

I. INTRODUCTION

T HE Korteweg-de Vries (KdV) equation

(1)

posed on a bounded domain can be seen as a nonlinearcontrol system where the inputs are the boundary data. Fromthe nature of this equation, one boundary condition at the leftend-point and two boundary conditions at the right end-pointhave to be imposed. The most studied case considers boundaryconditions on

(2)

Surprisingly, the control properties of this system are very dif-ferent depending on where the controls are located. If we act onthe left Dirichlet boundary condition and homogeneous data isconsidered at the right, then the system behaves like a heat equa-tion and only null-controllability can be proven [7], [22]. On theother hand, if we act on the two right data and homogeneousboundary condition is considered at the left, then the system be-haves like a wave equation with an infinite speed of propagation,in the sense that exact controllability holds for any time of con-trol [21]. Another fascinating phenomena occurs when we put

Manuscript received January 26, 2012; revised January 29, 2012; acceptedDecember 30, 2012. Date of publication January 21, 2013; date of current ver-sion June 19, 2013. This work was supported in part by the ERC advancedgrant FP7-266907 CPDENL of the 7th Research Framework Program (FP7), byFondecyt grant #11090161, by CMM Basal grant, and by MathAmsud projectCIP-PDE. This work has been done while E. Cerpa was visiting the LaboratoireJacques-Louis Lions at Université Pierre et Marie Curie in Paris. Recommendedby Associate Editor C. Prieur.E. Cerpa is with the Departamento de Matemática, Universidad Técnica Fed-

erico Santa María, 2340000 Valparaíso, Chile (e-mail: [email protected]).J.-M. Coron is with the Institut Universitaire de France and Laboratoire

Jacques-Louis Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05,France (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2241479

only one control input at the right end-point and keep homo-geneous the other two boundary conditions: there exist somespatial domains (intervals of some given lengths) for which thecorresponding linearized KdV equation is not any more con-trollable [8], [21]. In despite of that, in these critical cases thenonlinearity gives the exact controllability of the nonlinear KdVequation [1], [2], [5]. However, for some critical intervals theapplied method requires a minimal time of control, which is notknown to be really necessary.Due to the existence of these critical intervals for the linear

system, we do not know if the energy of the KdV equation isdecreasing when the three boundary conditions considered arehomogeneous (free control case). For instance, if , thetime-independent function given by satisfies

(3)(4)

Therefore, we find out a stationary solution of the linear KdVequation with homogeneous boundary conditions and conse-quently a solution with nondecreasing energy. This implies thatfor the free control case with , the linearized system as-sociated to the KdV (1) is not exponentially stable.With a feedback control law acting at the left-hand side such

phenomenon does not appear and the method we propose in thiscase allows to address the problem of rapid exponential stabi-lization: given a desired decay rate, we find a feedback law ex-ponentially stabilizing the system at that rate.Based on the hyperbolic nature of the KdV equation con-

trolled from the right, a method introduced in [10] and [30]is used in [3] to get the rapid stabilization of the linear KdVequation from the Neumann boundary condition on the right.This gramian-based approach comes from the finite dimensionaltheory [9], [18] and was first introduced for PDE in the contextof internal control [25]. This method requires the controllabilityof the linear system and therefore a non critical interval has tobe considered.In this paper, we applied the backstepping method to de-

sign the feedback control law. The backstepping method is wellknown as an ODE control method (see [11] and [4, Sec. 12.5]).The first extensions to PDE have appeared in [6] and [17]. Lateron, Krstic and his collaborators introduced a modification of themethod by means of an integral transformation of the PDE. Thisinvertible transformation maps the original PDE into an asymp-totically stable one. In this context, the first continuous back-stepping designs were proposed for the heat equation [16], [27].The applications to wave equation appeared later in [12], [26],[29]. An excellent starting point to get inside this method is thebook [13] by Krstic and Smyshlyaev.

0018-9286/$31.00 © 2013 IEEE

CERPA AND CORON: RAPID STABILIZATION FOR A KORTEWEG-DE VRIES EQUATION FROM THE LEFT DIRICHLET BOUNDARY CONDITION 1689

This paper is organized as follows. In Section II, we formulatethe problem and state the main result. Section III contains thebackstepping design of the feedback control law. Section IV isdevoted to prove the exponential decay of the -norm of the so-lutions for the linearized system around the origin. In Section V,we prove that this result still holds for the nonlinear KdV controlsystem when the initial condition is small enough. Some exten-sions to the non-constant coefficient case and different boundaryconditions are considered in Section VI. Finally, some final re-marks are given in Section VII.Remark 1: The stabilization of the KdV equation by using

an internal feedback control law was addressed in [19], [20](see also [15], [23] for KdV equations with other nonlinearities).They have proved the following semi-global stabilization. Let

, and a damping term satisfying, for every where is nonempty open subset of. Then, there exist and

such that

(5)

for any solution of

(6)

with . This result is different from oursin the sense that the exponential decay rate can not be imposedas large as desired. A key role in their design is played by thedamping term , which prevents the existence of criticaldomains and allows to work with a dissipative system for any. Other internal feedback control laws (static or time-varyingones) for the KdV equation with periodic boundary conditionscan be found in [14], [24].

II. PROBLEM STATEMENT AND MAIN RESULT

Given , we consider the following nonlinear controlsystem on the interval

(7)

For any positive , we address the problem of building somefeedback control law such that the origin isexponentially stable for the corresponding closed-loop system(7) and the exponential decay rate is .By using the backstepping method, we are able to find such a

control law. The design is based on the linearized system aroundthe origin. The linear closed-loop system is exponentially stableand the same result is obtained for the nonlinear KdV equationby adding a smallness condition on the initial data.Our main theorem is the following.Theorem 1: For any , there exist a feedback control

law , and such that

(8)

for any solution of (7) satisfying .

The feedback law is explicitly defined as follows:

(9)

where the function is characterized in Section IIIas the solution of a given partial differential equation dependingon .Remark 2: As we shall see in Section VI, Theorem 1 also

holds if system (7) is replaced by

(10)

or

(11)

where are given functions in and ,respectively.

III. CONTROL DESIGN

The backstepping method applied here is based on the linearpart of the equation. In this way, we consider the control systemlinearized around the origin

(12)

Given a positive parameter , we look for a transformationdefined by

(13)

such that the trajectory , solution of (12) with

(14)

is mapped into the trajectory , solution of the linearsystem

(15)

For system (15), called the target system, we have for any

(16)

and therefore we easily obtain for the exponentialdecay at rate

(17)

In Section IV, we prove that, thanks to the invertibility of themap , the exponential decay (17) also holds for system (12).


Fig. 1. First five eigenvalues of system (18) for (a) , (b) , and(c) , respectively.

Naturally, we can wonder if this decay rate is sharp. Let usnotice that the eigenvalues of system (15) are the eigenvaluesof

(18)

shifted to the left units. Thus, we are led to study the eigen-values of

(19)

Surprisingly, the location of the eigenvalues of (19) dependson the length of the interval. From [21], we know that there existsome eigenvalues located on the imaginary axis if and only if

, which is called theset of critical lengths for this problem. In Fig. 1, the first fiveeigenvalues of system (18) are plot for different values of .In case (a), (noncritical) and the first eigenvalue isapproximately . The system behaves like a dissipative one.In (b), (critical) and we have . The systemhas one conservative component given by the eigenfunction

. In (c), and the first two eigen-values are imaginary numbers and . Thisexamples show the different behaviors system (15) can have andthe important role played by the parameter in our design. Inconclusion, the decay in (17) is optimal for some values of .Let us focus in the key step, which is finding the kernel

such that satisfies (15). For that,we perform the following computations:

and

Thus, given and using (12), we have

After the above computations and since

we obtain that one has (15) for every solution of (12) with(14) if the kernel defined in the triangle

satisfiesinininin .

Let us make the following change of variable:

(20)

and define . We have

and therefore

Now, the function , defined in, satisfies

inininin .


Let us transform this system into an integral one. We write theequation in variables , integrate in and use that

. Next, we integrate in and use that. Finally, we integrate in and use

that . Thus, we can write the following integralform for :

(21)

To prove that such a function exists, we usethe method of successive approximations. We take as an initialguess

(22)

and define the recursive formula as follows:

(23)

Performing some computations, we get for instance

(24)

and more generally the following formula:

(25)

where the coefficients satisfy and more importantly,there exist positive constants such that, for anyand any

(26)

This implies that the series is uniformly con-vergent in . Therefore, the series defines a continuous func-tion

(27)

and we get a solution of our integral equation. Indeed, we canwrite

Fig. 2. Gain kernel corresponding to for (a) , (b) ,and (c) , respectively.

(28)

where we have used that the corresponding series andare also uniformly convergent.

Once we have found the function , we get theexistence of the kernel . It is easy to see that themap , defined by (13), is continuousand consequently we have the existence of a positive constant

such that

(29)

In Fig. 2, we plot the gain kernel [see (13)] as a func-tion of for different lengths (a) (noncritical),(b) (critical), and (c) (critical). Thekernel functions are defined with . This illustrates the factthat case (a) is easier to stabilize than case (b), which is easierto stabilize than case (c). This is due to the location of the cor-responding open-loop eigenvalues as shown in Fig. 1.

IV. STABILITY OF THE LINEAR SYSTEM

We know that the target system (15) is exponentially stable. Inorder to get the same conclusion for the linear system (12), themethod we are applying uses the inverse transformation .For that, we introduce a kernel function which satisfies

inininin .

(30)

The existence and uniqueness of such a kernelare proven in the same way as for the kernel inSection III. Once we have defined , it is easy to seethat the transformation is characterized by

(31)

Let us see that and are related by the formula

(32)

which in fact proves that maps a trajectory of (15) into atrajectory of (12) with control defined by (14). Indeed, by


plugging (31) into (13) and using Fubini’s theorem we get, forany , that for any

(33)

which proves (32) for any .The map is continuous and

therefore we get the existence of a positive constant suchthat

(34)

Let us prove that system (12)–(14) is exponentially stable. Infact, given , we define

(35)

The solution of (15) with initial conditionsatisfies (16), i.e.,

(36)

Moreover, the solution of (12) is given by. Thus, from (29), (34) and (36) we have for any

(37)

which proves the exponential decay at rate for system (12)with feedback law (14).

V. STABILITY OF THE NONLINEAR SYSTEM

Let be a solution of the nonlinear (7) with thecontrol given by (14). Then, satisfies

(38)

with homogeneous boundary conditions

(39)

We multiply (38) by and integrate in to obtain

(40)

where the term is given by

(41)

We can prove that there exists a positive constantsuch that

(42)

and therefore, if there exists such that

(43)

then we obtain

(44)

Thus, we get

(45)

provided that

(46)

As we did for the linear system, by using the continuity ofthe transformations and (see (29) and (34)) and (45), weobtain the exponential decay of the nonlinear (7). From (46), wehave to add a smallness condition on the initial data of system(7). This concludes the proof of Theorem 1.

VI. SOME EXTENSIONS

As mentioned in Remark 2, this method can be applied tostabilize other related KdV systems. In this section, we focusin the linear control design because the nonlinear part of theargument is the same as in Section V. More precisely, we showthe equations that define the kernel functions corresponding toeach case.

A. Different Boundary Conditions

In order to stabilize system

(47)


we have to consider a kernel satisfying on thetriangle the equation

inininin .

(48)

The transformation

(49)

maps the solution into the trajectory ,solution of the target system

(50)

which is exponentially asymptotically stable for , with adecay rate at least equals to .By making the change of variable

(51)

and defining , we get that the function, defined in ,

satisfies

(52)(53)(54)

(55)

Let us transform this system into an integral one. We write (52)in variables as follows:

(56)

We integrate in and use (53). Next, we integratein and use (54) and (55). Finally, we integrate inand use again (54). Thus, we can write the following integralform for :

(57)

that can be studied by applying the method of successive ap-proximations. In fact, we take as an initial guess

(58)

and define the recursive formula as follows:

(59)

Performing some computations, we get

(60)

and more generally the following formula

(61)

where the coefficients have appropriate decay properties sothat the series is uniformly convergent in .Therefore the series defines a continuous function

(62)

and we get a solution of our integral (57) which defines a kernel. The invertibility of the transformation (49) is ob-

tained as before in Section IV. This property and the exponentialstability of the target system (50) allow us to show that the so-lutions of system (47) with a feedback control law

(63)

exponentially decay to zero with a decay rate at least equals to.

B. Nonconstant Coefficient Case

Given two functions and , weconsider the non-constant coefficient linear KdV equation

(64)

and the target system

(65)

which is exponentially asymptotically stable for large enough.In fact


is a sufficient condition to ensure an exponential decay rateequals to for solutions of (65).In this case, the kernel to be considered in order

to define the corresponding transformation (49) is the solutionof

(66)(67)(68)

(69)

where is defined by

(70)

By using the change of variable (51), system (66)–(69) can beled to the equation

(71)

where

and

This equation, which is similar to (21), can be studied byapplying again the method of successive approximations.

C. A Nonzero Equilibrium StateWe consider solution of (7). If instead of lin-

earizing this system around zero, we do that around a nonzeroequilibrium state solution of

(72)

with , we have to consider the control system

(73)

where and are the first-order approxima-tion of the state and the control ,respectively. Thus, by using the control design in Section VI-Bfor nonconstant coefficients, we can locally stabilizes system (7)around a nonzero equilibrium.

VII. CONCLUSIONWe have applied the backstepping method to build some

boundary feedback laws, which locally stabilize the Ko-rteweg-de Vries equation posed on a finite interval. Our controlacts on the Dirichlet boundary condition at the left-hand side of

the interval where the system evolves. The closed-loop systemis proven to be locally exponentially stable with a decay ratethat can be chosen to be as large as we want. This approachallows to consider the KdV (7) with other boundary conditionsand low order terms as or with non-constant coefficientsdepending on the space variable.The situation where we act on the right end-point is different.

If we consider homogeneous Dirichlet condition on the left andone or two control inputs at the right-hand side of the interval,then we are not able to prove the backstepping method workswith the transformation (13). Indeed, when imposing

(74)

on the target system, we get the following expression atto vanish:

(75)

As we do not have to our disposal , the first term abovearises the condition . Even if we do not care aboutthe two last terms in (75), in order to keep ,we have to impose for any . With thesefour boundary restrictions (the other two are on ), thethird-order kernel equation satisfied by becomesoverdetermined. Therefore, it is not clear if such a function

exists.A natural idea to deal with controls at is to use the

transformation

(76)

instead of (13). However, it is not clear if that approach works.In fact, if we do that, we have to deal now with the extra con-dition for any . This is due to the factthat when imposing on the targetsystem, we get the extra term to be canceled.As previously, this fourth restriction gives an overdeterminedkernel equation for . Moreover, the existence ofcritical lengths when only one control is considered at the rightend-point suggests that either the existence of the kernel or theinvertibility of the corresponding map should fail for somespatial domains.Two related problems that remain still open are the boundary

global or semi-global stabilization and the output feedback con-trol problem. The boundary global stabilization is hard becauseit is needed a really nonlinear design as in [14] for the KdV equa-tion with periodic boundary conditions. Concerning the outputfeedback control problem, we believe it could be solved by ap-plying the backstepping approach in order to built some ob-servers as done in [12], [28] for the heat and the wave equationsrespectively.

ACKNOWLEDGMENT

E. Cerpa would like to thank the Laboratoire Jacques-LouisLions at Université Pierre et Marie Curie in Paris for itshospitality.


REFERENCES[1] E. Cerpa, “Exact controllability of a nonlinear Korteweg-de Vries equa-

tion on a critical spatial domain,” SIAM J. Control Optim., vol. 46, pp.877–899, 2007.

[2] E. Cerpa and E. Crépeau, “Boundary controllability for the nonlinearKorteweg-de Vries equation on any critical domain,” Ann. Inst. H.Poincaré Anal. Non Linéaire, vol. 26, pp. 457–475, 2009.

[3] E. Cerpa and E. Crépeau, “Rapid exponential stabilization for a linearKorteweg-de Vries equation,” Discrete Contin. Dyn. Syst. Ser. B, vol.11, no. 3, pp. 655–668, 2009.

[4] J.-M. Coron, Control and Nonlinearity. Providence, RI, USA: Amer-ican Mathematical Society, 2007.

[5] J.-M. Coron and E. Crépeau, “Exact boundary controllability of a non-linear KdV equation with critical lengths,” J. Eur. Math. Soc. (JEMS),vol. 6, pp. 367–398, 2004.

[6] J.-M. Coron and B. d’Andréa-Novel, “Stabilization of a rotating bodybeam without damping,” IEEE Trans. Autom. Control, vol. 43, no. 5,pp. 608–618, May 1998.

[7] O. Glass and S. Guerrero, “Some exact controllability results for thelinear KdV equation and uniform controllability in the zero-dispersionlimit,” Asymptot. Anal., vol. 60, pp. 61–100, 2008.

[8] O. Glass and S. Guerrero, “Controllability of the Korteweg-de Vriesequation from the right Dirichlet boundary condition,” Syst. ControlLett., vol. 59, pp. 390–395, 2010.

[9] D. L. Kleinman, “An easy way to stabilize a linear control system,”IEEE Trans. Autom. Control, vol. AC-15, p. 692, Dec. 1970.

[10] V. Komornik, “Rapid boundary stabilization of linear distributed sys-tems,” SIAM J. Control Optim., vol. 35, pp. 1591–1613, 1997.

[11] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adap-tive Control Design. New York, NY, USA: Wiley, 1995.

[12] M. Krstic, B. Z. Guo, A. Balogh, and A. Smyshlyaev, “Output-feed-back stabilization of an unstable wave equation,” Autom., vol. 44, pp.63–74, 2008.

[13] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course onBackstepping Designs. Philadelphia, PA, USA: Society for Industrialand Applied Mathematics (SIAM), 2008.

[14] C. Laurent, L. Rosier, and B.-Y. Zhang, “Control and stabilization ofthe Korteweg-de Vries equation on a periodic domain,” Comm. PartialDifferential Equat., vol. 35, pp. 707–744, 2010.

[15] F. Linares and A. F. Pazoto, “On the exponential decay of the crit-ical generalized Korteweg-de Vries with localized damping,” in Proc.Amer. Math. Soc., 2007, vol. 135, pp. 1515–1522.

[16] W. Liu, “Boundary feedback stabilization of an unstable heat equa-tion,” SIAM J. Control Optim., vol. 42, pp. 1033–1043, 2003.

[17] W. Liu and M. Krstic, “Backstepping boundary control of Burgers’equation with actuator dynamics,” Syst. Control Lett., vol. 41, no. 4,pp. 291–303, 2000.

[18] D. L. Lukes, “Stabilizability and optimal control,” Funkcial. Ekvac.,vol. 11, pp. 39–50, 1968.

[19] A. F. Pazoto, “Unique continuation and decay for the Korteweg-deVries equation with localized damping,” ESAIM Control Optim. Calc.Var., vol. 11, pp. 473–486, 2005.

[20] G. Perla Menzala, C. F. Vasconcellos, and E. Zuazua, “Stabilization ofthe Korteweg-de Vries equation with localized damping,”Quarterly ofAppl. Math., vol. LX, pp. 111–129, 2002.

[21] L. Rosier, “Exact boundary controllability for the Korteweg-de Vriesequation on a bounded domain,” ESAIM Control Optim. Calc. Var.,vol. 2, pp. 33–55, 1997.

[22] L. Rosier, “Control of the surface of a fluid by a wavemaker,” ESAIMControl Optim. Calc. Var., vol. 10, pp. 346–380, 2004.

[23] L. Rosier and B.-Y. Zhang, “Global Stabilization of the generalizedKorteweg-de Vries Equation Posed on a finite Domain,” SIAM J. Con-trol Optim., vol. 45, pp. 927–956, 2006.

[24] D. L. Russell and B.-Y. Zhang, “Exact controllability and stabiliz-ability of the Korteweg-de Vries equation,” Trans. Amer. Math. Soc.,vol. 348, pp. 3643–3672, 1996.

[25] M. Slemrod, “A note on complete controllability and stabilizability forlinear control systems in Hilbert space,” SIAM J. Control, vol. 12, pp.500–508, 1974.

[26] A. Smyshlyaev, E. Cerpa, and M. Krstic, “Boundary stabilization ofa 1-D wave equation with in-domain antidamping,” SIAM J. ControlOptim., vol. 48, no. 6, pp. 4014–4031, 2010.

[27] A. Smyshlyaev and M. Krstic, “Closed-form boundary state feedbacksfor a class of 1-D partial integro-differential equations,” IEEE Trans.Autom. Control, vol. 49, no. 12, pp. 2185–2202, Dec. 2004.

[28] A. Smyshlyaev and M. Krstic, “Backstepping observers for a class ofparabolic PDEs,” Syst. Control Lett., vol. 54, no. 7, pp. 613–625, 2005.

[29] A. Smyshlyaev and M. Krstic, “Boundary control of an anti-stablewave equation with anti-damping on the uncontrolled boundary,” Syst.Control Lett., vol. 58, no. 8, pp. 617–623, 2009.

[30] J. M. Urquiza, “Rapid exponential feedback stabilization with un-bounded control operators,” SIAM J. Control Optim., vol. 43, pp.2233–2244, 2005.

Eduardo Cerpa was born in Santiago, Chile, in1979. He received the Diplôme of engineer degreefrom the Universidad de Chile, Santiago, in 2004,the M.S. degree in mathematics from UniversitéPierre et Marie Curie, Paris, France, in 2005, andthe Ph.D. degree in mathematics from the UniversitéParis-Sud 11, Paris, in 2008.He was a Research Scholar at University of

California, San Diego, CA, USA. He is currently anAssistant Professor at Universidad Técnica FedericoSanta María, Valparaíso, Chile. His research interests

include nonlinear partial differential equations, nonlinear control theory, andinverse problems.

Jean-Michel Coron was born in Paris, France, in1956. He received the Diplôme of engineer degreefrom the Ecole Polytechnique, Paris, France, in 1978and from the Corps des Mines in 1981. He receivedthe Thèse d’ état in 1982.He has been a Researcher at Ecole Nationale

Supérieure des Mines de Paris, then Associate Pro-fessor at the Ecole Polytechnique, and Professor atUniversité Paris-Sud 11. He is currently a Professorat Université Pierre et Marie Curie and at the InstitutUniversitaire de France. His research interests

include nonlinear partial differential equations and nonlinear control theory.

Date post:	09-Oct-2020
Category:	Documents
Upload:	others
View:	4 times
Download:	0 times

1688 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58,...

Documents