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

Transcription

1 1688 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013 Rapid Stabilization for a Korteweg-de Vries Equation From the Left Dirichlet Boundary Condition Eduardo Cerpa and Jean-Michel Coron Abstract This paper deals with the stabilization problem for the Korteweg-de Vries equation posed on a bounded terval. The control acts on the left Dirichlet boundary condition. At the right end-pot, Dirichlet and Neumann homogeneous boundary conditions are considered. The proposed feedback law forces the exponential decay of the system under a smallness condition on the itial data. Moreover, the decay rate can be tuned to be as large as desired. The feedback control law is designed by usg the backsteppg method. Index Terms Backsteppg, Korteweg-de Vries equation, stabilization by feedback. I. INTRODUCTION T HE Korteweg-de Vries (KdV) equation (1) posed on a bounded doma can be seen as a nonlear control system where the puts are the boundary data. From the nature of this equation, one boundary condition at the left end-pot and two boundary conditions at the right end-pot have to be imposed. The most studied case considers boundary conditions on Surprisgly, the control properties of this system are very different dependg on where the controls are located. If we act on the left Dirichlet boundary condition and homogeneous data is considered at the right, then the system behaves like a heat equation and only null-controllability can be proven [7], [22]. On the other hand, if we act on the two right data and homogeneous boundary condition is considered at the left, then the system behaves like a wave equation with an fite speed of propagation, the sense that exact controllability holds for any time of control [21]. Another fascatg phenomena occurs when we put Manuscript received January 26, 2012; revised January 29, 2012; accepted December 30, Date of publication January 21, 2013; date of current version June 19, This work was supported part by the ERC advanced grant FP CPDENL of the 7th Research Framework Program (FP7), by Fondecyt grant # , by CMM Basal grant, and by MathAmsud project CIP-PDE. This work has been done while E. Cerpa was visitg the Laboratoire Jacques-Louis Lions at Université Pierre et Marie Curie Paris. Recommended by Associate Editor C. Prieur. E. Cerpa is with the Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile ( eduardo.cerpa@usm.cl). J.-M. Coron is with the Institut Universitaire de France and Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris Cedex 05, France ( coron@ann.jussieu.fr). Color versions of one or more of the figures this paper are available onle at Digital Object Identifier /TAC (2) only one control put at the right end-pot and keep homogeneous the other two boundary conditions: there exist some spatial domas (tervals of some given lengths) for which the correspondg learized KdV equation is not any more controllable [8], [21]. In despite of that, these critical cases the nonlearity gives the exact controllability of the nonlear KdV equation[1],[2],[5].however,for some critical tervals the applied method requires a mimal time of control, which is not known to be really necessary. Due to the existence of these critical tervals for the lear system, we do not know if the energy of the KdV equation is decreasg when the three boundary conditions considered are homogeneous (free control case). For stance, if,the time-dependent function given by satisfies Therefore, we fd out a stationary solution of the lear KdV equation with homogeneous boundary conditions and consequently a solution with nondecreasg energy. This implies that for the free control case with, the learized system associated to the KdV (1) is not exponentially stable. With a feedback control law actg at the left-hand side such phenomenon does not appear and the method we propose this case allows to address the problem of rapid exponential stabilization: given a desired decay rate, we fd a feedback law exponentially stabilizg the system at that rate. Basedonthehyperbolic nature of the KdV equation controlled from the right, a method troduced [10] and [30] is used [3] to get the rapid stabilization of the lear KdV equation from the Neumann boundary condition on the right. This gramian-based approach comes from the fite dimensional theory [9], [18] and was first troduced for PDE the context of ternal control [25]. This method requires the controllability of the lear system and therefore a non critical terval has to be considered. In this paper, we applied the backsteppg method to design the feedback control law. The backsteppg method is well known as an ODE control method (see [11] and [4, Sec. 12.5]). The first extensions to PDE have appeared [6] and [17]. Later on, Krstic and his collaborators troduced a modification of the method by means of an tegral transformation of the PDE. This vertible transformation maps the origal PDE to an asymptotically stable one. In this context, the first contuous backsteppg designs were proposed for the heat equation [16], [27]. The applications to wave equation appeared later [12], [26], [29]. An excellent startg pot to get side this method is the book [13] by Krstic and Smyshlyaev. (3) (4) /$ IEEE

2 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 formulate the problem and state the ma result. Section III contas the backsteppg design of the feedback control law. Section IV is devoted to prove the exponential decay of the -norm of the solutions for the learized system around the orig. In Section V, we prove that this result still holds for the nonlear KdV control system when the itial condition is small enough. Some extensions to the non-constant coefficient case and different boundary conditions are considered Section VI. Fally, some fal remarksaregivensectionvii. Remark 1: The stabilization of the KdV equation by usg an ternal feedback control law was addressed [19], [20] (see also [15], [23] for KdV equations with other nonlearities). They have proved the followg semi-global stabilization. Let, and a dampg term satisfyg, for every where is nonempty open subset of. Then, there exist and for any solution of with. This result is different from ours the sense that the exponential decay rate can not be imposed as large as desired. A key role their design is played by the dampg term, which prevents the existence of critical domas and allows to work with a dissipative system for any. Other ternal feedback control laws (static or time-varyg ones) for the KdV equation with periodic boundary conditions can be found [14], [24]. (5) (6) The feedback law is explicitly defed as follows: where the function is characterized Section III as the solution of a given partial differential equation dependg on. Remark 2: As we shall see Section VI, Theorem 1 also holds if system (7) is replaced by or (9) (10) (11) where are given functions and, respectively. III. CONTROL DESIGN The backsteppg method applied here is based on the lear part of the equation. In this way, we consider the control system learized around the orig (12) Given a positive parameter,welookforatransformation defed by (13) II. PROBLEM STATEMENT AND MAIN RESULT Given, we consider the followg nonlear control system on the terval (7) the trajectory is mapped to the trajectory system, solution of (12) with (14), solution of the lear For any positive, we address the problem of buildg some feedback control law the orig is exponentially stable for the correspondg closed-loop system (7) and the exponential decay rate is. By usg the backsteppg method, we are able to fd such a control law. The design is based on the learized system around the orig. The lear closed-loop system is exponentially stable and the same result is obtaed for the nonlear KdV equation by addg a smallness condition on the itial data. Our ma theorem is the followg. Theorem 1: For any, there exist a feedback control law, and for any solution of (7) satisfyg. (8) For system (15), called the target system, we have for any and therefore we easily obta for decay at rate (15) (16) the exponential (17) In Section IV, we prove that, thanks to the vertibility of the map, the exponential decay (17) also holds for system (12).

3 1690 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013 and 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 us notice that the eigenvalues of system (15) are the eigenvalues of (18) Thus, given and usg (12), we have units. Thus, we are led to study the eigen- shiftedtotheleft values of (19) Surprisgly, the location of the eigenvalues of (19) depends on the length of the terval. From [21], we know that there exist some eigenvalues located on the imagary axis if and only if, which is called the set of critical lengths for this problem. In Fig. 1, the first five eigenvalues of system (18) are plot for different values of. In case (a), (noncritical) and the first eigenvalue is approximately. The system behaves like a dissipative one. In (b), (critical) and we have. The system has one conservative component given by the eigenfunction.in(c), and the first two eigenvalues are imagary numbers and.this examples show the different behaviors system (15) can have and the important role played by the parameter our design. In conclusion, the decay (17) is optimal for some values of. Let us focus the key step, which is fdg the kernel satisfies (15). For that, we perform the followg computations: After the above computations and sce we obta that one has (15) for every solution of (12) with (14) if the kernel defed the triangle satisfies Let us make the followg change of variable:. (20) and defe.wehave and therefore Now, the function,satisfies,defed.

4 CERPA AND CORON: RAPID STABILIZATION FOR A KORTEWEG-DE VRIES EQUATION FROM THE LEFT DIRICHLET BOUNDARY CONDITION 1691 Let us transform this system to an tegral one. We write the equation variables,tegrate and use that. Next, we tegrate and use that. Fally, we tegrate and use that. Thus, we can write the followg tegral form for : Fig. 2. Ga kernel correspondg to for (a),(b), and (c), respectively. (21) To prove that such a function exists, we use the method of successive approximations. We take as an itial guess and defe the recursive formula as follows: Performg some computations, we get for stance and more generally the followg formula: where the coefficients satisfy there exist positive constants and any (22) (23) (24) (25) and more importantly,, for any (26) This implies that the series is uniformly convergent. Therefore, the series defes a contuous function (27) and we get a solution of our tegral equation. Indeed, we can write (28) where we have used that the correspondg series and are also uniformly convergent. Once we have found the function,wegetthe existence of the kernel. It is easy to see that the map,defed by (13), is contuous and consequently we have the existence of a positive constant (29) In Fig. 2, we plot the ga kernel [see (13)] as a function of for different lengths (a) (noncritical), (b) (critical), and (c) (critical). The kernel functions are defed with. This illustrates the fact that case (a) is easier to stabilize than case (b), which is easier to stabilize than case (c). This is due to the location of the correspondg open-loop eigenvalues as shown Fig. 1. IV. STABILITY OF THE LINEAR SYSTEM We know that the target system (15) is exponentially stable. In order to get the same conclusion for the lear system (12), the method we are applyg uses the verse transformation. For that, we troduce a kernel function which satisfies. (30) The existence and uniqueness of such a kernel are proven the same way as for the kernel Section III. Once we have defed, it is easy to see that the transformation is characterized by Let us see that and are related by the formula (31) (32) which fact proves that trajectory of (12) with control maps a trajectory of (15) to a defed by (14). Indeed, by

5 1692 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013 pluggg (31) to (13) and usg Fubi s theorem we get, for any, that for any (40) where the term is given by (33) which proves (32) for any. The map is contuous and therefore we get the existence of a positive constant such that (34) Let us prove that system (12) (14) is exponentially stable. In fact, given,wedefe (35) We can prove that there exists a positive constant (41) (42) The solution of (15) with itial condition satisfies (16), i.e., and therefore, if there exists (43) (36) then we obta Moreover, the solution of (12) is given by. Thus, from (29), (34) and (36) we have for any (44) Thus, we get (45) (37) which proves the exponential decay at rate for system (12) with feedback law (14). provided that (46) V. STABILITY OF THE NONLINEAR SYSTEM Let be a solution of the nonlear (7) with the control given by (14). Then, satisfies As we did for the lear system, by usg the contuity of the transformations and (see (29) and (34)) and (45), we obta the exponential decay of the nonlear (7). From (46), we have to add a smallness condition on the itial data of system (7). This concludes the proof of Theorem 1. with homogeneous boundary conditions We multiply (38) by and tegrate to obta (38) (39) VI. SOME EXTENSIONS As mentioned Remark 2, this method can be applied to stabilize other related KdV systems. In this section, we focus the lear control design because the nonlear part of the argument is the same as Section V. More precisely, we show the equations that defe the kernel functions correspondg to each case. A. Different Boundary Conditions In order to stabilize system (47)

6 CERPA AND CORON: RAPID STABILIZATION FOR A KORTEWEG-DE VRIES EQUATION FROM THE LEFT DIRICHLET BOUNDARY CONDITION 1693 we have to consider a kernel triangle The transformation satisfyg on the the equation. (48) and defe the recursive formula as follows: (49) maps the solution to the trajectory, solution of the target system (50) Performg some computations, we get (59) (60) which is exponentially asymptotically stable for decay rate at least equals to. By makg the change of variable,witha (51) and more generally the followg formula (61) and defg, we get that the function,defed, satisfies (52) (53) (54) (55) Let us transform this system to an tegral one. We write (52) variables as follows: (56) We tegrate anduse (53). Next, we tegrate and use (54) and (55). Fally, we tegrate and use aga (54). Thus, we can write the followg tegral form for : where the coefficients have appropriate decay properties so that the series is uniformly convergent. Therefore the series defes a contuous function (62) and we get a solution of our tegral (57) which defes a kernel. The vertibility of the transformation (49) is obtaed as before Section IV. This property and the exponential stability of the target system (50) allow us to show that the solutions 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,we consider the non-constant coefficient lear KdV equation (64) and the target system (57) that can be studied by applyg the method of successive approximations. In fact, we take as an itial guess (58) which is exponentially asymptotically stable for In fact (65) large enough.

7 1694 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 7, JULY 2013 is a sufficient condition to ensure an exponential decay rate equals to for solutions of (65). In this case, the kernel to be considered order to defe the correspondg transformation (49) is the solution of (66) (67) (68) (69) the terval where the system evolves. The closed-loop system is proven to be locally exponentially stable with a decay rate that can be chosen to be as large as we want. This approach allows to consider the KdV (7) with other boundary conditions and low order terms as or with non-constant coefficients dependg on the space variable. The situation where we act on the right end-pot is different. If we consider homogeneous Dirichlet condition on the left and one or two control puts at the right-hand side of the terval, then we are not able to prove the backsteppg method works with the transformation (13). Indeed, when imposg where is defed by (74) (70) By usg the change of variable (51), system (66) (69) can be led to the equation on the target system, we get the followg expression at to vanish: (75) where (71) As we do not have to our disposal,thefirst term above arises the condition. Even if we do not care about the two last terms (75), order to keep, we have to impose for any. With these four boundary restrictions (the other two are on ), the third-order kernel equation satisfied by becomes overdetermed. Therefore, it is not clear if such a function exists. A natural idea to deal with controls at is to use the transformation and This equation, which is similar to (21), can be studied by applyg aga the method of successive approximations. C. A Nonzero Equilibrium State We consider solution of (7). If stead of learizg this system around zero, we do that around a nonzero equilibrium state solution of with,wehavetoconsiderthecontrolsystem (72) (73) where and are the first-order approximation of the state and the control, respectively. Thus, by usg the controldesignsectionvi-b for nonconstant coefficients, we can locally stabilizes system (7) around a nonzero equilibrium. VII. CONCLUSION We have applied the backsteppg method to build some boundary feedback laws, which locally stabilize the Korteweg-de Vries equation posed on a fite terval. Our control acts on the Dirichlet boundary condition at the left-hand side of (76) stead 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 condition for any.thisisduetothefact that when imposg on the target system, we get the extra term to be canceled. As previously, this fourth restriction gives an overdetermed kernel equation for. Moreover, the existence of critical lengths when only one control is considered at the right end-pot suggests that either the existence of the kernel or the vertibility of the correspondg map should fail for some spatial domas. Two related problems that rema still open are the boundary global or semi-global stabilization and the output feedback control problem. The boundary global stabilization is hard because it is needed a really nonlear design as [14] for the KdV equation with periodic boundary conditions. Concerng the output feedback control problem, we believe it could be solved by applyg the backsteppg approach order to built some observers as done [12], [28] for the heat and the wave equations respectively. ACKNOWLEDGMENT E. Cerpa would like to thank the Laboratoire Jacques-Louis Lions at Université Pierre et Marie Curie Paris for its hospitality.

8 CERPA AND CORON: RAPID STABILIZATION FOR A KORTEWEG-DE VRIES EQUATION FROM THE LEFT DIRICHLET BOUNDARY CONDITION 1695 REFERENCES [1] E. Cerpa, Exact controllability of a nonlear Korteweg-de Vries equation on a critical spatial doma, SIAM J. Control Optim., vol. 46, pp , [2] E. Cerpa and E. Crépeau, Boundary controllability for the nonlear Korteweg-de Vries equation on any critical doma, Ann. Inst. H. Pocaré Anal. Non Léaire, vol. 26, pp , [3] E. Cerpa and E. Crépeau, Rapid exponential stabilization for a lear Korteweg-de Vries equation, Discrete Cont. Dyn. Syst. Ser. B, vol. 11, no. 3, pp , [4] J.-M. Coron, Control and Nonlearity. Providence, RI, USA: American Mathematical Society, [5] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), vol. 6, pp , [6] J.-M. Coron and B. d Andréa-Novel, Stabilization of a rotatg body beam without dampg, IEEE Trans. Autom. Control, vol. 43, no. 5, pp , May [7] O. 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Krstic, Closed-form boundary state feedbacks for a class of 1-D partial tegro-differential equations, IEEE Trans. Autom. Control, vol. 49, no. 12, pp , Dec [28] A. Smyshlyaev and M. Krstic, Backsteppg observers for a class of parabolic PDEs, Syst. Control Lett., vol. 54, no. 7, pp , [29] A. Smyshlyaev and M. Krstic, Boundary control of an anti-stable wave equation with anti-dampg on the uncontrolled boundary, Syst. Control Lett., vol. 58, no. 8, pp , [30] J. M. Urquiza, Rapid exponential feedback stabilization with unbounded control operators, SIAM J. Control Optim., vol. 43, pp , Eduardo Cerpa was born Santiago, Chile, He received the Diplôme of engeer degree from the Universidad de Chile, Santiago, 2004, the M.S. degree mathematics from Université Pierre et Marie Curie, Paris, France, 2005, and the Ph.D. degree mathematics from the Université Paris-Sud 11, Paris, He was a Research Scholar at University of California, San Diego, CA, USA. He is currently an Assistant Professor at Universidad Técnica Federico Santa María, Valparaíso, Chile. His research terests clude nonlear partial differential equations, nonlear control theory, and verse problems. Jean-Michel Coron wasbornparis,france, He received the Diplôme of engeer degree from the Ecole Polytechnique, Paris, France, 1978 and from the Corps des Mes He received the Thèse d état He has been a Researcher at Ecole Nationale Supérieure des Mes de Paris, then Associate Professor at the Ecole Polytechnique, and Professor at Université Paris-Sud 11. He is currently a Professor at Université Pierre et Marie Curie and at the Institut Universitaire de France. His research terests clude nonlear partial differential equations and nonlear control theory.