Geodesic
On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control
ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 237-263.

Voir la notice de l'article provenant de la source Numdam

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2017001
Classification : 49N35, 93C20, 93D20
Keywords: Receding horizon control, model predictive control, asymptotic stability, infinite-dimensional systems

Azmi, Behzad 1 ; Boulanger, Anne-Céline 1 ; Kunisch, Karl 1

1 
@article{COCV_2018__24_1_237_0,
     author = {Azmi, Behzad and Boulanger, Anne-C\'eline and Kunisch, Karl},
     title = {On the semi-global stabilizability of the {Korteweg-de} {Vries} {Equation} via model predictive control},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {237--263},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {1},
     year = {2018},
     doi = {10.1051/cocv/2017001},
     zbl = {1396.93102},
     mrnumber = {3843184},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017001/}
}
TY  - JOUR
AU  - Azmi, Behzad
AU  - Boulanger, Anne-Céline
AU  - Kunisch, Karl
TI  - On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 237
EP  - 263
VL  - 24
IS  - 1
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017001/
DO  - 10.1051/cocv/2017001
LA  - en
ID  - COCV_2018__24_1_237_0
ER  - 
%0 Journal Article
%A Azmi, Behzad
%A Boulanger, Anne-Céline
%A Kunisch, Karl
%T On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 237-263
%V 24
%N 1
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017001/
%R 10.1051/cocv/2017001
%G en
%F COCV_2018__24_1_237_0
Azmi, Behzad; Boulanger, Anne-Céline; Kunisch, Karl. On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 237-263. doi : 10.1051/cocv/2017001. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2017001/

[1] F. Allgöwer, T.A. Badgwell, J.S. Qin, J.B. Rawlings and S.J. Wright, Nonlinear predictive control and moving horizon estimationan introductory overview, in Advances in control. Springer (1999) 391–449. | DOI

[2] F. Allgöwer and A. Zheng, Nonlinear model predictive control, Vol. 26. Birkhäuser Basel (2000). | MR | Zbl | DOI

[3] D.N. Arnold and R. Winther, A superconvergent finite element method for the Korteweg-de Vries equation. Math. Comput. 38 (1982) 23–36. | MR | Zbl | DOI

[4] B. Azmi and K. Kunisch, On the Stabilizability of the Burgers Equation by Receding Horizon Control. SIAM J. Control Optim. 54 (2016) 1378–1405. | MR | Zbl | DOI

[5] J. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc. 63 (1977) 370–373. | MR | Zbl

[6] J. Barzilai and J.M. Borwein, Two-point step size gradient methods. IMA J. Numer. Anal. 8 (1988) 141–148. | MR | Zbl | DOI

[7] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite dimensional systems, Springer Science and Business Media (2007). | MR | Zbl | DOI

[8] J.L. Bona, V.A. Dougalis and O.A. Karakashian, Fully discrete Galerkin methods for the Korteweg-de Vries equation. Comput. Math. Appl. 12 (1986) 859–884. | MR | Zbl | DOI

[9] J.L. Bona, S.-M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. Commun. Partial Differ. Equ. 28 (2003) 1391–1436. | MR | Zbl | DOI

[10] J.L. Bona, S.M. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain II. J. Differ. Equ. 247 (2009) 2558–2596. | MR | Zbl | DOI

[11] A.-C. Boulanger and P. Trautmann, Sparse optimal control of the Korteweg-de Vries-Burgers equation on a bounded domain. SIAM J. Control Optim. 55 (2017) 3673–3706. | MR | Zbl | DOI

[12] J. Boussinesq, Essai sur la théories des eaux courantes, Mémoires présentés par divers savants à l’Académie des Sciences de l’Institut Nationale de France 23 (1877). | JFM

[13] J.P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation (2001). | MR | Zbl

[14] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations. Springer Science and Business Media (2010). | MR | Zbl

[15] R.A. Capistrano-Filho, A.F. Pazoto and L. Rosier, Internal controllability of the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 21 (2015) 1076–1107. | MR | Zbl | mathdoc-id

[16] E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optimiz. 46 (2007) 877–899. | MR | Zbl | DOI

[17] H. Chen and F. Allgöwer, A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34 (1998) 1205–1217. | MR | Zbl

[18] T. Colin, J.-M. Ghidaglia, An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval. Adv. Differ. Equ. 6 (2001) 1463–1492. | MR | Zbl

[19] A. Constantin and R. Johnson, On the Non-Dimensionalisation, Scaling and Resulting Interpretation of the Classical Governing Equations for Water Waves. J. Nonlin. Math. Phys. 15 (2008) 58–73. | MR | Zbl | DOI

[20] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. (JEMS) 6 (2004) 367–398. | MR | Zbl | DOI

[21] Y.-H. Dai and H. Zhang, Adaptive two-point stepsize gradient algorithm. Numer. Algorithms 27 (2001) 377–385. | MR | Zbl | DOI

[22] K. Djidjeli, W.G. Price, E.H. Twizell and Y. Wang, Numerical methods for the solution of the third-and fifth-order dispersive Korteweg-de Vries equations. J. Comput. Appl. Math. 58 (1995) 307–336. | MR | Zbl | DOI

[23] D. Dutykh, T. Katsaounis and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models. Inter. J. Numer. Methods Fluids 71 (2013) 717–736. | MR | Zbl | DOI

[24] A. Faminskii et al., Global well-posedness of two initial-boundary-value problems for the Korteweg-de Vries equation. Differ. Integral Equ. 20 (2007) 601–642. | MR | Zbl

[25] A.V. Faminskii and N.A. Larkin, Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval. Electron. J. Differ. Equ. 2010 (2010) 1–20. | MR | Zbl

[26] O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right dirichlet boundary condition. Syst. Control Lett. 59 (2010) 390–395. | MR | Zbl | DOI

[27] G. Grimm, M.J. Messina, S.E. Tuna and A.R. Teel, Model predictive control: for want of a local control Lyapunov function, all is not lost. Automatic Control, IEEE Trans. 50 (2005) 546–558. | MR | Zbl | DOI

[28] L. Grüne, Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems. SIAM J. Control Optimiz. 48 (2009) 1206–1228. | MR | Zbl | DOI

[29] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Springer London (2011). | MR | Zbl | DOI

[30] L. Grüne and A. Rantzer, On the infinite horizon performance of receding horizon controllers. Automatic Control, IEEE Transactions 53 (2008) 2100–2111. | MR | Zbl | DOI

[31] J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation. Commun. Partial Differ. Equ. 31 (2006) 1151–1190. | MR | Zbl | DOI

[32] K. Ito and K. Kunisch, Receding horizon optimal control for infinite dimensional systems. ESAIM: COCV 8 (2002) 741–760. | MR | Zbl | mathdoc-id

[33] K. Ito and K. Kunisch, Receding horizon control with incomplete observations. SIAM J. Control Optimiz. 45 (2006) 207–225. | MR | Zbl | DOI

[34] A. Jadbabaie and J. Hauser, On the stability of receding horizon control with a general terminal cost. Automatic Control, IEEE Trans. 50 (2005) 674–678. | MR | Zbl | DOI

[35] A. Jadbabaie, J. Yu and J. Hauser, Unconstrained receding-horizon control of nonlinear systems. Automatic Control, IEEE Trans. 46 (2001) 776–783. | MR | Zbl | DOI

[36] C. Jia and B.-Y. Zhang, Boundary stabilization of the Korteweg-de Vries equation and the Korteweg-de Vries-Burgers equation. Acta Appl. Math. 118 (2012) 25–47. | MR | Zbl | DOI

[37] D. Kordeweg and G. De Vries, On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary wave. Phil. Mag 39 (1895) 422–443. | MR | JFM | DOI

[38] D.J. Korteweg and G. De Vries Xli. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. The London, Edinburgh, and Dublin Philosophical Magazine J. Sci. 39 1895 422–443. | MR | JFM | DOI

[39] E.F. Kramer and B. Zhang, Nonhomogeneous boundary value problems for the Korteweg-de Vries equation on a bounded domain. J. Syst. Sci. Complexity 23 (2010) 499–526. | MR | Zbl | DOI

[40] F. Linares and A. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping.Proc. Amer. Math. Soc. 135 (2007) 1515–1522. | MR | Zbl | DOI

[41] H. Ma and W. Sun, A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential equations. SIAM J. Numer. Anal. 38 (2000) 1425–1438. | MR | Zbl | DOI

[42] H. Ma and W. Sun, Optimal error estimates of the Legendre-Petrov-Galerkin method for the Korteweg-de Vries equation. SIAM J. Numer. Anal. 39 (2001) 1380–1394. | MR | Zbl | DOI

[43] C.P. Massarolo, G.P. Menzala and A.F. Pazoto, On the uniform decay for the Korteweg-de Vries equation with weak damping. Math. Methods Appl. Sci. 30 (2007) 1419–1435. | MR | Zbl | DOI

[44] D.Q. Mayne, J.B. Rawlings, C.V. Rao and P.O. Scokaert, Constrained model predictive control: Stability and optimality. Automatica 36 (2000) 789–814. | MR | Zbl | DOI

[45] G.P. Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping. Quarterly Appl. Math. 60 (2002) 111–129. | MR | Zbl | DOI

[46] A.F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM: COCV 11 (2005) 473–486. | MR | Zbl | mathdoc-id

[47] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Vol. 44. Springer Science and Business Media (2012). | MR | Zbl

[48] M. Reble and F. Allgöwer, Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems. Automatica 48 (2012) 1812–1817. | MR | Zbl | DOI

[49] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. | MR | Zbl | mathdoc-id

[50] L. Rosier, Control of the surface of a fluid by a wavemaker. ESAIM: COCV 10 (2004) 346–380. | Zbl | MR | mathdoc-id

[51] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain. SIAM J. Control Optimiz. 45 (2006) 927–956. | MR | Zbl | DOI

[52] L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: recent progresses. J. Syst. Sci. Complexity 22 (2009) 647–682. | MR | Zbl | DOI

[53] J. Shen, A new dual-Petrov-Galerkin method for third and higher odd-order differential equations: application to the KDV equation. SIAM J. Numer. Anal. 41 (2003) 1595–1619. | MR | Zbl | DOI

[54] J. Simon, Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. 146 (1987) 65–96. | MR | Zbl | DOI

[55] R. Winther, A conservative finite element method for the Korteweg-de Vries equation. Math. Comput. (1980) 23–43. | MR | Zbl | DOI

[56] J. Yan and C.-W. Shu, A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40 (2002) 769–791. | MR | Zbl | DOI

[57] N.J. Zabusky and M.D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15 (1965) 240–243. | Zbl | DOI

[58] B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optimiz. 37 (1999) 543–565. | MR | Zbl | DOI

Cité par Sources :