Problèmes de contrôle pour des équations dispersives unidimensionnelles

Glass, Olivier

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 3, 15 p.

[1] Banks S. P.,Exact boundary controllability and optimal control for a generalised Korteweg de Vries equation, Nonlinear Anal. 47 (2001), no. 8, pp. 5537–5546. | MR | Zbl

[2] Bona J., Sun S. M., Zhang B.-Y., A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations 28 (2003), no. 7-8, pp. 1391–1436. | MR | Zbl

[3] Bubnov B. A., General boundary value problems for the Korteweg-de Vries equation in a bounded domain, Differentsial’nye Uravneniya 15 (1979), no. 1, pp. 26–31 & 185–186. | MR | Zbl

[4] Cattabriga L., Un problema al contorno per una equazione parabolica di ordine dispari, Ann. Scuola Norm. Sup. Pisa 13 (1959), pp. 163–203. | Numdam | MR | Zbl

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

[6] Cerpa E., Crépeau E., Boundary controlability for the non linear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 2, 457–475. | Numdam | MR | Zbl

[7] Chapouly M., Global controllability of a nonlinear Korteweg-de Vries equation, Communications in Contemporary Mathematics, à paraître.

[8] Colin T., Ghidaglia J.-M., An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval, Adv. Differential Equations 6 (2001), no. 12, pp. 1463–1492. | MR | Zbl

[9] Colliander J. E. and Kenig C. E., The generalized Korteweg-de Vries equation on the half line, Comm. Partial Diff. Eq. 27 (2002), no. 11-12, pp. 2187–2266. | MR | Zbl

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

[11] Coron J.-M., Guerrero S., Singular optimal control : A linear 1-D parabolic-hyperbolic example, Asymp. Anal. 44 3,4 (2005), pp. 237–257. | MR | Zbl

[12] Danchin R., Poches de tourbillon visqueuses, J. Math. Pures Appl. 76 (1997), pp. 609–647. | MR | Zbl

[13] Dawson L., Uniqueness properties of higher order dispersive equations, J. Differential Equations 236 (2007), no. 1, pp. 199–236. | MR | Zbl

[14] Doronin G. G., Larkin N A., Kawahara equation in a bounded domain. Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 4, pp. 783–799. | MR | Zbl

[15] Faminskii A. V., On Two Initial Boundary Value Problems for the Generalized KdV Equation, Nonlinear Boundary Problems 14 (2004), pp. 58–71. | Zbl

[16] Fernández-Cara E., Zuazua E., The cost of approximate controllability for heat equations : the linear case. Adv. Differential Equations 5 (2000), no. 4-6, pp. 465–514. | MR | Zbl

[17] Fursikov A., Imanuvilov O. Yu., Controllability of Evolution Equations, Lecture Notes #34, Seoul National University, Korea, 1996. | MR | Zbl

[18] Glass O., Guerrero S., On the uniform controllability of the Burgers equation, SIAM J. Control Optim. 46 (2007), no. 4, pp. 1211–1238. | MR | Zbl

[19] Glass O., Guerrero S., Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymp. Anal. 60 (2008), no. 1-2, pp. 61–100. | MR | Zbl

[20] Glass O., Guerrero S., Uniform controllability of a transport equation in zero diffusion-dispersion limit, Math. Models Methods Appl. Sci., à paraître.

[21] Glass O., Guerrero S., On the controllability of the fifth-order Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, à paraître.

[22] Guerrero S., Lebeau G., Singular optimal control for a transport-diffusion equation, Comm. Partial Differential Equations 32 (2007), no. 12, pp. 1813–1836. | MR | Zbl

[23] Holmer J., The initial-boundary value problem for the Korteweg-de Vries equation, Communications in Partial Differential Equations, 31 (2006), pp. 1151-1190. | MR | Zbl

[24] Kawahara R., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), pp. 260–264.

[25] Kenig C., Ponce G., Vega L., Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc. 122 (1994), no. 1, pp. 157–166. | MR | Zbl

[26] Kichenassamy S., Olver P. J., Existence and nonexistence of solitary wave solutions to higher-order model evolution equations, SIAM J. Math. Anal. 23 (1992), no. 5, pp. 1141–1166. | MR | Zbl

[27] Kwon S., Well-posedness and ill-posedness of the fifth-order modified KdV equation, Electron. J. Differential Equations 2008, No. 01, pp. 1–15. | MR | Zbl

[28] Laurent C., Rosier L., Zhang B.-Y., Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain, preprint 2009.

[29] Lax P. D., Levermore C. D., The zero dispersion limit for the Korteweg-de Vries KdV equation, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), pp. 3602–3606. | MR | Zbl

[30] LeFloch P. G., Hyperbolic systems of conservation laws : The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2002. | MR | Zbl

[31] Lions J.-L., Exact controllability, stabilizability and perturbations for distributed systems, SIAM Review, 30 (1988), pp. 1–68. | MR | Zbl

[32] Miller L., How violent are fast controls for Schrödinger and plate vibrations ?, Arch. Ration. Mech. Anal. 172 (2004), no. 3, pp. 429–456. | MR | Zbl

[33] Ponce G., Lax pairs and higher order models for water waves, J. Differential Equations 102 (1993), no. 2, pp. 360–381. | MR | Zbl

[34] Rosier L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 2 (1997), pp. 33–55. | Numdam | MR | Zbl

[35] Rosier L., Control of the surface of a fluid by a wavemaker. ESAIM Control Optim. Calc. Var. 10 (2004), no. 3, pp. 346–380 | Numdam | MR | Zbl

[36] Russell D. L., Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Review 20 (1978), pp. 639–739. | MR | Zbl

[37] Russell D. L., Zhang B. Y., Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31 (1993), no. 3, pp. 659–676. | MR | Zbl

[38] Russell D. L., Zhang B. Y., Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996), no. 9, pp. 3643–3672. | MR | Zbl

[39] Seidman T. I., Two results on exact boundary control of parabolic equations, Appl. Math. Optim. 11 (1984), no. 2, pp. 145–152. | MR | Zbl

[40] Zhang B.-Y., Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Control Optim. 37 (1999), no. 2, pp. 543–565. | MR | Zbl