On the controllability and stabilization of the Benjamin equation on a periodic domain

Panthee, M.; Vielma Leal, F.

Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1605-1652.

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain $\mathbb{T}$. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in $H_p^s(\mathbb{T})$, with $s\ge 0$. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32].

[1] Albert, J.P.; Bona, J.L.; Restrepo, J.M. Solitary waves solutions of the Benjamin equation, SIAM J. Appl. Math., Volume 59 (1999) no. 6, pp. 2139-2161 | DOI | MR | Zbl

[2] Angulo, J. Existence and stability of solitary wave solutions of the Benjamin equation, J. Differ. Equ., Volume 152 (1999) no. 1, pp. 136-159 | DOI | MR | Zbl

[3] Aubin, T. Nonlinear Analysis on Manifolds Monge-Ampere Equations, Grundlehren Mathematischen Wissenschaften, vol. 252, Springer Verlag, New York Inc., Heidelberg Berlin, 1982 | MR | Zbl

[4] Bergh, J.; Lofstrom, J. Interpolation Spaces an Introduction, A Series of Comprehensive Studies in Mathematics, Springer-Verlag, Berlin Heidelberg New York, 1976 | MR | Zbl

[5] Benjamin, T.B. A new kind of solitary waves, J. Fluid Mech., Volume 245 (1992), pp. 401-411 | DOI | MR | Zbl

[6] Bona, J.; Ridgway, S. Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., Volume 43 (1976) no. 1, pp. 87-99 | DOI | MR | Zbl

[7] Bourgain, J. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part II: the KdV-equation, Geom. Funct. Anal., Volume 3 (1993), pp. 209-262 | DOI | MR | Zbl

[8] Chen, H.; Bona, J.L. Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations, Adv. Differ. Equ., Volume 3 (1998) no. 1, pp. 51-84 | MR | Zbl

[9] Chen, W.; Xiao, J. A sharp bilinear estimate for the Bourgain-type space with application to the Benjamin equation, Commun. Partial Differ. Equ., Volume 35 (2010) no. 10, pp. 1739-1762 | DOI | MR | Zbl

[10] Chen, W.; Guo, Z.; Xiao, J. Sharp well-posedness for the Benjamin equation, Nonlinear Anal., Volume 74 (2011) no. 17, pp. 6209-6230 | DOI | MR | Zbl

[11] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Sharp global well-posedness for KdV and modified KdV on R and $\mathbb{T}$ , J. Am. Math. Soc., Volume 16 (2003), pp. 705-749 | DOI | MR | Zbl

[12] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Multilinear estimates for periodic KdV equations, and applications, J. Funct. Anal., Volume 211 (2004), pp. 173-218 | DOI | MR | Zbl

[13] Coron, J.-M. Control and Nonlinearity, Amer. Math. Soc., Math. Surveys and Monographs, vol. 136, 2007 | MR | Zbl

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

[15] Coron, J.-M.; Rosier, L. Relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Syst. Estim. Control, Volume 4 (1994), pp. 67-84 | MR | Zbl

[16] Dehman, B.; Lebeau, G.; Zuazua, E. Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. Éc. Norm. Supér., Volume 36 (2003) no. 4, pp. 525-551 | DOI | Numdam | MR | Zbl

[17] Dehman, B.; Gérard, P.; Lebeau, G. Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Mat. Ž., Volume 254 (2006), pp. 729-749 | MR | Zbl

[18] Erdogan, B.; Tzirakis, N. Dispersive Partial Differential Equations Well-Posedness and Applications, Cambridge University Press, 2016

[19] Kappeler, T.; Topalov, P. Global well-posedness of KdV in $H^{-1}(T,R)$ , Duke Math. J., Volume 135 (2006) no. 2, pp. 327-360 | DOI | MR | Zbl

[20] Kenig, C.E.; Ponce, G.; Vega, L. A bilinear estimate with applications to the KdV equation, J. Am. Math. Soc., Volume 9 (1996) no. 2, pp. 573-603 | DOI | MR | Zbl

[21] Kozono, H.; Ogawa, T.; Tanisaka, H. Well-posedness for the Benjamin equations, J. Korean Math. Soc., Volume 38 (2001) no. 6, pp. 1205-1234 | MR | Zbl

[22] Laurent, C. Global controllability and stabilization for the nonlinear Schrödinger equation on an interval, ESAIM Control Optim. Calc. Var., Volume 16 (2010) no. 2, pp. 356-379 | DOI | Numdam | MR | Zbl

[23] Laurent, C. Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3, SIAM J. Math. Anal., Volume 42 (2010) no. 2, pp. 785-832 | DOI | MR | Zbl

[24] Laurent, C.; Rosier, L.; Zhang, B. Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Commun. Partial Differ. Equ., Volume 35 (2010) no. 4, pp. 707-744 | DOI | MR | Zbl

[25] Laurent, C.; Linares, F.; Rosier, L. Control and stabilization of the Benjamin-Ono equation in $L^2(\mathbb{T})$ , Arch. Ration. Mech. Anal., Volume 218 (2015) no. 3, pp. 1531-1575 | DOI | MR

[26] Linares, F.; Ortega, J.H. On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM Control Optim. Calc. Var., Volume 11 (2005) no. 2, pp. 204-218 | DOI | Numdam | MR | Zbl

[27] Lions, J.-L. Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., Volume 30 (1988) no. 1, pp. 1-68 | DOI | MR | Zbl

[28] Linares, F. $L^2$-global well-posedness of the initial value problem associated to the Benjamin equation, J. Differ. Equ., Volume 152 (1999) no. 2, pp. 377-393 | DOI | MR | Zbl

[29] Linares, F.; Rosier, L. Control and stabilization of the Benjamin-Ono equation on a periodic domain, Trans. Am. Math. Soc., Volume 367 (2015) no. 7, pp. 4595-4626 | DOI | MR

[30] Linares, F.; Scialom, M. On generalized Benjamin type equations, Discrete Contin. Dyn. Syst., Volume 12 (2005) no. 1, pp. 161-174 | DOI | MR | Zbl

[31] Menzala, G.P.; Vasconcellos, C.F.; Zuazua, E. Stabilization of the Korteweg-de Vries equation with localized damping, Q. Appl. Math., Volume 60 (2002) no. 1, pp. 111-129 | DOI | MR | Zbl

[32] Panthee, M.; Vielma, F. On the controllability and stabilization of the linearized Benjamin equation on a periodic domain, Nonlinear Anal., Real World Appl., Volume 51 (2020) | DOI | MR

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

[34] Rosier, L.; Zhang, B.-Y. Exact controllability and stabilization of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., Volume 48 (2009) no. 2, pp. 972-992 | DOI | MR | Zbl

[35] Rosier, L.; Zhang, B.-Y. Global stabilization of the generalized Korteweg-de Vries equation, SIAM J. Control Optim., Volume 45 (2006) no. 3, pp. 927-956 | DOI | MR | Zbl

[36] Rosier, L.; Zhang, B.-Y. Control and stabilization of the nonlinear Schrödinger equation on rectangles, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 12, pp. 2293-2347 | DOI | MR | Zbl

[37] Russell, D.L. Controllability and stabilization theory for linear partial differential equations: recent progress and open questions, SIAM Rev., Volume 20 (1978) no. 4, pp. 639-739 | DOI | MR | Zbl

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

[39] Russell, D.L.; Zhang, B. Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Am. Math. Soc., Volume 348 (1996) no. 9, pp. 3643-3672 | DOI | MR | Zbl

[40] Shi, S.; Junfeng, L. Local well-posedness for periodic Benjamin equation with small data, Bound. Value Probl., Volume 2015 (2015) | MR

[41] Tao, T. Multilinear weighted convolution of $L^2$ functions, and applications to nonlinear dispersive equations, Am. J. Math., Volume 123 (2001) no. 5, pp. 839-908 | DOI | MR | Zbl

[42] Tao, T. Nonlinear Dispersive Equations Local and Global Analysis, Conference Board of the Mathematical Sciences, 2005

[43] Tartar, L. Interpolation non lineaire et regularite, J. Funct. Anal., Volume 9 (1972), pp. 469-489 | DOI | MR | Zbl

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