Boundary linear stabilization of the modified generalized Korteweg-de Vries-Burgers equation. (English) Zbl 1485.35341
Summary: The linear stabilization problem of the modified generalized Korteweg-de Vries-Burgers equation (MGKdVB) is considered when the spatial variable lies in \([0,1]\). First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in \(L^2 (0,1)\). Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35Q51 | Soliton equations |
| 93C20 | Control/observation systems governed by partial differential equations |
Keywords:
modified generalized Korteweg-de Vries-Burgers equation; well-posedness; exponential stability; boundary controlReferences:
| [1] | Amick, C.J., Bona, J.L., Schonbek, M.E.: Decay of solutions of some nonlinear wave equations. J. Differ. Equ. 81, 1-49 (1989) · Zbl 0689.35081 · doi:10.1016/0022-0396(89)90176-9 |
| [2] | Antar, N.: The Korteweg-de Vries-Burgers hierarchy in fluid-filled elastic tubes. Int. J. Eng. Sci. 40, 1179-1198 (2001) · Zbl 1211.74087 · doi:10.1016/S0020-7225(02)00011-3 |
| [3] | Balogh, A., Gilliam, D.S., Shubov, V.I.: Stationary solutions for a boundary controlled Burgers’ equation. Math. Comput. Model. 33, 21-37 (2001) · Zbl 0967.93051 · doi:10.1016/S0895-7177(00)00226-0 |
| [4] | Balogh, A.; Krstic, M., Global boundary stabilization and regularization of Burgers’ equation, San Diego |
| [5] | Balogh, A., Krstic, M.: Burgers’ equation with nonlinear boundary feedback: \(H1H_1\) stability, well-posedness and simulation. Math. Probl. Eng. 6, 189-200 (2000) · Zbl 1058.93050 |
| [6] | Biagioni, H.A., Gramchev, T.: Multidimensional Kuramoto-Sivashinsky type equations: singular initial data and analytic regularity. Mat. Contemp. 15, 21-42 (1998) · Zbl 0921.35067 |
| [7] | Biler, P.: Large-time behavior of periodic solutions to dissipative equations of Korteweg-de Vries-Burgers type. Bull. Pol. Acad. Sci., Math. 32, 401-405 (1984) · Zbl 0561.35065 |
| [8] | Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, G.: Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries Burgers equation. Appl. Numer. Math. 10, 335-355 (1992) · Zbl 0757.65123 · doi:10.1016/0168-9274(92)90049-J |
| [9] | Bona, J.L., Sun, S.M., Zhang, B.-Y.: A non-homogeneous boundary-value problem for Korteweg-de Vries equation posed on a finite domain. Commun. Partial Differ. Equ. 28, 1391-1436 (2003) · Zbl 1057.35049 · doi:10.1081/PDE-120024373 |
| [10] | Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011) · Zbl 1220.46002 |
| [11] | Bui, A.T.: Initial boundary value problems for the Korteweg-de Vries equation. J. Differ. Equ. 25, 288-300 (1977) · Zbl 0372.35070 · doi:10.1016/0022-0396(77)90046-8 |
| [12] | Burgers, T.M.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech. 1, 171-199 (1948) · doi:10.1016/S0065-2156(08)70100-5 |
| [13] | Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Tata McGraw-Hill, New Delhi (1977) · Zbl 0064.33002 |
| [14] | Cousin, A.T., Larkin, N.A.: Kuramoto-Sivashisky equation in domains with moving boundaries. Port. Math. 59, 336-349 (2002) · Zbl 1008.35065 |
| [15] | Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1989) · Zbl 0661.35001 · doi:10.1017/CBO9781139172059 |
| [16] | Efe, M.O., Ozbay, H.: Low dimensional modeling and Dirichlet boundary controller design for Burgers equation. Int. J. Control 77, 895-906 (2004) · Zbl 1064.93020 · doi:10.1080/00207170412331270532 |
| [17] | Gomes, S.N., Papageorgiou, D.T., Pavliotis, G.A.: Stabilizing nontrivial solutions of the generalized Kuramoto-Sivashinsky equation using feedback and optimal control. IMA J. Appl. Math., 1-37 (2016) |
| [18] | Guo, B.: The existence and nonexistence of a global solution for the initial value problem of generalized Kuramoto-Sivashinsky equations. J. Math. Res. Expo. 11, 57-69 (1991) · Zbl 0938.35167 |
| [19] | Guo, B., Xiang, X.M.: The large time convergence of spectral method for Kuramoto-Sivashinsky equations. J. Comput. Math. 15, 1-13 (1997) · Zbl 0870.65114 |
| [20] | He, J. W.; Glowinski, R.; Gorman, M.; Periaux, J., Some results on the controllability and the stabilization of the Kuramoto-Sivashinsky equation, 571-590 (1998), Paris · Zbl 0912.35021 |
| [21] | Hublov, V. V., On a boundary value problems for the Korteweg-de Vries equation in bounded regions, 137-141 (1979), Novosibirsk · Zbl 0472.35072 |
| [22] | Krstic, M.: On global stabilization of Burgers’ equation by boundary control. Syst. Control Lett. 37, 123-142 (1999) · Zbl 1074.93552 · doi:10.1016/S0167-6911(99)00013-4 |
| [23] | Kuramoto, Y.: Diffusion-induced chaos in reaction systems. Prog. Theor. Phys. Suppl. 64, 346-367 (1978) · doi:10.1143/PTPS.64.346 |
| [24] | Larkin, N.A.: Korteweg-de Vries and Kuramoto-Sivashinsky equations in bounded domains. J. Math. Anal. Appl. 297, 169-185 (2004) · Zbl 1075.35070 · doi:10.1016/j.jmaa.2004.04.053 |
| [25] | Lions, J.L., Magenes, E.: Problèmes aux Limites Non-Homogènes et Applications, vol. 1. Dunod, Paris (1968) · Zbl 0165.10801 |
| [26] | Liu, G., Duan, G.: Stationary wave solution for equations of longitudinal wave in a nonlinear elastic rod. J. Henan Norm. Univ. Nat. Sci. 29, 101-103 (2001) · Zbl 1016.35063 |
| [27] | Messaoudi, S.A., Soufyane, A.: General decay of solutions of a wave equation with a boundary control of memory type. Nonlinear Anal., Real World Appl. 11, 2896-2904 (2010) · Zbl 1194.35064 · doi:10.1016/j.nonrwa.2009.10.013 |
| [28] | Mikhailov, V.P.: Partial Differential Equations. Nauka, Moscow (1976) · Zbl 0342.35052 |
| [29] | Russell, J.S.: Experimental researches into the laws of certain hydrodynamical phenomena that accompany the motion of floating bodies and have not previously been reduced into conformity with the laws of resistance of fluids. Transactions of the Royal Society of London, Edinburgh, XIV 1840, 47-109 |
| [30] | Santos, M.L., Ferreira, J., Pereira, D.C., Raposo, C.A.: Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary. Nonlinear Anal. 54, 959-976 (2003) · Zbl 1032.35140 · doi:10.1016/S0362-546X(03)00121-4 |
| [31] | Santos, M.L., Junior, F.: A boundary condition with memory for Kirchhoff plates equations. Appl. Math. Comput. 148, 475-496 (2004) · Zbl 1046.74028 |
| [32] | Senouf, D.: Dynamics and condensation of complex singularities for Burgers’ equation. SIAM J. Math. Anal. 28, 1457-1489 (1997) · Zbl 0889.35088 · doi:10.1137/S0036141095289373 |
| [33] | Sivashinsky, G.: Nonlinear analysis for hydrodynamic instability in laminar flames. Derivation of basic equations. Acta Astronaut. 4, 1177-1206 (1977) · Zbl 0427.76047 · doi:10.1016/0094-5765(77)90096-0 |
| [34] | Smaoui, N.: Nonlinear boundary control of the generalized Burgers equation. Nonlinear Dyn. 37, 75-86 (2004) · Zbl 1078.76026 · doi:10.1023/B:NODY.0000040023.92220.09 |
| [35] | Smaoui, N.: Controlling the dynamics of Burgers equation with a higher order nonlinearity. Int. J. Math. Math. Sci. 62, 3321-3332 (2004) · Zbl 1070.35072 · doi:10.1155/S0161171204404116 |
| [36] | Smaoui, N.: Boundary and distributed control of the viscous Burgers equation. J. Comput. Appl. Math. 182, 91-104 (2005) · Zbl 1074.93023 · doi:10.1016/j.cam.2004.10.020 |
| [37] | Smaoui, N., Al-Jamal, R.: Boundary control of the generalized Korteweg-de Vries-Burgers equation. Nonlinear Dyn. 51, 439-446 (2008) · Zbl 1170.93018 · doi:10.1007/s11071-007-9222-5 |
| [38] | Smaoui, N., El-Kadri, E., Zribi, M.: Adaptive boundary control of the forced generalized Korteweg-de Vries-Burgers equation. Eur. J. Control 16, 72-84 (2010) · Zbl 1291.93164 · doi:10.3166/ejc.16.72-84 |
| [39] | Smaoui, N., El-Kadri, E., Zribi, M.: Nonlinear boundary control of the unforced generalized Korteweg-de Vries-Burgers equation. Nonlinear Dyn. 60, 561-574 (2010) · Zbl 1194.35389 · doi:10.1007/s11071-009-9615-8 |
| [40] | Tadmor, E.: The well-posedness of the Kuramoto-Sivashinsky equation. SIAM J. Math. Anal. 17, 884-893 (1986) · Zbl 0606.35073 · doi:10.1137/0517063 |
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