On the stability of KdV equation with time-dependent delay on the boundary feedback in presence of saturated source term
DOI:
https://doi.org/10.24193/subbmath.2025.2.12Keywords:
KdV equation, Stability, Saturated source term, Time-varying delayAbstract
The current paper investigate the question of stabilizability of the Korteweg–de Vries equation with time-varying delay on the boundary feedback in the presence of a saturated source term. Under specific assumptions regarding the time-varying delay, we have established that the studied system is well-posed. Moreover, using an appropriate Lyapunov functional, we prove the exponential stability result. Finally, we give some conclusions.
References
[1] Abdallah, C., Dorato, P., Benites-Read, J., Byrne, R., Delayed positive feedback can stabilize oscillatory systems, Proc. Amer. Control Conf., (1993) 3106-3107.
[2] Baudouin, L., Cr epeau, E., Valein, J., Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback, IEEE Trans. Autom. Control, 64(2018), no. 4, 1403-1414.
[3] Capistrano-Filho, R.A., Pazoto, A.F., Rosier, L., Internal controllability of the Korteweg-de Vries equation on a bounded domain, ESAIM: Control Optim. Calc. Var., 21(215), no. 4, 1076-1107.
[4] Cerpa, E., Coron, J.-M., Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Autom. Control, 58(2013), no. 7, 1688-1695.
[5] Coron, J.-M., Cr epeau, E., Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc., 6(2004), no. 3, 367-398.
[6] Coutinho, D.F., Da Silva, J.M.G., Computing estimates of the region of attraction for rational control systems with saturating actuators, IET Control Theory Appl., 4(2010), no. 3, 315-325.
[7] Datko, R., Lagnese, J., Polis, M.P., An example on the e ect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24(1986), no. 1, 152-156.
[8] Fridman, E., Nicaise, S., Valein, J., Stabilization of second order evolution equations with unbounded feedback with time-dependent delay, SIAM J. Control Optim., 48(2010), no. 8, 5028-5052.
[9] Glass, O., Guerrero, S., Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition, Syst. Control Lett., 59(2010), no. 7, 390-395.
[10] Grimm, G., Hat eld, J., Postlethwaite, I., Teel, A.R., Turner, M.C., Zaccarian, L., An-tiwindup for stable linear systems with input saturation: An LMI-based synthesis, IEEE Trans. Autom. Control, 48(2003), no. 9, 1509-1525.
[11] Guzman, P., Marx, S., Cerpa, E., Stabilization of the linear Kuramoto-Sivashinsky equation with a delayed boundary control, IFAC-PapersOnLine, 52(2019), no. 2, 70-75.
[12] Hale, J.K., Lunel, S.M.V., Introduction to Functional Di erential Equations, Springer Sci. Business Media, 2013.
[13] He, Y., Wang, Q.-G., Lin, C., Wu, M., Delay-range-dependent stability for systems with time-varying delay, Automatica, 43(2007), no. 2, 371-376.
[14] Kato, T., Quasi-linear equations of evolution, with applications to partial di erential equations, in Spectral Theory and Di erential Equations: Proceedings of the Symposium held at Dundee, Springer, Scotland, 2006.
[15] Komornik, V., Pignotti, C., Well-posedness and exponential decay estimates for a Korteweg-de Vries-Burgers equation with time-delay, Nonlinear Anal., 191(2020), 111646.
[16] Lasiecka, I., Seidman, T.I., Strong stability of elastic control systems with dissipative saturating feedback, Syst. Control Lett., 48(2003), no. 3-4, 243-252.
[17] Liu, W., Chitour, Y., Sontag, E., On nite-gain stabilizability of linear systems subject to input saturation, SIAM J. Control Optim., 34(1996), no. 4, 1190-1219.
[18] Marx, S., Cerpa, E., Output feedback control of the linear Korteweg-de Vries equation, 53rd IEEE Conf. Decis. Control, 2014, 2083-2087.
[19] Marx, S., Cerpa, E., Prieur, C., Andrieu, V., Stabilization of a linear Korteweg-de Vries equation with a saturated internal control, 2015 European Control Conf. (ECC), 2015, 867-872.
[20] Marx, S., Cerpa, E., Prieur, C., Andrieu, V., Global stabilization of a Korteweg-de Vries equation with saturating distributed control, SIAM J. Control Optim., 55(2017), no. 3, 1452-1480.
[21] Nicaise, S., Pignotti, C., Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45(2006), no. 5, 1561-1585.
[22] Nicaise, S., Valein, J., Fridman, E., Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2(2009), no. 3, 559-581.
[23] Parada, H., Cr epeau, E., Prieur, C., Delayed stabilization of the Korteweg-de Vries equation on a star-shaped network, Math. Control Signals Syst., 34(2022), no. 3, 559-605.
[24] Parada, H., Cr epeau, E., Prieur, C., Global well-posedness of the KDV equation on a star-shaped network and stabilization by saturated controllers, SIAM J. Control Optim., 60(2022), no. 4, 2268-2296.
[25] Parada, H., Timimoun, C., Valein, J., Stability results for the KdV equation with time-varying delay, Syst. Control Lett., 177(2023), 105547.
[26] Park, P., Ko, J.W., Stability and robust stability for systems with a time-varying delay, Automatica, 43(2007), no. 10, 1855-1858.
[27] Pazoto, A.F., Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11(2005), no. 3, 473-486.
[28] Pazy, A., Semigroups of Linear Operators and Applications to Partial Di erential Equations, Springer Sci. Business Media, 1983.
[29] Perla Menzala, G., Vasconcellos, C.F., Zuazua, E., Stabilization of the Korteweg-de Vries equation with localized damping, Q. Appl. Math., 60(2002), no. 1, 111-129.
[30] Pyragas, K., Delayed feedback control of chaos, Philos. Trans. R. Soc. A, 364(2006), no. 1846, 2309-2334.
[31] Rosier, L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2(1997), 33-55.
[32] Russell, D., Zhang, B.-Y., Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Am. Math. Soc., 348(1996), no. 9, 3643-3672.
[33] Slemrod, M., Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control, Math. Control Signals Syst., 2(1989), 265-285.
[34] Taboye, A.M., Laabissi, M., Exponential stabilization of a linear Korteweg-de Vries equation with input saturation, Evol. Equ. Control Theory, 11(2022), no. 5, 1519-1532.
[35] Tang, X.-Y., Liu, S.-J., Liang, Z.-F., Wang, J.-Y., A general nonlocal variable coefficient KdV equation with shifted parity and delayed time reversal, Nonlinear Dyn., 94(2018), 693-702.
[36] Valein, J., On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback, Math. Control Relat. Fields, 12(2022), 667-694.
[37] Zhang, B.-Y., Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Control Optim., 37(1999), no. 2, 543-565.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Transfer of copyright agreement: When the article is accepted for publication, the authors and the representative of the coauthors, hereby agree to transfer to Studia Universitatis Babeș-Bolyai Mathematica all rights, including those pertaining to electronic forms and transmissions, under existing copyright laws, except for the following, which the authors specifically retain: the authors can use the material however they want as long as it fits the NC ND terms of the license. The authors have all rights for reuse according to the license.
