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.
Mathematics Subject Classification (2010): 93B05, 35R02, 93C20.
Received 30 March 2024; Accepted 22 May 2024.
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., Crépeau, 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., Crépeau, 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 effect 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., Hatfield, J., Postlethwaite, I., Teel, A.R., Turner, M.C., Zaccarian, L., Antiwindup 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 Differential 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 differential equations, in Spectral Theory and Differential 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 finite-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., Crépeau, 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., Crépeau, 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 Differential 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.
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