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doi: 10.3934/mcrf.2021039
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On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Institut Elie Cartan de Lorraine, Université de Lorraine & Inria (Project-Team SPHINX), BP 70 239, F-54506 Vandoeuvre-les-Nancy Cedex, France

* Corresponding author: Julie Valein

Received  May 2019 Revised  February 2021 Early access September 2021

Fund Project: This research was partially funded by the French Grant ANR ISDEEC (ANR-16-CE40-0013) and ODISSE (ANR-19-CE48-0004-01), and by MathAmsud project ICoPS (17-MATH-04).
This article was recruited by Birgit Jacob.

The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.

Citation: Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021039
References:
[1]

L. BaudouinE. Crépeau and J. Valein, Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback, IEEE Trans. Automat. Control, 64 (2019), 1403-1414.  doi: 10.1109/TAC.2018.2849564.  Google Scholar

[2]

J. Bona and R. Winther, The Korteweg–de Vries equation, posed in a quarter-plane, SIAM J. Math. Anal., 14 (1983), 1056-1106.  doi: 10.1137/0514085.  Google Scholar

[3]

E. Cerpa, Control of a Korteweg-de Vries equation: A tutorial, Math. Control Relat. Fields, 4 (2014), 45-99.  doi: 10.3934/mcrf.2014.4.45.  Google Scholar

[4]

E. Cerpa and J.-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.  doi: 10.1109/TAC.2013.2241479.  Google Scholar

[5]

E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 655-668.  doi: 10.3934/dcdsb.2009.11.655.  Google Scholar

[6]

J. ChuJ.-M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg–de Vries equation with critical lengths, J. Differential Equations, 259 (2015), 4045-4085.  doi: 10.1016/j.jde.2015.05.010.  Google Scholar

[7]

T. Colin and M. Gisclon, An initial-boundary value probleme that approximate the quarter-plane problem for the Korteweg-de Vries equation, Nonlinear Anal., 46 (2001), 869-892.  doi: 10.1016/S0362-546X(00)00155-3.  Google Scholar

[8]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), 6 (2004), 367-398.   Google Scholar

[9]

J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl., 102 (2014), 1080-1120.  doi: 10.1016/j.matpur.2014.03.004.  Google Scholar

[10]

J.-M. CoronI. Rivas and S. Xiang, Local exponential stabilization for a class of Korteweg–de Vries equations by means of time-varying feedback laws, Anal. PDE, 10 (2017), 1089-1122.  doi: 10.2140/apde.2017.10.1089.  Google Scholar

[11]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar

[12]

R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar

[13]

W. Kang and E. Fridman, Distributed stabilization of Korteweg–de Vries–Burgers equation in the presence of input delay, Automatica J. IFAC, 100 (2019), 260-273.  doi: 10.1016/j.automatica.2018.11.025.  Google Scholar

[14]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005.  Google Scholar

[15]

S. MarxE. CerpaC. Prieur and V. Andrieu, Global stabilization of a Korteweg-de Vries equation with saturating distributed control, SIAM J. Control Optim., 55 (2017), 1452-1480.  doi: 10.1137/16M1061837.  Google Scholar

[16]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[17]

S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1.  Google Scholar

[18]

S. Nicaise and S. Rebiai, Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback, Port. Math., 68 (2011), 19-39.  doi: 10.4171/PM/1879.  Google Scholar

[19]

S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var., 16 (2010), 420-456.  doi: 10.1051/cocv/2009007.  Google Scholar

[20]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.  Google Scholar

[21]

A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486.  doi: 10.1051/cocv:2005015.  Google Scholar

[22]

A. F. PazotoM. Sepúlveda and O. V. Villagrán, Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping, Numer. Math., 116 (2010), 317-356.  doi: 10.1007/s00211-010-0291-x.  Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

G. Perla MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.  Google Scholar

[25]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33–55 (electronic). doi: 10.1051/cocv:1997102.  Google Scholar

[26]

L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: Recent progresses, J. Syst. Sci. Complex., 22 (2009), 647-682.  doi: 10.1007/s11424-009-9194-2.  Google Scholar

[27]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[28]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[29]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

S. TangJ. ChuP. Shang and J.-M. Coron, Asymptotic stability of a Korteweg–de Vries equation with a two-dimensional center manifold, Adv. Nonlinear Anal., 7 (2018), 497-515.  doi: 10.1515/anona-2016-0097.  Google Scholar

[31]

S. Xiang, Small-time local stabilization for a Korteweg–de Vries equation, Systems Control Lett., 111 (2018), 64-69.  doi: 10.1016/j.sysconle.2017.11.003.  Google Scholar

[32]

B. Y. Zhang, Boundary stabilization of the Korteweg-de Vries equation, in Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena (Vorau, 1993), Internat. Ser. Numer. Math., Birkhäuser, Basel, 118 (1994), 371–389.  Google Scholar

[33]

Z. Zhao, E. Rong and X. Zhao, Existence for Korteweg-de Vries-type equation with delay, Adv. Difference Equ., 2012 (2012), 12 pp. doi: 10.1186/1687-1847-2012-64.  Google Scholar

show all references

References:
[1]

L. BaudouinE. Crépeau and J. Valein, Two approaches for the stabilization of nonlinear KdV equation with boundary time-delay feedback, IEEE Trans. Automat. Control, 64 (2019), 1403-1414.  doi: 10.1109/TAC.2018.2849564.  Google Scholar

[2]

J. Bona and R. Winther, The Korteweg–de Vries equation, posed in a quarter-plane, SIAM J. Math. Anal., 14 (1983), 1056-1106.  doi: 10.1137/0514085.  Google Scholar

[3]

E. Cerpa, Control of a Korteweg-de Vries equation: A tutorial, Math. Control Relat. Fields, 4 (2014), 45-99.  doi: 10.3934/mcrf.2014.4.45.  Google Scholar

[4]

E. Cerpa and J.-M. Coron, Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition, IEEE Trans. Automat. Control, 58 (2013), 1688-1695.  doi: 10.1109/TAC.2013.2241479.  Google Scholar

[5]

E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 655-668.  doi: 10.3934/dcdsb.2009.11.655.  Google Scholar

[6]

J. ChuJ.-M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg–de Vries equation with critical lengths, J. Differential Equations, 259 (2015), 4045-4085.  doi: 10.1016/j.jde.2015.05.010.  Google Scholar

[7]

T. Colin and M. Gisclon, An initial-boundary value probleme that approximate the quarter-plane problem for the Korteweg-de Vries equation, Nonlinear Anal., 46 (2001), 869-892.  doi: 10.1016/S0362-546X(00)00155-3.  Google Scholar

[8]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), 6 (2004), 367-398.   Google Scholar

[9]

J.-M. Coron and Q. Lü, Local rapid stabilization for a Korteweg-de Vries equation with a Neumann boundary control on the right, J. Math. Pures Appl., 102 (2014), 1080-1120.  doi: 10.1016/j.matpur.2014.03.004.  Google Scholar

[10]

J.-M. CoronI. Rivas and S. Xiang, Local exponential stabilization for a class of Korteweg–de Vries equations by means of time-varying feedback laws, Anal. PDE, 10 (2017), 1089-1122.  doi: 10.2140/apde.2017.10.1089.  Google Scholar

[11]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar

[12]

R. DatkoJ. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar

[13]

W. Kang and E. Fridman, Distributed stabilization of Korteweg–de Vries–Burgers equation in the presence of input delay, Automatica J. IFAC, 100 (2019), 260-273.  doi: 10.1016/j.automatica.2018.11.025.  Google Scholar

[14]

M. Krstic and A. Smyshlyaev, Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays, Systems Control Lett., 57 (2008), 750-758.  doi: 10.1016/j.sysconle.2008.02.005.  Google Scholar

[15]

S. MarxE. CerpaC. Prieur and V. Andrieu, Global stabilization of a Korteweg-de Vries equation with saturating distributed control, SIAM J. Control Optim., 55 (2017), 1452-1480.  doi: 10.1137/16M1061837.  Google Scholar

[16]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[17]

S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1.  Google Scholar

[18]

S. Nicaise and S. Rebiai, Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback, Port. Math., 68 (2011), 19-39.  doi: 10.4171/PM/1879.  Google Scholar

[19]

S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM Control Optim. Calc. Var., 16 (2010), 420-456.  doi: 10.1051/cocv/2009007.  Google Scholar

[20]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.  Google Scholar

[21]

A. F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486.  doi: 10.1051/cocv:2005015.  Google Scholar

[22]

A. F. PazotoM. Sepúlveda and O. V. Villagrán, Uniform stabilization of numerical schemes for the critical generalized Korteweg-de Vries equation with damping, Numer. Math., 116 (2010), 317-356.  doi: 10.1007/s00211-010-0291-x.  Google Scholar

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[24]

G. Perla MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.  Google Scholar

[25]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33–55 (electronic). doi: 10.1051/cocv:1997102.  Google Scholar

[26]

L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation: Recent progresses, J. Syst. Sci. Complex., 22 (2009), 647-682.  doi: 10.1007/s11424-009-9194-2.  Google Scholar

[27]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-739.  doi: 10.1137/1020095.  Google Scholar

[28]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[29]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

S. TangJ. ChuP. Shang and J.-M. Coron, Asymptotic stability of a Korteweg–de Vries equation with a two-dimensional center manifold, Adv. Nonlinear Anal., 7 (2018), 497-515.  doi: 10.1515/anona-2016-0097.  Google Scholar

[31]

S. Xiang, Small-time local stabilization for a Korteweg–de Vries equation, Systems Control Lett., 111 (2018), 64-69.  doi: 10.1016/j.sysconle.2017.11.003.  Google Scholar

[32]

B. Y. Zhang, Boundary stabilization of the Korteweg-de Vries equation, in Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena (Vorau, 1993), Internat. Ser. Numer. Math., Birkhäuser, Basel, 118 (1994), 371–389.  Google Scholar

[33]

Z. Zhao, E. Rong and X. Zhao, Existence for Korteweg-de Vries-type equation with delay, Adv. Difference Equ., 2012 (2012), 12 pp. doi: 10.1186/1687-1847-2012-64.  Google Scholar

Figure 1.  Representation of $ t\mapsto \ln(E(t)) $ for different values of $ a $ and $ b $
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