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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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