doi: 10.3934/dcdsb.2022032
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On the stabilization of a flexible structure via a nonlinear delayed boundary control

1. 

Kuwait University, Faculty of Science, Department of Mathematics, Safat 13060, Kuwait

2. 

Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

*Corresponding author: Boumediène Chentouf

Received  March 2021 Revised  January 2022 Early access March 2022

The main concern of this article is to investigate the stabilization problem of a flexible structure with one dynamical boundary condition subject to two nonlinearities in the proposed boundary control. Specifically, the first nonlinearity is related to the velocity term, while the second one arises from the delayed term. Despite such a situation, it is shown that the system has a global solution by means of semigroups theory of nonlinear operators. More importantly, we prove that the system energy decays. An estimate of the decay rate of the energy is explicitly provided depending on the type of the nonlinearities.

Citation: Boumediène Chentouf, Baowei Feng. On the stabilization of a flexible structure via a nonlinear delayed boundary control. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022032
References:
[1]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245.

[2]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differ. Eqs., 248 (2010), 1473-1517.  doi: 10.1016/j.jde.2009.12.005.

[3]

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, Control of Partial Differential Equations, 2048 (2012), 1-100.  doi: 10.1007/978-3-642-27893-8_1.

[4]

F. Alabau-Boussouira and K. Ammari, Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system, J. Funct. Anal., 260 (2011), 2424-2450.  doi: 10.1016/j.jfa.2011.01.003.

[5]

F. Alabau-BoussouiraP. Cannarsa and G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55 (2017), 2052-2087.  doi: 10.1137/15M1020538.

[6]

F. Al-MusallamK. Ammari and B. Chentouf, Asymptotic behavior of a 2D overhead crane with input delays in the boundary control, ZAMM Z. Angew. Math. Mech., 98 (2018), 1103-1122.  doi: 10.1002/zamm.201700208.

[7]

K. Ammari, A. Bchatnia and K. E. Mufti, Stabilization of the nonlinear damped wave equation via linear weak observability, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 6, 18 pp. doi: 10.1007/s00030-016-0363-4.

[8]

K. Ammari and B. Chentouf, Further results on the long-time behavior of a 2D overhead crane with a boundary delay: Exponential convergence, Appl. Math. Comput., 365 (2020), 124698, 17 pp. doi: 10.1016/j.amc.2019.124698.

[9]

B. Augner, Well-posedness and stability of infinite-dimensional linear Port-Hamiltonian systems with nonlinear boundary feedback, SIAM J. Control Optim., 57 (2019), 1818-1844.  doi: 10.1137/15M1024901.

[10]

B. d'Andréa-NovelF. Boustany and B. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane, Math. Control. Signals Systems., 7 (1994), 1-22.  doi: 10.1007/BF01211483.

[11]

B. d'Andréa-Novel and J. M. Coron, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 36 (2000), 587-593.  doi: 10.1016/S0005-1098(99)00182-X.

[12]

B. d'Andréa-Novel, I. Moyano and L. Rosier, Finite-time stabilization of an overhead crane with a flexible cable, Math. Control. Signals Systems., 31 (2019), Art. 6, 19 pp. doi: 10.1007/s00498-019-0235-7.

[13]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[14]

A. Benaissa, A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 123514, 19 pp. doi: 10.1063/1.4765046.

[15]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, Mathematics Studies, 1973.

[16]

M. M. CavalcantiV. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.

[17]

B. Chentouf and Z. Han, On the stabilization of an overhead crane system with dynamic and delayed boundary conditions, IEEE Trans. Automat. Control, 65 (2020), 4273-4280. 

[18]

B. Chentouf and S. Mansouri, Exponential decay rate for the energy of a flexible structure with dynamic delayed boundary conditions and a local interior damping, Appl. Math. Lett., 103 (2020), 106185, 7 pp. doi: 10.1016/j.aml.2019.106185.

[19]

B. Chentouf and S. Mansouri, On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only, J. Franklin Inst., 358 (2021), 934-962.  doi: 10.1016/j.jfranklin.2020.10.027.

[20]

B. ChentoufC. Z. Xu and G. Sallet, On the stabilization of a vibrating equation, Nonlinear Anal., 39 (2000), 537-558.  doi: 10.1016/S0362-546X(98)00220-X.

[21]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Inc., New York-Toronto-London, 1955.

[22]

F. Conrad and A. Mifdal, Strong stability of a model of an overhead crane, Control Cybernet., 27 (1998), 363-394. 

[23]

F. ConradG. O'Dowd and F. Z. Saouri, Asymptotic behavior for a model of flexible cable with tip masses, Asymptot. Anal., 30 (2002), 313-330. 

[24]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[25]

C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Func. Anal., 13 (1973), 97-106.  doi: 10.1016/0022-1236(73)90069-4.

[26]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[27]

A. Elharfi, Control design of an overhead crane system from the perspective of stabilizing undesired oscillations, IMA J. Math. Control Info., 28 (2011), 267-278.  doi: 10.1093/imamci/dnr007.

[28]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[29]

A. Elharfi, Exponential stabilization of a class of 1-D hyperbolic PDEs, J. Evol. Equ., 16 (2016), 665-679.  doi: 10.1007/s00028-015-0317-z.

[30]

M. Gueye, Exact boundary controllability of 1-d parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.

[31]

A. Haraux, Systèms dynamique dissipatifs et applications, Masson, Paris, 1991.

[32]

A. HastirF. Califano and H. Zwart, Well-posedness of infinite-dimensional linear systems with nonlinear feedback, Systems Control Lett., 128 (2019), 19-25.  doi: 10.1016/j.sysconle.2019.04.002.

[33]

V. Komornik and E. Zuazua, A direct dethod for boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. 

[34]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. 

[35]

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. 

[36]

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0419-3.

[37]

A. Mifdal, Stabilisation uniforme d'un système hybride, C. R. Acad. Sci. Paris. Série I., 324 (1997), 37-42.  doi: 10.1016/S0764-4442(97)80100-0.

[38]

O. MorgulB. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145.  doi: 10.1109/9.328811.

[39]

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.

[40]

S. NicaiseC. Pignotti and J. Valein, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[41]

B. Rao, Decay estimate of solution for hybrid system of flexible structures, Euro. J. Appl. Math., 4 (1993), 303-319.  doi: 10.1017/S0956792500001133.

[42]

H. Sano, Boundary stabilization of hyperbolic systems related to overhead cranes, IMA J Math Control Info, 25 (2008), 353-366.  doi: 10.1093/imamci/dnm031.

[43]

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

[44]

G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Anal. Hybrid Syst., 1 (2007), 383-397.  doi: 10.1016/j.nahs.2006.07.003.

[45]

G. Q. XuS. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

[46]

W. H. Young, On classes of summable functions and their Fourier series, Proc. Roy. Soc. London Ser. A, 87 (1912), 225-229.  doi: 10.1098/rspa.1912.0076.

[47]

A. Zeidler, Nonlinear Functional Analysis and its Applications, vol. 2, Springer Verlag, New York, 1986.

[48]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-477.  doi: 10.1137/0328025.

show all references

References:
[1]

F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61-105.  doi: 10.1007/s00245.

[2]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differ. Eqs., 248 (2010), 1473-1517.  doi: 10.1016/j.jde.2009.12.005.

[3]

F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equations, Control of Partial Differential Equations, 2048 (2012), 1-100.  doi: 10.1007/978-3-642-27893-8_1.

[4]

F. Alabau-Boussouira and K. Ammari, Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system, J. Funct. Anal., 260 (2011), 2424-2450.  doi: 10.1016/j.jfa.2011.01.003.

[5]

F. Alabau-BoussouiraP. Cannarsa and G. Leugering, Control and stabilization of degenerate wave equations, SIAM J. Control Optim., 55 (2017), 2052-2087.  doi: 10.1137/15M1020538.

[6]

F. Al-MusallamK. Ammari and B. Chentouf, Asymptotic behavior of a 2D overhead crane with input delays in the boundary control, ZAMM Z. Angew. Math. Mech., 98 (2018), 1103-1122.  doi: 10.1002/zamm.201700208.

[7]

K. Ammari, A. Bchatnia and K. E. Mufti, Stabilization of the nonlinear damped wave equation via linear weak observability, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 6, 18 pp. doi: 10.1007/s00030-016-0363-4.

[8]

K. Ammari and B. Chentouf, Further results on the long-time behavior of a 2D overhead crane with a boundary delay: Exponential convergence, Appl. Math. Comput., 365 (2020), 124698, 17 pp. doi: 10.1016/j.amc.2019.124698.

[9]

B. Augner, Well-posedness and stability of infinite-dimensional linear Port-Hamiltonian systems with nonlinear boundary feedback, SIAM J. Control Optim., 57 (2019), 1818-1844.  doi: 10.1137/15M1024901.

[10]

B. d'Andréa-NovelF. Boustany and B. Rao, Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane, Math. Control. Signals Systems., 7 (1994), 1-22.  doi: 10.1007/BF01211483.

[11]

B. d'Andréa-Novel and J. M. Coron, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 36 (2000), 587-593.  doi: 10.1016/S0005-1098(99)00182-X.

[12]

B. d'Andréa-Novel, I. Moyano and L. Rosier, Finite-time stabilization of an overhead crane with a flexible cable, Math. Control. Signals Systems., 31 (2019), Art. 6, 19 pp. doi: 10.1007/s00498-019-0235-7.

[13]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[14]

A. Benaissa, A. Benaissa and S. A. Messaoudi, Global existence and energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks, J. Math. Phys., 53 (2012), 123514, 19 pp. doi: 10.1063/1.4765046.

[15]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland, Mathematics Studies, 1973.

[16]

M. M. CavalcantiV. D. Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differ. Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.

[17]

B. Chentouf and Z. Han, On the stabilization of an overhead crane system with dynamic and delayed boundary conditions, IEEE Trans. Automat. Control, 65 (2020), 4273-4280. 

[18]

B. Chentouf and S. Mansouri, Exponential decay rate for the energy of a flexible structure with dynamic delayed boundary conditions and a local interior damping, Appl. Math. Lett., 103 (2020), 106185, 7 pp. doi: 10.1016/j.aml.2019.106185.

[19]

B. Chentouf and S. Mansouri, On the exponential stabilization of a flexible structure with dynamic delayed boundary conditions via one boundary control only, J. Franklin Inst., 358 (2021), 934-962.  doi: 10.1016/j.jfranklin.2020.10.027.

[20]

B. ChentoufC. Z. Xu and G. Sallet, On the stabilization of a vibrating equation, Nonlinear Anal., 39 (2000), 537-558.  doi: 10.1016/S0362-546X(98)00220-X.

[21]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Inc., New York-Toronto-London, 1955.

[22]

F. Conrad and A. Mifdal, Strong stability of a model of an overhead crane, Control Cybernet., 27 (1998), 363-394. 

[23]

F. ConradG. O'Dowd and F. Z. Saouri, Asymptotic behavior for a model of flexible cable with tip masses, Asymptot. Anal., 30 (2002), 313-330. 

[24]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[25]

C. M. Dafermos and M. Slemrod, Asymptotic behavior of nonlinear contraction semigroups, J. Func. Anal., 13 (1973), 97-106.  doi: 10.1016/0022-1236(73)90069-4.

[26]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[27]

A. Elharfi, Control design of an overhead crane system from the perspective of stabilizing undesired oscillations, IMA J. Math. Control Info., 28 (2011), 267-278.  doi: 10.1093/imamci/dnr007.

[28]

A. Elharfi, Exponential stabilization and motion planning of an overhead crane system, IMA J. Math. Control Info., 34 (2017), 1299-1321.  doi: 10.1093/imamci/dnw026.

[29]

A. Elharfi, Exponential stabilization of a class of 1-D hyperbolic PDEs, J. Evol. Equ., 16 (2016), 665-679.  doi: 10.1007/s00028-015-0317-z.

[30]

M. Gueye, Exact boundary controllability of 1-d parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.

[31]

A. Haraux, Systèms dynamique dissipatifs et applications, Masson, Paris, 1991.

[32]

A. HastirF. Califano and H. Zwart, Well-posedness of infinite-dimensional linear systems with nonlinear feedback, Systems Control Lett., 128 (2019), 19-25.  doi: 10.1016/j.sysconle.2019.04.002.

[33]

V. Komornik and E. Zuazua, A direct dethod for boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. 

[34]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equations, 6 (1993), 507-533. 

[35]

W.-J. Liu and E. Zuazua, Decay rates for dissipative wave equations, Ricerche Mat., 48 (1999), 61-75. 

[36]

Z. H. Luo, B. Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer-Verlag, London, 1999. doi: 10.1007/978-1-4471-0419-3.

[37]

A. Mifdal, Stabilisation uniforme d'un système hybride, C. R. Acad. Sci. Paris. Série I., 324 (1997), 37-42.  doi: 10.1016/S0764-4442(97)80100-0.

[38]

O. MorgulB. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145.  doi: 10.1109/9.328811.

[39]

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.

[40]

S. NicaiseC. Pignotti and J. Valein, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[41]

B. Rao, Decay estimate of solution for hybrid system of flexible structures, Euro. J. Appl. Math., 4 (1993), 303-319.  doi: 10.1017/S0956792500001133.

[42]

H. Sano, Boundary stabilization of hyperbolic systems related to overhead cranes, IMA J Math Control Info, 25 (2008), 353-366.  doi: 10.1093/imamci/dnm031.

[43]

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

[44]

G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Anal. Hybrid Syst., 1 (2007), 383-397.  doi: 10.1016/j.nahs.2006.07.003.

[45]

G. Q. XuS. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

[46]

W. H. Young, On classes of summable functions and their Fourier series, Proc. Roy. Soc. London Ser. A, 87 (1912), 225-229.  doi: 10.1098/rspa.1912.0076.

[47]

A. Zeidler, Nonlinear Functional Analysis and its Applications, vol. 2, Springer Verlag, New York, 1986.

[48]

E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Control Optim., 28 (1990), 466-477.  doi: 10.1137/0328025.

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