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May  2022, 15(5): 1103-1125. doi: 10.3934/dcdss.2021139

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

University of Tunis El Manar, Faculty of Sciences of Tunis, Tunisia

Received  July 2021 Revised  September 2021 Published  May 2022 Early access  November 2021

This study is concerned with the pointwise stabilization for a star-shaped network of $ N $ variable coefficients strings connected at the common node by a point mass and subject to boundary feedback dampings at all extreme nodes. It is shown that the closed-loop system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. As a consequence, the spectrum-determined growth condition fulfills. In the meanwhile, the asymptotic expression of the spectrum is presented, and the exponential stability of the system is obtained by giving the optimal decay rate. We prove also that a phenomenon of lack of uniform stability occurs in the absence of damper at one extreme node. This paper reconfirmed the main stability results given by Hansen and Zuazua [SIAM J. Control Optim., 33 (1995), 1357-1391] in a very particular case.

Citation: Walid Boughamda. Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1103-1125. doi: 10.3934/dcdss.2021139
References:
[1]

N. Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space, vol. 9, 10 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1981.

[2]

K. AmmariA. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240. 

[3]

K. AmmariA. Henrot and M. Tucsnak, Optimal location of the actuator for the pointwise stabilization of a string, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 330 (2000), 275-280.  doi: 10.1016/S0764-4442(00)00113-0.

[4]

K. AmmariM. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Contin. Syst., 11 (2005), 177-193.  doi: 10.1007/s10883-005-4169-7.

[5]

K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings, Appl. Maths., 52 (2007), 327-343.  doi: 10.1007/s10492-007-0018-1.

[6]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differ. Integr. Equ., 17 (2004), 1395-1410. 

[7]

K. AmmariZ. Liu and F. Shel, Stability of the wave equations on a tree with local Kelvin–Voigt damping, Semigroup Forum, 100 (2020), 364-382.  doi: 10.1007/s00233-019-10064-7.

[8]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19.  doi: 10.3934/eect.2015.4.1.

[9]

K. AmmariD. Mercier and V. Régnier, Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications, J. Differ. Equ., 259 (2015), 6923-6959.  doi: 10.1016/j.jde.2015.08.017.

[10]

K. AmmariF. Shel and M. Vanninathan, Feedback stabilization of a simplified model of fluid-structure interaction on a tree, Asymptotic Analysis, 103 (2017), 33-55.  doi: 10.3233/ASY-171418.

[11]

R. AsselM. Jellouli and M. Khenissi, Optimal decay rate for the local energy of a unbounded network, J. Differ. Equ., 261 (2016), 4030-4054.  doi: 10.1016/j.jde.2016.06.016.

[12]

T. K. AugustinM. E. Patrice and T. M. Mathurin, Stabilization and Riesz basis property for an overhead crane model with feedback in velocity and rotating velocity, Journal of Nonlinear Analysis and Application, 2014 (2014), 1-14.  doi: 10.5899/2014/jnaa-00184.

[13]

S. Avdonin abd J. Edward, Exact controllability for string with attached masses, SIAM J. Control Optim., 56 (2018), 945-980.  doi: 10.1137/15M1029333.

[14]

J. Ben Amara and E. Beldi, Boundary controllability of two vibrating strings connected by a point mass with variable coefficients, SIAM J. Control Optim., 57 (2019), 3360-3387.  doi: 10.1137/16M1100496.

[15]

J. Ben Amara and W. Boughamda, Exponential stability of two strings under joint damping with variable coefficients, Syst. Cont. Lett., 141 (2020), 104709.  doi: 10.1016/j.sysconle.2020.104709.

[16]

J. Ben Amara and W. Boughamda, Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients, Math. Meth. Appl. Sci., 43 (2020), 2322-2336.  doi: 10.1002/mma.6043.

[17]

W. Boughamda, On the pointwise stability of a tree-shaped network of variable coefficients strings under joint damping, preprint.

[18]

Y. ChenZ. HanG. Xu and D. Liu, Exponential stability of string system with variable coefficients under non-collocated feedback controls, Asian Journal of Control, 13 (2011), 148-163.  doi: 10.1002/asjc.255.

[19]

J. Conway, Functions of One Complex Variable I, $2^{nd}$ edition, Graduate Texts in Mathematics, 11, Springer-Verlag, New York-Berlin, 1978. doi: 10.1007/978-1-4612-6313-5.

[20]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995. doi: 10.1007/978-1-4612-4224-6.

[21]

M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1.

[22]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monogr. 18, AMS, Providence, RI. 1969.

[23]

P. Grabowski, Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory, Opuscula Mathematica, 26 (2006), 45-97. 

[24]

B. Z. Guo, On the boundary control of a hybrid system with variable coefficients, Journal of Optimization Theory and Application, 114 (2002), 373-395.  doi: 10.1023/A:1016039819069.

[25]

B. Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 (2001), 1736-1747.  doi: 10.1137/S0363012999354880.

[26]

B. Z. Guo and J. M. Wang, Control of Wave and Beam PDEs: The Riesz Basis Approach, 596 Springer-Verlag, Cham, 2019. doi: 10.1007/978-3-030-12481-6.

[27]

B. Z. Guo and G. Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, Journal of Functional Analysis, 231 (2006), 245-268.  doi: 10.1016/j.jfa.2005.02.006.

[28]

Y. N. Guo and G. Q. Xu, Exponential stabilisation of a tree-shaped network of strings with variable coefficients, Glasgow Math. J., 53 (2011), 481-499.  doi: 10.1017/S0017089511000085.

[29]

B. Z. Guo and R. Yu, The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control, IMA Journal of Mathematical Control and Information, 18 (2001), 241-251.  doi: 10.1093/imamci/18.2.241.

[30]

Z. J. Han and G. Q. Xu, Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type, Discrete Continuous Dynam. Systems - B, 17 (2012), 57-77.  doi: 10.3934/dcdsb.2012.17.57.

[31]

Z. J. Han and G. Q. Xu, Dynamical behavior of a hybrid system of nonhomogeneous Timoshenko beam with partial non-collocated inputs, Journal of Dynamical and Control Systems, 17 (2011), 77-121.  doi: 10.1007/s10883-011-9111-6.

[32]

S. Hansen and E. Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM J. Control Optim., 33 (1995), 1357-1391.  doi: 10.1137/S0363012993248347.

[33]

E. B. Lee and Y. C. You, Stabilization of a hybrid (string/point mass) system, Proc. Fifth Int. Conf. Syst. Eng., (Dayton, Ohio, EUA), (1987).

[34]

E. B. Lee and Y. You, Stabilization of a vibrating string linked by point masses, Control of Boundaries and Stabilization, Lecture Notes in Control and Information Sciences, 125 (1989), 177-198.  doi: 10.1007/BFb0043361.

[35]

B. M. Levitan and I. C. Sargsjan, Introduction to Spectral Theory, AMS, 1975.

[36]

W. Littman and S. W. Taylor, Boundary feedback stabilization of a vibrating string with an interior point mass, Nonlinear Problems in Mathematical Physics and Related Topic I, in: Int. Math. Ser., 1 (2002), 271-287.  doi: 10.1007/978-1-4615-0777-2_16.

[37]

K. S. LiuF. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM J. Appl. Math., 49 (1989), 1694-1707.  doi: 10.1137/0149102.

[38]

Y. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.

[39]

A. Mifdal, Stabilisation uniforme d'un système hybride, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 324 (1997), 37-42.  doi: 10.1016/S0764-4442(97)80100-0.

[40]

Ö. MorgülB. P. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Transactions on Automatic Control, 39 (1994), 2140-2145.  doi: 10.1109/9.328811.

[41]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[42]

A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions, Journal of Mathematical Sciences, 33 (1986), 1311-1342.  doi: 10.1007/BF01084754.

[43]

G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim., 42 (2003), 966-984.  doi: 10.1137/S0363012901400081.

[44]

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

[45]

G. Q. Xu and S. Yung, The expansion of semigroup and a Riesz basis criterion, J. Differ. Equ., 210 (2005), 1-24.  doi: 10.1016/j.jde.2004.09.015.

[46] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, 1980. 

show all references

References:
[1]

N. Akhiezer and I. Glazman, Theory of Linear Operators in Hilbert Space, vol. 9, 10 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1981.

[2]

K. AmmariA. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240. 

[3]

K. AmmariA. Henrot and M. Tucsnak, Optimal location of the actuator for the pointwise stabilization of a string, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 330 (2000), 275-280.  doi: 10.1016/S0764-4442(00)00113-0.

[4]

K. AmmariM. Jellouli and M. Khenissi, Stabilization of generic trees of strings, J. Dyn. Contin. Syst., 11 (2005), 177-193.  doi: 10.1007/s10883-005-4169-7.

[5]

K. Ammari and M. Jellouli, Remark in stabilization of tree-shaped networks of strings, Appl. Maths., 52 (2007), 327-343.  doi: 10.1007/s10492-007-0018-1.

[6]

K. Ammari and M. Jellouli, Stabilization of star-shaped networks of strings, Differ. Integr. Equ., 17 (2004), 1395-1410. 

[7]

K. AmmariZ. Liu and F. Shel, Stability of the wave equations on a tree with local Kelvin–Voigt damping, Semigroup Forum, 100 (2020), 364-382.  doi: 10.1007/s00233-019-10064-7.

[8]

K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19.  doi: 10.3934/eect.2015.4.1.

[9]

K. AmmariD. Mercier and V. Régnier, Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications, J. Differ. Equ., 259 (2015), 6923-6959.  doi: 10.1016/j.jde.2015.08.017.

[10]

K. AmmariF. Shel and M. Vanninathan, Feedback stabilization of a simplified model of fluid-structure interaction on a tree, Asymptotic Analysis, 103 (2017), 33-55.  doi: 10.3233/ASY-171418.

[11]

R. AsselM. Jellouli and M. Khenissi, Optimal decay rate for the local energy of a unbounded network, J. Differ. Equ., 261 (2016), 4030-4054.  doi: 10.1016/j.jde.2016.06.016.

[12]

T. K. AugustinM. E. Patrice and T. M. Mathurin, Stabilization and Riesz basis property for an overhead crane model with feedback in velocity and rotating velocity, Journal of Nonlinear Analysis and Application, 2014 (2014), 1-14.  doi: 10.5899/2014/jnaa-00184.

[13]

S. Avdonin abd J. Edward, Exact controllability for string with attached masses, SIAM J. Control Optim., 56 (2018), 945-980.  doi: 10.1137/15M1029333.

[14]

J. Ben Amara and E. Beldi, Boundary controllability of two vibrating strings connected by a point mass with variable coefficients, SIAM J. Control Optim., 57 (2019), 3360-3387.  doi: 10.1137/16M1100496.

[15]

J. Ben Amara and W. Boughamda, Exponential stability of two strings under joint damping with variable coefficients, Syst. Cont. Lett., 141 (2020), 104709.  doi: 10.1016/j.sysconle.2020.104709.

[16]

J. Ben Amara and W. Boughamda, Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients, Math. Meth. Appl. Sci., 43 (2020), 2322-2336.  doi: 10.1002/mma.6043.

[17]

W. Boughamda, On the pointwise stability of a tree-shaped network of variable coefficients strings under joint damping, preprint.

[18]

Y. ChenZ. HanG. Xu and D. Liu, Exponential stability of string system with variable coefficients under non-collocated feedback controls, Asian Journal of Control, 13 (2011), 148-163.  doi: 10.1002/asjc.255.

[19]

J. Conway, Functions of One Complex Variable I, $2^{nd}$ edition, Graduate Texts in Mathematics, 11, Springer-Verlag, New York-Berlin, 1978. doi: 10.1007/978-1-4612-6313-5.

[20]

R. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, vol. 21 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995. doi: 10.1007/978-1-4612-4224-6.

[21]

M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58016-1.

[22]

I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monogr. 18, AMS, Providence, RI. 1969.

[23]

P. Grabowski, Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory, Opuscula Mathematica, 26 (2006), 45-97. 

[24]

B. Z. Guo, On the boundary control of a hybrid system with variable coefficients, Journal of Optimization Theory and Application, 114 (2002), 373-395.  doi: 10.1023/A:1016039819069.

[25]

B. Z. Guo, Riesz basis approach to the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 39 (2001), 1736-1747.  doi: 10.1137/S0363012999354880.

[26]

B. Z. Guo and J. M. Wang, Control of Wave and Beam PDEs: The Riesz Basis Approach, 596 Springer-Verlag, Cham, 2019. doi: 10.1007/978-3-030-12481-6.

[27]

B. Z. Guo and G. Q. Xu, Expansion of solution in terms of generalized eigenfunctions for a hyperbolic system with static boundary condition, Journal of Functional Analysis, 231 (2006), 245-268.  doi: 10.1016/j.jfa.2005.02.006.

[28]

Y. N. Guo and G. Q. Xu, Exponential stabilisation of a tree-shaped network of strings with variable coefficients, Glasgow Math. J., 53 (2011), 481-499.  doi: 10.1017/S0017089511000085.

[29]

B. Z. Guo and R. Yu, The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control, IMA Journal of Mathematical Control and Information, 18 (2001), 241-251.  doi: 10.1093/imamci/18.2.241.

[30]

Z. J. Han and G. Q. Xu, Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type, Discrete Continuous Dynam. Systems - B, 17 (2012), 57-77.  doi: 10.3934/dcdsb.2012.17.57.

[31]

Z. J. Han and G. Q. Xu, Dynamical behavior of a hybrid system of nonhomogeneous Timoshenko beam with partial non-collocated inputs, Journal of Dynamical and Control Systems, 17 (2011), 77-121.  doi: 10.1007/s10883-011-9111-6.

[32]

S. Hansen and E. Zuazua, Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM J. Control Optim., 33 (1995), 1357-1391.  doi: 10.1137/S0363012993248347.

[33]

E. B. Lee and Y. C. You, Stabilization of a hybrid (string/point mass) system, Proc. Fifth Int. Conf. Syst. Eng., (Dayton, Ohio, EUA), (1987).

[34]

E. B. Lee and Y. You, Stabilization of a vibrating string linked by point masses, Control of Boundaries and Stabilization, Lecture Notes in Control and Information Sciences, 125 (1989), 177-198.  doi: 10.1007/BFb0043361.

[35]

B. M. Levitan and I. C. Sargsjan, Introduction to Spectral Theory, AMS, 1975.

[36]

W. Littman and S. W. Taylor, Boundary feedback stabilization of a vibrating string with an interior point mass, Nonlinear Problems in Mathematical Physics and Related Topic I, in: Int. Math. Ser., 1 (2002), 271-287.  doi: 10.1007/978-1-4615-0777-2_16.

[37]

K. S. LiuF. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM J. Appl. Math., 49 (1989), 1694-1707.  doi: 10.1137/0149102.

[38]

Y. I. Lyubich and V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88 (1988), 37-42.  doi: 10.4064/sm-88-1-37-42.

[39]

A. Mifdal, Stabilisation uniforme d'un système hybride, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 324 (1997), 37-42.  doi: 10.1016/S0764-4442(97)80100-0.

[40]

Ö. MorgülB. P. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Transactions on Automatic Control, 39 (1994), 2140-2145.  doi: 10.1109/9.328811.

[41]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1.

[42]

A. A. Shkalikov, Boundary problems for ordinary differential equations with parameter in the boundary conditions, Journal of Mathematical Sciences, 33 (1986), 1311-1342.  doi: 10.1007/BF01084754.

[43]

G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim., 42 (2003), 966-984.  doi: 10.1137/S0363012901400081.

[44]

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

[45]

G. Q. Xu and S. Yung, The expansion of semigroup and a Riesz basis criterion, J. Differ. Equ., 210 (2005), 1-24.  doi: 10.1016/j.jde.2004.09.015.

[46] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, 1980. 
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