# American Institute of Mathematical Sciences

September  2019, 9(3): 453-494. doi: 10.3934/mcrf.2019021

## Controllability for a string with attached masses and Riesz bases for asymmetric spaces

 1 Department of Mathematics and Statistics, University of Alaska at Fairbanks, Fairbanks, AK 99775, USA 2 Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA

* Corresponding author

Received  April 2017 Revised  September 2018 Published  April 2019

Fund Project: The research of Sergei Avdonin was supported in part by the National Science Foundation, grant DMS 1411564 and by the Ministry of Education and Science of Republic of Kazakhstan, grant no. AP05136197.

We consider the problem of boundary control for a vibrating string with $N$ interior point masses. We assume the control of Dirichlet, or Neumann, or mixed type is at the left end, and the string is fixed at the right end. Singularities in waves are "smoothed" out to one order as they cross a point mass. We characterize the reachable set for an $L^2$ control. The control problem is reduced to a moment problem, which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results. The results are sharp with respect to both the regularity of the solution and with respect to time. The eigenfunctions of the associated Sturm--Liouville problem are used to construct Riesz bases for a family of asymmetric spaces that include the sets of reachable positions and velocities.

Citation: Sergei Avdonin, Julian Edward. Controllability for a string with attached masses and Riesz bases for asymmetric spaces. Mathematical Control and Related Fields, 2019, 9 (3) : 453-494. doi: 10.3934/mcrf.2019021
##### References:
 [1] F. Al-Musallam, S. A. Avdonin, N. Avdonina and J. Edward, Control and inverse problems for networks of vibrating strings with attached masses, Nanosystems: Physics, Chemistry, and Mathematics, 7 (2016), 835-841. [2] S. A. Avdonin, On the question of Riesz bases of exponential functions in $L^2$, Vestnik Leningrad Univ. Math., 7 (1979), 203-211. [3] S. A. Avdonin, N. Avdonina and J. Edward, Boundary inverse problems for networks of vibrating strings with attached masses, Proceedings of Dynamic Systems and Applications, 7 (2016), 41-44. [4] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Matrix inverse problem for the equation $u_tt - u_xx + Q(x)u = 0$, Math. USSR Sbornik, 7 (1992), 287-310.  doi: 10.1070/SM1992v072n02ABEH002141. [5] S. A. Avdonin, A. Choque and L. de Teresa, Exact boundary controllability of coupled hyperbolic equations, Int. J. Appl. Math. Comp. Sci., 23 (2013), 701-709.  doi: 10.2478/amcs-2013-0052. [6] S. A. Avdonin and J. Edward, Exact controllability for string with attached masses, SIAM J. Optim. Cont., 56 (2018), 945-980.  doi: 10.1137/15M1029333. [7] S. A. Avdonin and J. Edward, Spectral Clusters, Asymmetric Spaces, and Boundary Control for Schrödinger Equation with Strong Singularities, to be published in Operator Theory: Advances and Applications. [8] S. A. Avdonin and J. Edward,, work in progress. [9] S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, London, Melbourne, 1995. [10] S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2001), 339-351. [11] S. A. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schrödinger equation by the Boundary Control method, Inverse Problems, 18 (2002), 349-361.  doi: 10.1088/0266-5611/18/2/304. [12] S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21.  doi: 10.3934/ipi.2008.2.1. [13] S. A. Avdonin and V. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. doi: 10.1088/0266-5611/26/4/045009. [14] S. A. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of divided differences, International Journal of Applied Math. and Computer Science, 11 (2001), 803-820. [15] S. A. Avdonin, J. Park and L. de Teresa, Controllability of coupled hyperbolic equations in asymmetric spaces, submitted. [16] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956. [17] M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14 (2006), 29-46.  doi: 10.1515/156939406776237474. [18] J. Ben Amara and E. Beldi, Boundary controllability of two vibrating strings connected by interior point mass with variable coefficients, preprint. arXiv: 1706.04246 [19] J. Ben Amara and E. Beldi, Neumann boundary controllability of two vibrating strings connected by a point mass with variable coefficients, preprint. [20] C. Castro, Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass, ESAIM: Control, Optimization and Calculus of Variations, 2 (1997), 231-280.  doi: 10.1051/cocv:1997108. [21] C. Castro and E. Zuazua, Une remarque sur les séries de Fourier non-harmoniques et son application â la contrôlabilité des cordes avec densité singulière, C. R. Acad. Sci. Paris Ser. I. Math., 323 (1996), 365-370. [22] C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass, SIAM J. Control and Optimization, 36 (1998), 1576-1595.  doi: 10.1137/S0363012997316378. [23] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II, Interscience Publishers, New York, London, and Sydney, 1962. [24] R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, in Mathematiques and Applications (Berlin), 50. Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3. [25] S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data, Discrete and Continuous Dyn. Systems, 14 (2010), 1375-1401.  doi: 10.3934/dcdsb.2010.14.1375. [26] Z. J. Han and G. Q. Xu, Output feedback stabilization of a tree-shaped network of vibrating strings with non-collocated observation, Internat. J. Control, 84 (2011), 458-475.  doi: 10.1080/00207179.2011.561441. [27] S. Hansen, Exact Boundary Controllability of a Schrödinger Equation with an Internal Point Mass, American Control Conference (ACC), 2017, IEEE. doi: 10.23919/ACC.2017.7963538. [28] 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. [29] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. [30] B. Ja. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1964. [31] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis, and Control of Dynamical Elastic Multilink Structures, Birkhauser, Basel, 1994. doi: 10.1007/978-1-4612-0273-8. [32] J. Martinez, Modeling and Controllability of a Heat Equation with a Point Mass, Ph.D.Thesis, Iowa State University. 2015. 94 pp. ISBN: 978-1339-45983-7. [33] D. Mercier and V. Regnier, Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses, Collect. Math., 60 (2009), 307-334.  doi: 10.1007/BF03191374. [34] H. Mounier, J. Rudolph, M. Fliess and P. Rouchon, Tracking control of a vibrating string with an interior mass viewed as delay system., ESAIM Control Optim. Calc. Var., 3 (1998), 315-321.  doi: 10.1051/cocv:1998112. [35] A. A. Samarski and A. N. Tikhonov, Equations of Mathematical Physics, Dover Publications, N.Y. 1990. [36] D. Ullrich, Divided differences and systems of nonharmonic Fourier series, Proc. AMS, 80 (1980), 47-57.  doi: 10.1090/S0002-9939-1980-0574507-8.

show all references

##### References:
 [1] F. Al-Musallam, S. A. Avdonin, N. Avdonina and J. Edward, Control and inverse problems for networks of vibrating strings with attached masses, Nanosystems: Physics, Chemistry, and Mathematics, 7 (2016), 835-841. [2] S. A. Avdonin, On the question of Riesz bases of exponential functions in $L^2$, Vestnik Leningrad Univ. Math., 7 (1979), 203-211. [3] S. A. Avdonin, N. Avdonina and J. Edward, Boundary inverse problems for networks of vibrating strings with attached masses, Proceedings of Dynamic Systems and Applications, 7 (2016), 41-44. [4] S. A. Avdonin, M. I. Belishev and S. A. Ivanov, Matrix inverse problem for the equation $u_tt - u_xx + Q(x)u = 0$, Math. USSR Sbornik, 7 (1992), 287-310.  doi: 10.1070/SM1992v072n02ABEH002141. [5] S. A. Avdonin, A. Choque and L. de Teresa, Exact boundary controllability of coupled hyperbolic equations, Int. J. Appl. Math. Comp. Sci., 23 (2013), 701-709.  doi: 10.2478/amcs-2013-0052. [6] S. A. Avdonin and J. Edward, Exact controllability for string with attached masses, SIAM J. Optim. Cont., 56 (2018), 945-980.  doi: 10.1137/15M1029333. [7] S. A. Avdonin and J. Edward, Spectral Clusters, Asymmetric Spaces, and Boundary Control for Schrödinger Equation with Strong Singularities, to be published in Operator Theory: Advances and Applications. [8] S. A. Avdonin and J. Edward,, work in progress. [9] S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, London, Melbourne, 1995. [10] S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2001), 339-351. [11] S. A. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schrödinger equation by the Boundary Control method, Inverse Problems, 18 (2002), 349-361.  doi: 10.1088/0266-5611/18/2/304. [12] S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging, 2 (2008), 1-21.  doi: 10.3934/ipi.2008.2.1. [13] S. A. Avdonin and V. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. doi: 10.1088/0266-5611/26/4/045009. [14] S. A. Avdonin and W. Moran, Ingham type inequalities and Riesz bases of divided differences, International Journal of Applied Math. and Computer Science, 11 (2001), 803-820. [15] S. A. Avdonin, J. Park and L. de Teresa, Controllability of coupled hyperbolic equations in asymmetric spaces, submitted. [16] C. Baiocchi, V. Komornik and P. Loreti, Ingham-Beurling type theorems with weakened gap conditions, Acta Math. Hungar., 97 (2002), 55-95.  doi: 10.1023/A:1020806811956. [17] M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inv. Ill-Posed Problems, 14 (2006), 29-46.  doi: 10.1515/156939406776237474. [18] J. Ben Amara and E. Beldi, Boundary controllability of two vibrating strings connected by interior point mass with variable coefficients, preprint. arXiv: 1706.04246 [19] J. Ben Amara and E. Beldi, Neumann boundary controllability of two vibrating strings connected by a point mass with variable coefficients, preprint. [20] C. Castro, Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass, ESAIM: Control, Optimization and Calculus of Variations, 2 (1997), 231-280.  doi: 10.1051/cocv:1997108. [21] C. Castro and E. Zuazua, Une remarque sur les séries de Fourier non-harmoniques et son application â la contrôlabilité des cordes avec densité singulière, C. R. Acad. Sci. Paris Ser. I. Math., 323 (1996), 365-370. [22] C. Castro and E. Zuazua, Boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass, SIAM J. Control and Optimization, 36 (1998), 1576-1595.  doi: 10.1137/S0363012997316378. [23] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II, Interscience Publishers, New York, London, and Sydney, 1962. [24] R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, in Mathematiques and Applications (Berlin), 50. Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3. [25] S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data, Discrete and Continuous Dyn. Systems, 14 (2010), 1375-1401.  doi: 10.3934/dcdsb.2010.14.1375. [26] Z. J. Han and G. Q. Xu, Output feedback stabilization of a tree-shaped network of vibrating strings with non-collocated observation, Internat. J. Control, 84 (2011), 458-475.  doi: 10.1080/00207179.2011.561441. [27] S. Hansen, Exact Boundary Controllability of a Schrödinger Equation with an Internal Point Mass, American Control Conference (ACC), 2017, IEEE. doi: 10.23919/ACC.2017.7963538. [28] 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. [29] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. [30] B. Ja. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1964. [31] J. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis, and Control of Dynamical Elastic Multilink Structures, Birkhauser, Basel, 1994. doi: 10.1007/978-1-4612-0273-8. [32] J. Martinez, Modeling and Controllability of a Heat Equation with a Point Mass, Ph.D.Thesis, Iowa State University. 2015. 94 pp. ISBN: 978-1339-45983-7. [33] D. Mercier and V. Regnier, Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses, Collect. Math., 60 (2009), 307-334.  doi: 10.1007/BF03191374. [34] H. Mounier, J. Rudolph, M. Fliess and P. Rouchon, Tracking control of a vibrating string with an interior mass viewed as delay system., ESAIM Control Optim. Calc. Var., 3 (1998), 315-321.  doi: 10.1051/cocv:1998112. [35] A. A. Samarski and A. N. Tikhonov, Equations of Mathematical Physics, Dover Publications, N.Y. 1990. [36] D. Ullrich, Divided differences and systems of nonharmonic Fourier series, Proc. AMS, 80 (1980), 47-57.  doi: 10.1090/S0002-9939-1980-0574507-8.
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