# 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 & 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.   Google Scholar [2] S. A. Avdonin, On the question of Riesz bases of exponential functions in $L^2$, Vestnik Leningrad Univ. Math., 7 (1979), 203-211.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [8] S. A. Avdonin and J. Edward,, work in progress. Google Scholar [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.   Google Scholar [10] S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2001), 339-351.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [15] S. A. Avdonin, J. Park and L. de Teresa, Controllability of coupled hyperbolic equations in asymmetric spaces, submitted. Google Scholar [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.  Google Scholar [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.  Google Scholar [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 Google Scholar [19] J. Ben Amara and E. Beldi, Neumann boundary controllability of two vibrating strings connected by a point mass with variable coefficients, preprint. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [23] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II, Interscience Publishers, New York, London, and Sydney, 1962.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.  Google Scholar [30] B. Ja. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] A. A. Samarski and A. N. Tikhonov, Equations of Mathematical Physics, Dover Publications, N.Y. 1990.  Google Scholar [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.  Google Scholar

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.   Google Scholar [2] S. A. Avdonin, On the question of Riesz bases of exponential functions in $L^2$, Vestnik Leningrad Univ. Math., 7 (1979), 203-211.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [8] S. A. Avdonin and J. Edward,, work in progress. Google Scholar [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.   Google Scholar [10] S. A. Avdonin and S. A. Ivanov, Exponential Riesz bases of subspaces and divided differences, St. Petersburg Mathematical Journal, 13 (2001), 339-351.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [15] S. A. Avdonin, J. Park and L. de Teresa, Controllability of coupled hyperbolic equations in asymmetric spaces, submitted. Google Scholar [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.  Google Scholar [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.  Google Scholar [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 Google Scholar [19] J. Ben Amara and E. Beldi, Neumann boundary controllability of two vibrating strings connected by a point mass with variable coefficients, preprint. Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [23] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volume II, Interscience Publishers, New York, London, and Sydney, 1962.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944.  Google Scholar [30] B. Ja. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, RI, 1964.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [35] A. A. Samarski and A. N. Tikhonov, Equations of Mathematical Physics, Dover Publications, N.Y. 1990.  Google Scholar [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.  Google Scholar
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