American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Some worst-case datasets of deterministic first-order methods for solving binary logistic regression
  • IPI Home
  • This Issue
  • Next Article
    A parallel operator splitting algorithm for solving constrained total-variation retinex
doi: 10.3934/ipi.2020063

Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex

Lu Yang 1, , Guangsheng Wei 1,2, and  Vyacheslav Pivovarchik 1,2,,

1. 

College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China
2. 

South Ukrainian national Pedagogical University, Staroprtofrankovskaya str., 26, Odessa 65020, Ukraine

* Corresponding author: Vyacheslav Pivovarchik

Received  May 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported in part by NNSF grant 11971284

A spectral problem occurring in description of small transverse vibrations of a star graph of Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these vertices are clamped. One vertex (the root) can move with damping in the direction orthogonal to the equilibrium position of the strings. We describe the spectrum of such spectral problem. The corresponding inverse problem lies in recovering the values of point masses and the lengths of the intervals between the masses using the spectrum and some other parameters. We propose conditions on a sequence of complex numbers and a collection of real numbers to be the spectrum of a problem we consider and the lengths of the edges, correspondingly.

Keywords: star graph, Stieltjes string, inverse problems, $ S_0 $-functions, branching continued fraction, damped vibrations.
Mathematics Subject Classification: Primary: 34B45; Secondary: 30B70, 05C50, 15A18.
Citation: Lu Yang, Guangsheng Wei, Vyacheslav Pivovarchik. Direct and inverse spectral problems for a star graph of Stieltjes strings damped at a pendant vertex. Inverse Problems & Imaging, doi: 10.3934/ipi.2020063
References:
[1]

O. Boyko, O. Martynyuk and V. Pivovarchik, On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph, Methods of Funct. Anal. Topology, 25 (2019), 104-117.   Google Scholar

[2]

J. Genin and J. S. Maybee, Mechanical vibrations trees, J. Math. Anal. Appl., 45 (1974), 746-763.  doi: 10.1016/0022-247X(74)90065-1.  Google Scholar

[3]

G. Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.  Google Scholar

[4]

G. Gladwell, Matrix inverse eigenvalue problems, Dynamical Inverse Problems: Theory and Application, CISM Courses and Lect., SpringerWienNewYork, Vienna, 529 (2011), 1–28. doi: 10.1007/978-3-7091-0696-9_1.  Google Scholar

[5]

F. R. Gantmakher and M. G. Krein, Oscillating Matrices and Kernels and Small Vibrations of Mechanical Systems (in Russian), GITTL, Moscow-Leningrad, (1950), Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. doi: 10.1090/chel/345.  Google Scholar

[6]

V. A. Marchenko, Introduction to The Theory of Inverse Problems of Spectral Analysis (in Russian), Acta, Kharkov, 2005. Google Scholar

[7]

O. Martynyuk, V. Pivovarchik and C. Tretter, Inverse problem for a damped Stieltjes string from parts of spectra, Appl. Anal., 94 (2015), 2605-2619.  doi: 10.1080/00036811.2014.996874.  Google Scholar

[8]

M. Möller and V. Pivovarchik, Damped star graphs of Stieltjes strings, Proc. Amer. Math. Soc., 145 (2017), 1717-1728.  doi: 10.1090/proc/13367.  Google Scholar

[9]

M. Möller and V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-17070-1.  Google Scholar

[10]

V. Pivovarchik, Existence of a tree of Stieltjes strings corresponding to two given spectra,, J. Phys. A, 42 (2009), 375213, 16 pp. doi: 10.1088/1751-8113/42/37/375213.  Google Scholar

[11]

V. Pivovarchik, N. Rozhenko and C. Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra Appl., 439 (2013), 2263-2292.  doi: 10.1016/j.laa.2013.07.003.  Google Scholar

[12]

V. Pivovarchik and C. Tretter, Location and multiplicities of eigenvalues for a star graph of Stieltjes strings, J. Difference Equ. Appl., 21 (2015), 383-402.  doi: 10.1080/10236198.2014.992425.  Google Scholar

[13]

K. Veselić, On linear vibrational systems with one dimensional damping, Appl. Anal., 29 (1988), 1-18.  doi: 10.1080/00036818808839770.  Google Scholar

[14]

K. Veselić, On linear vibrational systems with one dimensional damping II, Integr. Equ. Oper. Theory, 13 (1990), 883-897.  doi: 10.1007/BF01198923.  Google Scholar

show all references

References:
[1]

O. Boyko, O. Martynyuk and V. Pivovarchik, On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph, Methods of Funct. Anal. Topology, 25 (2019), 104-117.   Google Scholar

[2]

J. Genin and J. S. Maybee, Mechanical vibrations trees, J. Math. Anal. Appl., 45 (1974), 746-763.  doi: 10.1016/0022-247X(74)90065-1.  Google Scholar

[3]

G. Gladwell, Inverse Problems in Vibration, Kluwer Academic Publishers, Dordrecht, 2004.  Google Scholar

[4]

G. Gladwell, Matrix inverse eigenvalue problems, Dynamical Inverse Problems: Theory and Application, CISM Courses and Lect., SpringerWienNewYork, Vienna, 529 (2011), 1–28. doi: 10.1007/978-3-7091-0696-9_1.  Google Scholar

[5]

F. R. Gantmakher and M. G. Krein, Oscillating Matrices and Kernels and Small Vibrations of Mechanical Systems (in Russian), GITTL, Moscow-Leningrad, (1950), Revised edition, AMS Chelsea Publishing, Providence, RI, 2002. doi: 10.1090/chel/345.  Google Scholar

[6]

V. A. Marchenko, Introduction to The Theory of Inverse Problems of Spectral Analysis (in Russian), Acta, Kharkov, 2005. Google Scholar

[7]

O. Martynyuk, V. Pivovarchik and C. Tretter, Inverse problem for a damped Stieltjes string from parts of spectra, Appl. Anal., 94 (2015), 2605-2619.  doi: 10.1080/00036811.2014.996874.  Google Scholar

[8]

M. Möller and V. Pivovarchik, Damped star graphs of Stieltjes strings, Proc. Amer. Math. Soc., 145 (2017), 1717-1728.  doi: 10.1090/proc/13367.  Google Scholar

[9]

M. Möller and V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and Their Applications, Birkhäuser, Cham, 2015. doi: 10.1007/978-3-319-17070-1.  Google Scholar

[10]

V. Pivovarchik, Existence of a tree of Stieltjes strings corresponding to two given spectra,, J. Phys. A, 42 (2009), 375213, 16 pp. doi: 10.1088/1751-8113/42/37/375213.  Google Scholar

[11]

V. Pivovarchik, N. Rozhenko and C. Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra Appl., 439 (2013), 2263-2292.  doi: 10.1016/j.laa.2013.07.003.  Google Scholar

[12]

V. Pivovarchik and C. Tretter, Location and multiplicities of eigenvalues for a star graph of Stieltjes strings, J. Difference Equ. Appl., 21 (2015), 383-402.  doi: 10.1080/10236198.2014.992425.  Google Scholar

[13]

K. Veselić, On linear vibrational systems with one dimensional damping, Appl. Anal., 29 (1988), 1-18.  doi: 10.1080/00036818808839770.  Google Scholar

[14]

K. Veselić, On linear vibrational systems with one dimensional damping II, Integr. Equ. Oper. Theory, 13 (1990), 883-897.  doi: 10.1007/BF01198923.  Google Scholar

[1]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020033
[2]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20
[3]

Svetlana Katok, Ilie Ugarcovici. Structure of attractors for $(a,b)$-continued fraction transformations. Journal of Modern Dynamics, 2010, 4 (4) : 637-691. doi: 10.3934/jmd.2010.4.637
[4]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281
[5]

Clelia Marchionna. Free vibrations in space of the single mode for the Kirchhoff string. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2947-2971. doi: 10.3934/cpaa.2013.12.2947
[6]

Sergei Avdonin, Pavel Kurasov, Marlena Nowaczyk. Inverse problems for quantum trees II: Recovering matching conditions for star graphs. Inverse Problems & Imaging, 2010, 4 (4) : 579-598. doi: 10.3934/ipi.2010.4.579
[7]

Alessia Berti, Maria Grazia Naso. Vibrations of a damped extensible beam between two stops. Evolution Equations & Control Theory, 2013, 2 (1) : 35-54. doi: 10.3934/eect.2013.2.35
[8]

Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221
[9]

Amin Boumenir, Vu Kim Tuan. Reconstruction of the coefficients of a star graph from observations of its vertices. Inverse Problems & Imaging, 2018, 12 (6) : 1293-1308. doi: 10.3934/ipi.2018054
[10]

Nemanja Kosovalić, Brian Pigott. Self-excited vibrations for damped and delayed higher dimensional wave equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2413-2435. doi: 10.3934/dcds.2019102
[11]

Serena Dipierro, Alessio Figalli, Giampiero Palatucci, Enrico Valdinoci. Asymptotics of the $s$-perimeter as $s\searrow 0$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2777-2790. doi: 10.3934/dcds.2013.33.2777
[12]

Sergei A. Avdonin, Boris P. Belinskiy. On the basis properties of the functions arising in the boundary control problem of a string with a variable tension. Conference Publications, 2005, 2005 (Special) : 40-49. doi: 10.3934/proc.2005.2005.40
[13]

Giuseppe Maria Coclite, Carlotta Donadello. Vanishing viscosity on a star-shaped graph under general transmission conditions at the node. Networks & Heterogeneous Media, 2020, 15 (2) : 197-213. doi: 10.3934/nhm.2020009
[14]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[15]

Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127
[16]

Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533
[17]

Bin Yu. Behavior $0$ nonsingular Morse Smale flows on $S^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 509-540. doi: 10.3934/dcds.2016.36.509
[18]

S. S. Dragomir, C. E. M. Pearce. Jensen's inequality for quasiconvex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 279-291. doi: 10.3934/naco.2012.2.279
[19]

Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic & Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[20]

Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems & Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059

2019 Impact Factor: 1.373

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences