June  2021, 16(2): 317-339. doi: 10.3934/nhm.2021008

An inverse problem for quantum trees with observations at interior vertices

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: Julian Edward

Received  April 2020 Revised  January 2021 Published  March 2021

Fund Project: The first author was supported in part by the National Science Foundation, grant DMS 1909869

In this paper we consider a non-standard dynamical inverse problem for the wave equation on a metric tree graph. We assume that positive masses may be attached to the internal vertices of the graph. Another specific feature of our investigation is that we use only one boundary actuator and one boundary sensor, all other observations being internal. Using the Dirichlet-to-Neumann map (acting from one boundary vertex to one boundary and all internal vertices) we recover the topology and geometry of the graph, the coefficients of the equations and the masses at the vertices.

Citation: Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008
References:
[1]

S. AdamE. H. HwangV. M. Galitski and S. Das Sarma, A self-consistent theory for graphene transports, Proc. Natl. Acad. Sci. USA, 104 (2007), 18392-18397.  doi: 10.1073/pnas.0704772104.  Google Scholar

[2]

F. Al-MusallamS. A. AvdoninN. 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.  doi: 10.17586/2220-8054-2016-7-5-835-841.  Google Scholar

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G. AlìA. Bartel and M. Günther, Parabolic differential-algebraic models in electrical network design, Multiscale Model. Simul., 4 (2005), 813-838.  doi: 10.1137/040610696.  Google Scholar

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S. Avdonin, Control, observation and identification problems for the wave equation on metric graphs, IFAC-PapersOnLine, 52 (2019), 52-57.  doi: 10.1016/j.ifacol.2019.08.010.  Google Scholar

[5]

S. Avdonin, Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks, in Analysis on Graphs and Its Applications (Proceedings of Symposia in Pure Mathematics), 77, AMS, Providence, RI, 2008,507–521. Google Scholar

[6]

S. Avdonin, N. Avdonina and J. Edward, Boundary inverse problems for networks of vibrating strings with attached masses, in Dynamic Systems and Applications, 7, Dynamic, Atlanta, GA, 2016, 41–44.  Google Scholar

[7]

S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Probl. Imaging, 9 (2015), 645-659.  doi: 10.3934/ipi.2015.9.645.  Google Scholar

[8]

S. Avdonin and J. Edward, An inverse problem for quantum trees with delta-prime vertex conditions, Vibration, 3 (2020), 448-463.  doi: 10.3390/vibration3040028.  Google Scholar

[9]

S. Avdonin and J. Edward, Controllability for a string with attached masses and Riesz bases for asymmetric spaces, Math. Control Relat. Fields, 9 (2019), 453-494.  doi: 10.3934/mcrf.2019021.  Google Scholar

[10]

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

[11]

S. 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

[12]

S. AvdoninP. Kurasov and M. Novaczyk, Inverse problems for quantum trees II: Recovering matching conditions for star graphs, Inverse Probl. Imaging, 4 (2010), 579-598.  doi: 10.3934/ipi.2010.4.579.  Google Scholar

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S. AvdoninG. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, ZAMM Z. Angew. Math. Mech., 90 (2010), 136-150.  doi: 10.1002/zamm.200900295.  Google Scholar

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S. Avdonin and V. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 19pp. doi: 10.1088/0266-5611/26/4/045009.  Google Scholar

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S. Avdonin and S. Nicaise, Source identification problems for the wave equation on graphs, Inverse Problems, 31 (2015), 29pp. doi: 10.1088/0266-5611/31/9/095007.  Google Scholar

[16]

S. Avdonin and Y. Zhao, Exact controllability of the 1-D wave equation on finite metric tree graphs, Appl. Math. Optim., (2019). doi: 10.1007/s00245-019-09629-3.  Google Scholar

[17]

S. Avdonin and Y. Zhao, Leaf peeling method for the wave equation on metric tree graphs, Inverse Probl. Imaging, 15 (2021), 185-199.  doi: 10.3934/ipi.2020060.  Google Scholar

[18]

S. A. AvdoninM. I. Belishev and S. A. Ivanov, Boundary control and an inverse matrix problem for the equation $u_tt-u_xx+V(x)u = 0$, Math. USSR-Sb., 72 (1992), 287-310.  doi: 10.1070/SM1992v072n02ABEH002141.  Google Scholar

[19] S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995.   Google Scholar
[20]

S. A. AvdoninV. Mikhaylov and K. B. Nurtazina, On inverse dynamical and spectral problems for the wave and Schrödinger equations on finite trees. The leaf peeling method, J. Math. Sci. (NY), 224 (2017), 1-10.  doi: 10.1007/s10958-017-3388-2.  Google Scholar

[21]

G. Bastin, J. M. Coron and B. d'Andrèa Novel, Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks, in Proceedings of the Lecture Notes for the Pre-Congress Workshop on Complex Embedded and Networked Control Systems, 17th IFAC World Congress, Seoul, Korea, 2008, 16–20. Google Scholar

[22]

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672.  doi: 10.1088/0266-5611/20/3/002.  Google Scholar

[23]

M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inverse Ill-Posed Probl., 14 (2006), 29-46.  doi: 10.1515/156939406776237474.  Google Scholar

[24]

J. Bell and G. Craciun, A distributed parameter identification problem in neuronal cable theory models, Math. Biosci., 194 (2005), 1-19.  doi: 10.1016/j.mbs.2004.07.001.  Google Scholar

[25]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.  Google Scholar

[26]

I. B. BourdonovA. S. Kossatchev and V. V. Kulyamin, Analysis of a graph by a set of automata, Program. Comput. Softw., 41 (2015), 307-310.  doi: 10.1134/S0361768815060031.  Google Scholar

[27]

I. B. BourdonovA. S. Kossatchev and V. V. Kulyamin, Parallel computations on a graph, Program. Comput. Softw., 41 (2015), 1-13.  doi: 10.1134/S0361768815010028.  Google Scholar

[28]

B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3231-3243.  doi: 10.1098/rspa.2005.1513.  Google Scholar

[29]

R. M. ColomboG. GuerraM. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[30]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-$d$ Flexible Multi-Structures, Mathematics & Applications, 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[31]

P. Exner, Vertex couplings in quantum graphs: Approximations by scaled Schrödinger operators, in Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011, 71–92. doi: 10.1142/9789814338820_0004.  Google Scholar

[32]

G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Appl. Anal., 86 (2007), 653-667.  doi: 10.1080/00036810701303976.  Google Scholar

[33]

M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 257-270.  doi: 10.1016/j.anihpc.2008.01.002.  Google Scholar

[34]

B. Gutkin and U. Smilansky, Can you hear the shape of a graph?, J. Phys. A., 34 (2001), 6061-6068.  doi: 10.1088/0305-4470/34/31/301.  Google Scholar

[35]

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

[36]

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

[37]

N. E. Hurt, Mathematical Physics of Quantum Wires and Devices, Mathematics and its Applications, 506, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9626-8.  Google Scholar

[38]

C. Joachim and S. Roth, Atomic and Molecular Wires, NATO Science Series E, 341, Springer Netherlands, 1997. Google Scholar

[39]

V. Kostrykin and R. Schrader, Kirchoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.  Google Scholar

[40]

V. Kostrykin and R. Schrader, Kirchoff's rule for quantum wires. II. The inverse problem with possible applications to quantum computers, Fortschr. Phys., 48 (2000), 703-716.  doi: 10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O.  Google Scholar

[41]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Physics, 274 (1999), 76-124.  doi: 10.1006/aphy.1999.5904.  Google Scholar

[42]

T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.  doi: 10.1103/PhysRevLett.79.4794.  Google Scholar

[43]

P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A., 38 (2005), 4901-4915.  doi: 10.1088/0305-4470/38/22/014.  Google Scholar

[44]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamical Elastic Multi-Link Structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[45]

Y. B. Melnikov and B. S. Pavlov, Two-body scattering on a graph and application to simple nanoelectronic devices, J. Math. Phys., 36 (1995), 2813-2825.  doi: 10.1063/1.531068.  Google Scholar

[46]

N. M. R. Peres, Scattering in one-dimensional heterostructures described by the Dirac equation, J. Phys. Condens. Matter, 21 (2009). doi: 10.1088/0953-8984/21/9/095501.  Google Scholar

[47]

N. M. R. Peres, J. N. B. Rodrigues, T. Stauber and J. M. B. Lopes dos Santos, Dirac electrons in graphene-based quantum wires and quantum dots, J. Phys. Condens. Matter, 21 (2009). doi: 10.1088/0953-8984/21/34/344202.  Google Scholar

[48]

W. Rall, Core conductor theory and cable properties of neurons, in Handbook of Physiology, The Nervous System, Cellular Biology of Neurons, American Physiological Society, Rockville, MD, 1977, 39–97. doi: 10.1002/cphy.cp010103.  Google Scholar

[49]

V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems, 21 (2005), 1075-1086.  doi: 10.1088/0266-5611/21/3/017.  Google Scholar

[50]

E. Zuazua, Control and stabilization of waves on 1-d networks, in Modelling and Optimisation of Flows on Networks, Lecture Notes in Math., 2062, Springer, Heidelberg, 2013,463–493. doi: 10.1007/978-3-642-32160-3_9.  Google Scholar

show all references

References:
[1]

S. AdamE. H. HwangV. M. Galitski and S. Das Sarma, A self-consistent theory for graphene transports, Proc. Natl. Acad. Sci. USA, 104 (2007), 18392-18397.  doi: 10.1073/pnas.0704772104.  Google Scholar

[2]

F. Al-MusallamS. A. AvdoninN. 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.  doi: 10.17586/2220-8054-2016-7-5-835-841.  Google Scholar

[3]

G. AlìA. Bartel and M. Günther, Parabolic differential-algebraic models in electrical network design, Multiscale Model. Simul., 4 (2005), 813-838.  doi: 10.1137/040610696.  Google Scholar

[4]

S. Avdonin, Control, observation and identification problems for the wave equation on metric graphs, IFAC-PapersOnLine, 52 (2019), 52-57.  doi: 10.1016/j.ifacol.2019.08.010.  Google Scholar

[5]

S. Avdonin, Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks, in Analysis on Graphs and Its Applications (Proceedings of Symposia in Pure Mathematics), 77, AMS, Providence, RI, 2008,507–521. Google Scholar

[6]

S. Avdonin, N. Avdonina and J. Edward, Boundary inverse problems for networks of vibrating strings with attached masses, in Dynamic Systems and Applications, 7, Dynamic, Atlanta, GA, 2016, 41–44.  Google Scholar

[7]

S. Avdonin and J. Bell, Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph, Inverse Probl. Imaging, 9 (2015), 645-659.  doi: 10.3934/ipi.2015.9.645.  Google Scholar

[8]

S. Avdonin and J. Edward, An inverse problem for quantum trees with delta-prime vertex conditions, Vibration, 3 (2020), 448-463.  doi: 10.3390/vibration3040028.  Google Scholar

[9]

S. Avdonin and J. Edward, Controllability for a string with attached masses and Riesz bases for asymmetric spaces, Math. Control Relat. Fields, 9 (2019), 453-494.  doi: 10.3934/mcrf.2019021.  Google Scholar

[10]

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

[11]

S. 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

[12]

S. AvdoninP. Kurasov and M. Novaczyk, Inverse problems for quantum trees II: Recovering matching conditions for star graphs, Inverse Probl. Imaging, 4 (2010), 579-598.  doi: 10.3934/ipi.2010.4.579.  Google Scholar

[13]

S. AvdoninG. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings, ZAMM Z. Angew. Math. Mech., 90 (2010), 136-150.  doi: 10.1002/zamm.200900295.  Google Scholar

[14]

S. Avdonin and V. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 19pp. doi: 10.1088/0266-5611/26/4/045009.  Google Scholar

[15]

S. Avdonin and S. Nicaise, Source identification problems for the wave equation on graphs, Inverse Problems, 31 (2015), 29pp. doi: 10.1088/0266-5611/31/9/095007.  Google Scholar

[16]

S. Avdonin and Y. Zhao, Exact controllability of the 1-D wave equation on finite metric tree graphs, Appl. Math. Optim., (2019). doi: 10.1007/s00245-019-09629-3.  Google Scholar

[17]

S. Avdonin and Y. Zhao, Leaf peeling method for the wave equation on metric tree graphs, Inverse Probl. Imaging, 15 (2021), 185-199.  doi: 10.3934/ipi.2020060.  Google Scholar

[18]

S. A. AvdoninM. I. Belishev and S. A. Ivanov, Boundary control and an inverse matrix problem for the equation $u_tt-u_xx+V(x)u = 0$, Math. USSR-Sb., 72 (1992), 287-310.  doi: 10.1070/SM1992v072n02ABEH002141.  Google Scholar

[19] S. A. Avdonin and S. A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995.   Google Scholar
[20]

S. A. AvdoninV. Mikhaylov and K. B. Nurtazina, On inverse dynamical and spectral problems for the wave and Schrödinger equations on finite trees. The leaf peeling method, J. Math. Sci. (NY), 224 (2017), 1-10.  doi: 10.1007/s10958-017-3388-2.  Google Scholar

[21]

G. Bastin, J. M. Coron and B. d'Andrèa Novel, Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks, in Proceedings of the Lecture Notes for the Pre-Congress Workshop on Complex Embedded and Networked Control Systems, 17th IFAC World Congress, Seoul, Korea, 2008, 16–20. Google Scholar

[22]

M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems, 20 (2004), 647-672.  doi: 10.1088/0266-5611/20/3/002.  Google Scholar

[23]

M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inverse Ill-Posed Probl., 14 (2006), 29-46.  doi: 10.1515/156939406776237474.  Google Scholar

[24]

J. Bell and G. Craciun, A distributed parameter identification problem in neuronal cable theory models, Math. Biosci., 194 (2005), 1-19.  doi: 10.1016/j.mbs.2004.07.001.  Google Scholar

[25]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/186.  Google Scholar

[26]

I. B. BourdonovA. S. Kossatchev and V. V. Kulyamin, Analysis of a graph by a set of automata, Program. Comput. Softw., 41 (2015), 307-310.  doi: 10.1134/S0361768815060031.  Google Scholar

[27]

I. B. BourdonovA. S. Kossatchev and V. V. Kulyamin, Parallel computations on a graph, Program. Comput. Softw., 41 (2015), 1-13.  doi: 10.1134/S0361768815010028.  Google Scholar

[28]

B. M. Brown and R. Weikard, A Borg-Levinson theorem for trees, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461 (2005), 3231-3243.  doi: 10.1098/rspa.2005.1513.  Google Scholar

[29]

R. M. ColomboG. GuerraM. Herty and V. Schleper, Optimal control in networks of pipes and canals, SIAM J. Control Optim., 48 (2009), 2032-2050.  doi: 10.1137/080716372.  Google Scholar

[30]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-$d$ Flexible Multi-Structures, Mathematics & Applications, 50, Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[31]

P. Exner, Vertex couplings in quantum graphs: Approximations by scaled Schrödinger operators, in Mathematics in Science and Technology, World Sci. Publ., Hackensack, NJ, 2011, 71–92. doi: 10.1142/9789814338820_0004.  Google Scholar

[32]

G. Freiling and V. Yurko, Inverse problems for differential operators on trees with general matching conditions, Appl. Anal., 86 (2007), 653-667.  doi: 10.1080/00036810701303976.  Google Scholar

[33]

M. Gugat and G. Leugering, Global boundary controllability of the Saint-Venant system for sloped canals with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 257-270.  doi: 10.1016/j.anihpc.2008.01.002.  Google Scholar

[34]

B. Gutkin and U. Smilansky, Can you hear the shape of a graph?, J. Phys. A., 34 (2001), 6061-6068.  doi: 10.1088/0305-4470/34/31/301.  Google Scholar

[35]

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

[36]

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

[37]

N. E. Hurt, Mathematical Physics of Quantum Wires and Devices, Mathematics and its Applications, 506, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9626-8.  Google Scholar

[38]

C. Joachim and S. Roth, Atomic and Molecular Wires, NATO Science Series E, 341, Springer Netherlands, 1997. Google Scholar

[39]

V. Kostrykin and R. Schrader, Kirchoff's rule for quantum wires, J. Phys. A, 32 (1999), 595-630.  doi: 10.1088/0305-4470/32/4/006.  Google Scholar

[40]

V. Kostrykin and R. Schrader, Kirchoff's rule for quantum wires. II. The inverse problem with possible applications to quantum computers, Fortschr. Phys., 48 (2000), 703-716.  doi: 10.1002/1521-3978(200008)48:8<703::AID-PROP703>3.0.CO;2-O.  Google Scholar

[41]

T. Kottos and U. Smilansky, Periodic orbit theory and spectral statistics for quantum graphs, Ann. Physics, 274 (1999), 76-124.  doi: 10.1006/aphy.1999.5904.  Google Scholar

[42]

T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79 (1997), 4794-4797.  doi: 10.1103/PhysRevLett.79.4794.  Google Scholar

[43]

P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A., 38 (2005), 4901-4915.  doi: 10.1088/0305-4470/38/22/014.  Google Scholar

[44]

J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modeling, Analysis and Control of Dynamical Elastic Multi-Link Structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. doi: 10.1007/978-1-4612-0273-8.  Google Scholar

[45]

Y. B. Melnikov and B. S. Pavlov, Two-body scattering on a graph and application to simple nanoelectronic devices, J. Math. Phys., 36 (1995), 2813-2825.  doi: 10.1063/1.531068.  Google Scholar

[46]

N. M. R. Peres, Scattering in one-dimensional heterostructures described by the Dirac equation, J. Phys. Condens. Matter, 21 (2009). doi: 10.1088/0953-8984/21/9/095501.  Google Scholar

[47]

N. M. R. Peres, J. N. B. Rodrigues, T. Stauber and J. M. B. Lopes dos Santos, Dirac electrons in graphene-based quantum wires and quantum dots, J. Phys. Condens. Matter, 21 (2009). doi: 10.1088/0953-8984/21/34/344202.  Google Scholar

[48]

W. Rall, Core conductor theory and cable properties of neurons, in Handbook of Physiology, The Nervous System, Cellular Biology of Neurons, American Physiological Society, Rockville, MD, 1977, 39–97. doi: 10.1002/cphy.cp010103.  Google Scholar

[49]

V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems, 21 (2005), 1075-1086.  doi: 10.1088/0266-5611/21/3/017.  Google Scholar

[50]

E. Zuazua, Control and stabilization of waves on 1-d networks, in Modelling and Optimisation of Flows on Networks, Lecture Notes in Math., 2062, Springer, Heidelberg, 2013,463–493. doi: 10.1007/978-3-642-32160-3_9.  Google Scholar

Figure 1.  A metric tree
Figure 2.  Sensors at vertex $ v_1 $ marked by arrows
Figure 3.  $ \Omega $ and subtree $ \Omega_{kj} $
Figure 4.  Star with coordinate system: $ e_j $ identified with $ [0,\ell_j] $
Figure 5.  Star as part of larger tree
Figure 6.  $ \Omega $ and subtree $ \Omega_{12} $
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