Shape, velocity, and exact controllability for the wave equation on a graph with cycle

Avdonin, S.; Edward, J.; Zhao, Y.

doi:10.1090/spmj/1791

Shape, velocity, and exact controllability for the wave equation on a graph with cycle
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by S. Avdonin, J. Edward and Y. Zhao

St. Petersburg Math. J. 35 (2024), 1-23

DOI: https://doi.org/10.1090/spmj/1791

Published electronically: April 12, 2024

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Abstract:

Exact controllability is proved on a graph with cycle. The controls can be a mix of controls applied at the boundary and interior vertices. The method of proof first applies a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated with exact controllability. In the case of a single control, either boundary or interior, it is shown that exact controllability fails.

References

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, Math. 7 (2016), 835–841.
Sergei Avdonin, Control problems on quantum graphs, Analysis on graphs and its applications, Proc. Sympos. Pure Math., vol. 77, Amer. Math. Soc., Providence, RI, 2008, pp. 507–521. MR 2459889, DOI 10.1090/pspum/077/2459889
Sergei Avdonin, Control, observation and identification problems for the wave equation on metric graphs, IFAC-PapersOnLine 52 (2019), no. 2, 52–57. MR 4153401, DOI 10.1016/j.ifacol.2019.08.010
S. A. Avdonin, M. I. Belishev, and S. A. Ivanov, Boundary control and an inverse matrix problem for the equation $u_{tt}-u_{xx}+V(x)u=0$, Mat. Sb. 182 (1991), no. 3, 307–331 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 287–310. MR 1110068, DOI 10.1070/SM1992v072n02ABEH002141
Sergei Avdonin and Julian Edward, Exact controllability for string with attached masses, SIAM J. Control Optim. 56 (2018), no. 2, 945–980. MR 3775124, DOI 10.1137/15M1029333
Sergei Avdonin and Julian Edward, Controllability for a string with attached masses and Riesz bases for asymmetric spaces, Math. Control Relat. Fields 9 (2019), no. 3, 453–494. MR 3956426, DOI 10.3934/mcrf.2019021
Sergei Avdonin and Julian Edward, An inverse problem for quantum trees with observations at interior vertices, Netw. Heterog. Media 16 (2021), no. 2, 317–339. MR 4247154, DOI 10.3934/nhm.2021008
Sergei A. Avdonin and Sergei A. Ivanov, Families of exponentials, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems; Translated from the Russian and revised by the authors. MR 1366650
Sergei Avdonin and Victor Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems 26 (2010), no. 4, 045009, 19. MR 2608622, DOI 10.1088/0266-5611/26/4/045009
Sergei Avdonin and Pavel Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging 2 (2008), no. 1, 1–21. MR 2375320, DOI 10.3934/ipi.2008.2.1
Sergei Avdonin and William Moran, Ingham-type inequalities and Riesz bases of divided differences, Int. J. Appl. Math. Comput. Sci. 11 (2001), no. 4, 803–820. Mathematical methods of optimization and control of large-scale systems (Ekaterinburg, 2000). MR 1874588
Sergei Avdonin and Serge Nicaise, Source identification problems for the wave equation on graphs, Inverse Problems 31 (2015), no. 9, 095007, 29. MR 3401998, DOI 10.1088/0266-5611/31/9/095007
Sergei Avdonin and Yuanyuan Zhao, Exact controllability of the 1-D wave equation on finite metric tree graphs, Appl. Math. Optim. 83 (2021), no. 3, 2303–2326. MR 4261290, DOI 10.1007/s00245-019-09629-3
M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems 20 (2004), no. 3, 647–672. MR 2067494, DOI 10.1088/0266-5611/20/3/002
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), no. 1, 29–46. MR 2218385, DOI 10.1163/156939406776237474
Gregory Berkolaiko and Peter Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. MR 3013208, DOI 10.1090/surv/186
Jan Boman, Pavel Kurasov, and Rune Suhr, Schrödinger operators on graphs and geometry II. Spectral estimates for $L_1$-potentials and an Ambartsumian theorem, Integral Equations Operator Theory 90 (2018), no. 3, Paper No. 40, 24. MR 3807959, DOI 10.1007/s00020-018-2467-1
René Dáger and Enrique Zuazua, Wave propagation, observation and control in $1\text {-}d$ flexible multi-structures, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 50, Springer-Verlag, Berlin, 2006. MR 2169126, DOI 10.1007/3-540-37726-3
P. Kurasov, Inverse problems for Aharonov-Bohm rings, Math. Proc. Cambridge Philos. Soc. 148 (2010), no. 2, 331–362. MR 2600145, DOI 10.1017/S030500410999034X
Pavel Kurasov, Inverse scattering for lasso graph, J. Math. Phys. 54 (2013), no. 4, 042103, 14. MR 3088223, DOI 10.1063/1.4799034
—, Quantum graphs: spectral theory and inverse problems, Springer. (in print)
J. E. Lagnese, Günter Leugering, and E. J. P. G. Schmidt, Modeling, analysis and control of dynamic elastic multi-link structures, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1279380, DOI 10.1007/978-1-4612-0273-8
Günter Leugering, Charlotte Rodriguez, and Yue Wang, Nodal profile control for networks of geometrically exact beams, J. Math. Pures Appl. (9) 155 (2021), 111–139 (English, with English and French summaries). MR 4324295, DOI 10.1016/j.matpur.2021.07.007
Irena Lasiecka and Roberto Triggiani, Control theory for partial differential equations: continuous and approximation theories. II, Encyclopedia of Mathematics and its Applications, vol. 75, Cambridge University Press, Cambridge, 2000. Abstract hyperbolic-like systems over a finite time horizon. MR 1745476, DOI 10.1017/CBO9780511574801.002
J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contrôlabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. MR 953547
David L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), no. 4, 639–739. MR 508380, DOI 10.1137/1020095
Kaili Zhuang, Günter Leugering, and Tatsien Li, Exact boundary controllability of nodal profile for Saint-Venant system on a network with loops, J. Math. Pures Appl. (9) 129 (2019), 34–60 (English, with English and French summaries). MR 3998789, DOI 10.1016/j.matpur.2018.10.001
Enrique Zuazua, Controllability and observability of partial differential equations: some results and open problems, Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, pp. 527–621. MR 2549374, DOI 10.1016/S1874-5717(07)80010-7
Enrique Zuazua, Control and stabilization of waves on 1-d networks, Modelling and optimisation of flows on networks, Lecture Notes in Math., vol. 2062, Springer, Heidelberg, 2013, pp. 463–493. MR 3050288, DOI 10.1007/978-3-642-32160-3_{9}