Simultaneous controllability in sharp time for two elastic strings
ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 259-273.

We study the simultaneously reachable subspace for two strings controlled from a common endpoint. We give necessary and sufficient conditions for simultaneous spectral and approximate controllability. Moreover we prove the lack of simultaneous exact controllability and we study the space of simultaneously reachable states as a function of the position of the joint. For each type of controllability result we give the sharp controllability time.

Classification: 93B,  35L,  42
Keywords: exact controllability, spectral controllability, approximate controllability, simultaneous controllability, string equation, boundary control, Riesz basis
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     author = {Avdonin, Sergei and Tucsnak, Marius},
     title = {Simultaneous controllability in sharp time for two elastic strings},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {259--273},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
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     language = {en},
     url = {http://archive.numdam.org/item/COCV_2001__6__259_0/}
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Avdonin, Sergei; Tucsnak, Marius. Simultaneous controllability in sharp time for two elastic strings. ESAIM: Control, Optimisation and Calculus of Variations, Volume 6 (2001), pp. 259-273. http://archive.numdam.org/item/COCV_2001__6__259_0/

[1] S.A. Avdonin, Simultaneous controllability of several elastic strings, in Proc. CD of the Fourteenth International Symposium on Mathematical Theory of Networks and Systems. Perpignan, France, June 19-23 (2000).

[2] 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 (1995). | MR | Zbl

[3] C. Baiocchi, V. Komornik and P. Loreti, Ingham type theorems and applications to control theory. Boll. Un. Mat. Ital. B 2 (1999) 33-63. | MR | Zbl

[4] C. Baiocchi, V. Komornik and P. Loreti, Généralisation d'un théorème de Beurling et application à la théorie du contrôle. C. R. Acad. Sci. Paris Sér. I Math. 330 (2000) 281-286. | Zbl

[5] J.W.S. Cassels, An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge (1965). | MR | Zbl

[6] S. Dolecki and D.L. Russell, A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. | MR | Zbl

[7] S. Jaffard, M. Tucsnak and E. Zuazua, Singular internal stabilization of the wave equation. J. Differential Equations 145 (1998) 184-215. | MR | Zbl

[8] S. Jaffard, M. Tucsnak and E. Zuazua, On a theorem of Ingham. J. Fourier Anal. Appl. 3 (1997) 577-582. | MR | Zbl

[9] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley & Sons, New York (1974). | MR | Zbl

[10] J.E. Lagnese and J.L. Lions, Modelling, Analysis and Control of Thin Plates. Masson, Paris (1988). | MR | Zbl

[11] S. Lang, Introduction to Diophantine Approximations. Addison Wesley, New York (1966). | MR | Zbl

[12] J.-L. Lions, Controlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Volume 1. Masson, Paris (1988).

[13] N.K. Nikol'Skii, A Treatise on the Shift Operator. Moscow, Nauka, 1980 (Russian); Engl. Transl., Springer, Berlin (1986).

[14] B.S. Pavlov, Basicity of an exponential systems and Muckenhoupt's condition. Dokl. Akad. Nauk SSSR 247 (1979) 37-40 (Russian); English transl. in Soviet Math. Dokl. 20 (1979) 655-659. | Zbl

[15] W. Rudin, Real and complex analysis. McGraw-Hill, New York (1987). | MR | Zbl

[16] D.L. Russell, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region. SIAM J. Control Optim. 24 (1986) 199-229. | Zbl

[17] M. Tucsnak and G. Weiss, Simultaneous exact controllability and some applications. SIAM J. Control Optim. 38 (2000) 1408-1427. | MR | Zbl

[18] R. Young, An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980). | MR | Zbl