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New results on controllability of fractional evolution systems with order $ \alpha\in (1,2) $
A canonical model of the one-dimensional dynamical Dirac system with boundary control
1. | St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023 Russia |
2. | St. Petersburg State University, 7–9 Universitetskaya nab., St. Petersburg, 199034 Russia |
$ \Sigma $ |
$ \begin{align*} & iu_t+i\sigma_{\!_3}\, u_x+Vu = 0, \, \, \, \, x, t>0;\, \, \, u|_{t = 0} = 0, \, \, x>0;\, \, \, \, u_1|_{x = 0} = f, \, \, t>0, \end{align*} $ |
$ \sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix} $ |
$ V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix} $ |
$ p = p(x) $ |
$ u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix} $ |
$ \mathscr H = L_2(\mathbb R_+;\mathbb C^2) $ |
$ f\in\mathscr F = L_2([0, \infty);\mathbb C) $ |
$ \Sigma $ |
$ \mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $ |
$ \mathscr H $ |
$ \Sigma_u $ |
$ \Sigma_a $ |
$ L_2(\mathbb R_+;\mathbb C) $ |
$ \Sigma_u $ |
$ \Sigma_a $ |
$ \overline{\mathscr U} $ |
$ \mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\} $ |
$ \Sigma_a $ |
$ \Sigma $ |
$ \Sigma $ |
$ \Sigma_a $ |
References:
[1] |
M. I. Belishev,
A unitary invariant of a semi-bounded operator in reconstruction of manifolds, Journal of Operator Theory, 69 (2013), 299-326.
doi: 10.7900/jot.2010oct22.1925. |
[2] |
M. I. Belishev, Boundary control method, in Encyclopedia of Applied and Computational Mathematics, (2015), 142–146.
doi: 10.1007/978-3-540-70529-1_7. |
[3] |
M. I. Belishev,
Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.
doi: 10.4213/rm9768. |
[4] |
M. I. Belishev and V. S. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 1–26.
doi: 10.1088/0266-5611/30/12/125013. |
[5] |
M. I. Belishev and S. A. Simonov,
Wave model of the Sturm-Liouville operator on the half-line, St. Petersburg Math. J., 29 (2018), 227-248.
doi: 10.1090/spmj/1491. |
[6] |
M. I. Belishev and S. A. Simonov,
A wave model of metric spaces, Functional Analysis and Its Applications, 53 (2019), 79-85.
doi: 10.1134/S0016266319020011. |
[7] |
M. I. Belishev and S. A. Simonov, A wave model of metric space with measure, Sbornik: Mathematics, 211 (2020), 521-538. Google Scholar |
[8] |
M. I. Belishev and S. A. Simonov, On evolutionary first-order dynamical system with boundary control, Zapiski Nauchnykh Seminarov POMI, in Russian, 483 (2019), 41–54. |
[9] |
I. C. Gohberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Transl. of Monographs No. 24, Amer. Math. Soc, Providence. Rhode Island, 1970. |
[10] |
R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969. |
show all references
References:
[1] |
M. I. Belishev,
A unitary invariant of a semi-bounded operator in reconstruction of manifolds, Journal of Operator Theory, 69 (2013), 299-326.
doi: 10.7900/jot.2010oct22.1925. |
[2] |
M. I. Belishev, Boundary control method, in Encyclopedia of Applied and Computational Mathematics, (2015), 142–146.
doi: 10.1007/978-3-540-70529-1_7. |
[3] |
M. I. Belishev,
Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.
doi: 10.4213/rm9768. |
[4] |
M. I. Belishev and V. S. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 1–26.
doi: 10.1088/0266-5611/30/12/125013. |
[5] |
M. I. Belishev and S. A. Simonov,
Wave model of the Sturm-Liouville operator on the half-line, St. Petersburg Math. J., 29 (2018), 227-248.
doi: 10.1090/spmj/1491. |
[6] |
M. I. Belishev and S. A. Simonov,
A wave model of metric spaces, Functional Analysis and Its Applications, 53 (2019), 79-85.
doi: 10.1134/S0016266319020011. |
[7] |
M. I. Belishev and S. A. Simonov, A wave model of metric space with measure, Sbornik: Mathematics, 211 (2020), 521-538. Google Scholar |
[8] |
M. I. Belishev and S. A. Simonov, On evolutionary first-order dynamical system with boundary control, Zapiski Nauchnykh Seminarov POMI, in Russian, 483 (2019), 41–54. |
[9] |
I. C. Gohberg and M. G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Transl. of Monographs No. 24, Amer. Math. Soc, Providence. Rhode Island, 1970. |
[10] |
R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969. |
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