# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021003
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## 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

Received  July 2020 Early access January 2021

Fund Project: The first author is supported by the RFBR grant 20-01 627A and Volkswagen Foundation. The second author is supported by the RFBR grant 19-01-00565A

The one-dimensional Dirac dynamical system
 $\Sigma$
is
 \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*}
where
 $\sigma_{\!_3} = \begin{pmatrix}1&0 \\ 0&-1\end{pmatrix}$
is the Pauli matrix;
 $V = \begin{pmatrix}0&p\\ \bar p&0\end{pmatrix}$
with
 $p = p(x)$
is a potential;
 $u = \begin{pmatrix}u_1^f(x, t) \\ u_2^f(x, t)\end{pmatrix}$
is the trajectory in
 $\mathscr H = L_2(\mathbb R_+;\mathbb C^2)$
;
 $f\in\mathscr F = L_2([0, \infty);\mathbb C)$
is a boundary control. System
 $\Sigma$
is not controllable: the total reachable set
 $\mathscr U = {\rm span}_{t>0}\{u^f(\cdot, t)\, |\, \, f\in \mathscr F\}$
is not dense in
 $\mathscr H$
, but contains a controllable part
 $\Sigma_u$
. We construct a dynamical system
 $\Sigma_a$
, which is controllable in
 $L_2(\mathbb R_+;\mathbb C)$
and connected with
 $\Sigma_u$
via a unitary transform. The construction is based on geometrical optics relations: trajectories of
 $\Sigma_a$
are composed of jump amplitudes that arise as a result of projecting in
 $\overline{\mathscr U}$
onto the reachable sets
 $\mathscr U^t = \{u^f(\cdot, t)\, |\, \, f\in \mathscr F\}$
. System
 $\Sigma_a$
, which we call the amplitude model of the original
 $\Sigma$
, has the same input/output correspondence as system
 $\Sigma$
. As such,
 $\Sigma_a$
provides a canonical completely reachable realization of the Dirac system.
Citation: Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, doi: 10.3934/eect.2021003
##### 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.  Google Scholar [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.  Google Scholar [3] M. I. Belishev, Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.  doi: 10.4213/rm9768.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] M. I. Belishev, Boundary control and tomography of Riemannian manifolds, Russian Mathematical Surveys, 72 (2017), 581-644.  doi: 10.4213/rm9768.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] R. E. Kalman, P. L. Falb and M. A. Arbib, Topics in Mathematical System Theory, New-York: McGraw-Hill, 1969.  Google Scholar
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