American Institute of Mathematical Sciences

Advanced
February  2022, 11(1): 283-300. doi: 10.3934/eect.2021003

A canonical model of the one-dimensional dynamical Dirac system with boundary control

Mikhail I. Belishev 1,, and  Sergey A. Simonov 1,2,

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 Published  February 2022 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.
Keywords: 1D dynamical Dirac system, reachable sets, lack of controllability, propagation of singularities, canonical controllable model.
Mathematics Subject Classification: Primary: 35B30, 47N70, 46N20; Secondary: 93B28.
Citation: Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations and Control Theory, 2022, 11 (1) : 283-300. 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.

[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. 

[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. 

[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.

[1]

Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042
[2]

Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic and Related Models, 2019, 12 (2) : 269-302. doi: 10.3934/krm.2019012
[3]

Rinaldo M. Colombo, Graziano Guerra. A coupling between a non--linear 1D compressible--incompressible limit and the 1D $p$--system in the non smooth case. Networks and Heterogeneous Media, 2016, 11 (2) : 313-330. doi: 10.3934/nhm.2016.11.313
[4]

Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control and Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125
[5]

Franck Boyer, Guillaume Olive. Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients. Mathematical Control and Related Fields, 2014, 4 (3) : 263-287. doi: 10.3934/mcrf.2014.4.263
[6]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations and Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[7]

Andrei Halanay, Luciano Pandolfi. Lack of controllability of thermal systems with memory. Evolution Equations and Control Theory, 2014, 3 (3) : 485-497. doi: 10.3934/eect.2014.3.485
[8]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations and Control Theory, 2022, 11 (2) : 373-397. doi: 10.3934/eect.2021004
[9]

Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
[10]

Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete and Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[11]

Yu. Dabaghian, R. V. Jensen, R. Blümel. Integrability in 1D quantum chaos. Conference Publications, 2003, 2003 (Special) : 206-212. doi: 10.3934/proc.2003.2003.206
[12]

Rachel Clipp, Brooke Steele. An evaluation of dynamic outlet boundary conditions in a 1D fluid dynamics model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 61-74. doi: 10.3934/mbe.2012.9.61
[13]

Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075
[14]

Christian Rohde, Wenjun Wang, Feng Xie. Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2145-2171. doi: 10.3934/cpaa.2013.12.2145
[15]

Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect. Networks and Heterogeneous Media, 2013, 8 (2) : 501-527. doi: 10.3934/nhm.2013.8.501
[16]

Alexander Blokhin, Alesya Ibragimova. 1D numerical simulation of the mep mathematical model in ballistic diode problem. Kinetic and Related Models, 2009, 2 (1) : 81-107. doi: 10.3934/krm.2009.2.81
[17]

Chaker Jammazi, Souhila Loucif. On the global controllability of the 1-D Boussinesq equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1499-1523. doi: 10.3934/dcdss.2022096
[18]

François Delarue, Franco Flandoli. The transition point in the zero noise limit for a 1D Peano example. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4071-4083. doi: 10.3934/dcds.2014.34.4071
[19]

Harish S. Bhat, Razvan C. Fetecau. Lagrangian averaging for the 1D compressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 979-1000. doi: 10.3934/dcdsb.2006.6.979
[20]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (279)
  • HTML views (408)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy