doi: 10.3934/eect.2022019
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Controllability for Schrödinger type system with mixed dispersion on compact star graphs

1. 

Departamento de Matemática, Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil

2. 

Instituto de Matemática, Universidade Federal de Alagoas, Maceió, Alagoas, Brazil

3. 

Departamento de Matematicas y Estadística, Universidad Nacional de Colombia - Sede Manizales, Manizales, Colombia

* Corresponding author: Roberto de A. Capistrano–Filho

Received  December 2021 Revised  March 2022 Early access April 2022

Fund Project: Capistrano–Filho was supported by CNPq grants 408181/2018-4 and 307808/2021-1, CAPES grants 88881.311964/2018-01 and 88881.520205/2020-01, MATHAMSUD 21- MATH-03 and Propesqi (UFPE). Cavalcante was supported by CNPq 310271/2021-5 and CAPES-MATHAMSUD 88887.368708/2019-00. Gallego was supported by MATHAMSUD 21-MATH-03 and the 100.000 Strong in the Americas Innovation Fund

In this work we are concerned with solutions to the linear Schrödinger type system with mixed dispersion, the so-called biharmonic Schrödinger equation. Precisely, we are able to prove an exact control property for these solutions with the control in the energy space posed on an oriented star graph structure
$ \mathcal{G} $
for
$ T>T_{min} $
, with
$ T_{min} = \sqrt{ \frac{ \overline{L} (L^2+\pi^2)}{\pi^2\varepsilon(1- \overline{L} \varepsilon)}}, $
when the couplings and the controls appear only on the Neumann boundary conditions.
Citation: Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Controllability for Schrödinger type system with mixed dispersion on compact star graphs. Evolution Equations and Control Theory, doi: 10.3934/eect.2022019
References:
[1]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008.

[2]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.  doi: 10.1016/j.jde.2016.01.029.

[3]

J. Angulo Pava and N. Goloshchapova, On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discrete Contin. Dyn. Syst., 38 (2018), 5039-5066.  doi: 10.3934/dcds.2018221.

[4]

J. Angulo Pava and N. Goloshchapova, Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, Adv. Differential Equations, 23 (2018), 793-846. 

[5]

K. Ammari and H. Bouzidi, Exact boundary controllability of the linear biharmonic Schrödinger equation with variable coefficients, arXiv: 2112.15196 [math.AP] (2021).

[6]

K. Ammari and E. Crépeau, Feedback stabilization and boundary controllability of the Korteweg–de Vries equation on a star-shaped network, SIAM J. Control Optim., 56 (2018), 1620-1639.  doi: 10.1137/17M113959X.

[7]

K. Ammari and E. Crépeau, Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems Control Lett., 127 (2019), 39-43.  doi: 10.1016/j.sysconle.2019.03.005.

[8]

L. Baudouin and M. Yamamoto, Inverse problem on a tree-shaped network: Unified approach for uniqueness, Appl. Anal., 94 (2015), 2370-2395.  doi: 10.1080/00036811.2014.985214.

[9]

M. Ben-ArtziH. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.

[10]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/186.

[11]

J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.

[12]

J. L. Bona and R. C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q., 16 (2008), 1-18. 

[13]

R. BurioniD. CassiM. RasettiP. Sodano and A. Vezzani, Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34 (2001), 4697-4710. 

[14]

R. de A. Capistrano-FilhoM. Cavalcante and F. A. Gallego, Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, Pacific J. Math., 309 (2020), 35-70.  doi: 10.2140/pjm.2020.309.35.

[15]

R. de A. Capistrano-Filho, M. Cavalcante and F. A. Gallego, Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation, Discrete & Continuous Dynamical Systems - B, 2021. doi: 10.3934/dcdsb.2021190.

[16]

M. Cavalcante, The Korteweg–de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69 (2018), Paper No. 124, 22 pp. doi: 10.1007/s00033-018-1018-6.

[17]

E. CerpaE. Crépeau and C. Moreno, On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network, IMA J. Math. Control Inform., 37 (2020), 226-240.  doi: 10.1093/imamci/dny047.

[18]

E. CerpaE. Crépeau and J. Valein, Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network, Evol. Equ. Control Theory, 9 (2020), 673-692.  doi: 10.3934/eect.2020028.

[19]

A. Duca, Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability, SIAM J. Control Optim., 58 (2020), 3092-3129.  doi: 10.1137/18M1212768.

[20]

A. Duca, Bilinear quantum systems on compact graphs: Well-posedness and global exact controllability, Automatica J. IFAC, 123 (2021), 109324, 13 pp. doi: 10.1016/j.automatica.2020.109324.

[21]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.

[22]

F. Gregorio and D. Mugnolo, Bi-Laplacians on graphs and networks, J. Evol. Equ., 20 (2020), 191-232.  doi: 10.1007/s00028-019-00523-7.

[23]

L. I. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the Schrödinger equation on a tree, Inverse Problems, 28 (2012), 015011, 30 pp. doi: 10.1088/0266-5611/28/1/015011.

[24]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. 

[25]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.

[26]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method Collection, RMA, vol 36, (Paris Masson), 1994.

[27]

D. Mugnolo, Mathematical Technology of Networks, Bielefeld, Springer Proceedings in Mathematics & Statistics, 128, 2015. doi: 10.1007/978-3-319-16619-3.

[28]

D. MugnoloD. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625.

[29]

D. Mugnolo and J.-F. Rault, Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks, Bull. Belg. Math. Soc. Simon Stevin., 21 (2014), 415-436. 

[30]

T. Tsutsumi, Strichartz estimates for Schrödinger equation of fourth order with periodic boundary condition, Kyoto University, (2014), 11 pp.

show all references

References:
[1]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Variational properties and orbital stability of standing waves for NLS equation on a star graph, J. Differential Equations, 257 (2014), 3738-3777.  doi: 10.1016/j.jde.2014.07.008.

[2]

R. AdamiC. CacciapuotiD. Finco and D. Noja, Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy, J. Differential Equations, 260 (2016), 7397-7415.  doi: 10.1016/j.jde.2016.01.029.

[3]

J. Angulo Pava and N. Goloshchapova, On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph, Discrete Contin. Dyn. Syst., 38 (2018), 5039-5066.  doi: 10.3934/dcds.2018221.

[4]

J. Angulo Pava and N. Goloshchapova, Extension theory approach in the stability of the standing waves for the NLS equation with point interactions on a star graph, Adv. Differential Equations, 23 (2018), 793-846. 

[5]

K. Ammari and H. Bouzidi, Exact boundary controllability of the linear biharmonic Schrödinger equation with variable coefficients, arXiv: 2112.15196 [math.AP] (2021).

[6]

K. Ammari and E. Crépeau, Feedback stabilization and boundary controllability of the Korteweg–de Vries equation on a star-shaped network, SIAM J. Control Optim., 56 (2018), 1620-1639.  doi: 10.1137/17M113959X.

[7]

K. Ammari and E. Crépeau, Well-posedness and stabilization of the Benjamin-Bona-Mahony equation on star-shaped networks, Systems Control Lett., 127 (2019), 39-43.  doi: 10.1016/j.sysconle.2019.03.005.

[8]

L. Baudouin and M. Yamamoto, Inverse problem on a tree-shaped network: Unified approach for uniqueness, Appl. Anal., 94 (2015), 2370-2395.  doi: 10.1080/00036811.2014.985214.

[9]

M. Ben-ArtziH. Koch and J.-C. Saut, Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 87-92.  doi: 10.1016/S0764-4442(00)00120-8.

[10]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/surv/186.

[11]

J. Blank, P. Exner and M. Havlicek, Hilbert Space Operators in Quantum Physics, 2nd edition, Theoretical and Mathematical Physics, Springer, New York, 2008.

[12]

J. L. Bona and R. C. Cascaval, Nonlinear dispersive waves on trees, Can. Appl. Math. Q., 16 (2008), 1-18. 

[13]

R. BurioniD. CassiM. RasettiP. Sodano and A. Vezzani, Bose-Einstein condensation on inhomogeneous complex networks, J. Phys. B: At. Mol. Opt. Phys., 34 (2001), 4697-4710. 

[14]

R. de A. Capistrano-FilhoM. Cavalcante and F. A. Gallego, Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane, Pacific J. Math., 309 (2020), 35-70.  doi: 10.2140/pjm.2020.309.35.

[15]

R. de A. Capistrano-Filho, M. Cavalcante and F. A. Gallego, Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation, Discrete & Continuous Dynamical Systems - B, 2021. doi: 10.3934/dcdsb.2021190.

[16]

M. Cavalcante, The Korteweg–de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69 (2018), Paper No. 124, 22 pp. doi: 10.1007/s00033-018-1018-6.

[17]

E. CerpaE. Crépeau and C. Moreno, On the boundary controllability of the Korteweg-de Vries equation on a star-shaped network, IMA J. Math. Control Inform., 37 (2020), 226-240.  doi: 10.1093/imamci/dny047.

[18]

E. CerpaE. Crépeau and J. Valein, Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network, Evol. Equ. Control Theory, 9 (2020), 673-692.  doi: 10.3934/eect.2020028.

[19]

A. Duca, Global exact controllability of bilinear quantum systems on compact graphs and energetic controllability, SIAM J. Control Optim., 58 (2020), 3092-3129.  doi: 10.1137/18M1212768.

[20]

A. Duca, Bilinear quantum systems on compact graphs: Well-posedness and global exact controllability, Automatica J. IFAC, 123 (2021), 109324, 13 pp. doi: 10.1016/j.automatica.2020.109324.

[21]

G. FibichB. Ilan and G. Papanicolaou, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462.  doi: 10.1137/S0036139901387241.

[22]

F. Gregorio and D. Mugnolo, Bi-Laplacians on graphs and networks, J. Evol. Equ., 20 (2020), 191-232.  doi: 10.1007/s00028-019-00523-7.

[23]

L. I. Ignat, A. F. Pazoto and L. Rosier, Inverse problem for the heat equation and the Schrödinger equation on a tree, Inverse Problems, 28 (2012), 015011, 30 pp. doi: 10.1088/0266-5611/28/1/015011.

[24]

V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E, 53 (1996), 1336-1339. 

[25]

V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger type equations with higher-order dispersion, Physica D, 144 (2000), 194-210.  doi: 10.1016/S0167-2789(00)00078-6.

[26]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method Collection, RMA, vol 36, (Paris Masson), 1994.

[27]

D. Mugnolo, Mathematical Technology of Networks, Bielefeld, Springer Proceedings in Mathematics & Statistics, 128, 2015. doi: 10.1007/978-3-319-16619-3.

[28]

D. MugnoloD. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625.

[29]

D. Mugnolo and J.-F. Rault, Construction of exact travelling waves for the Benjamin-Bona-Mahony equation on networks, Bull. Belg. Math. Soc. Simon Stevin., 21 (2014), 415-436. 

[30]

T. Tsutsumi, Strichartz estimates for Schrödinger equation of fourth order with periodic boundary condition, Kyoto University, (2014), 11 pp.

Figure 1.  A compact graph with $ N+1 $ edges
[1]

Zhongwei Tang. Segregated peak solutions of coupled Schrödinger systems with Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5299-5323. doi: 10.3934/dcds.2014.34.5299

[2]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367

[3]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[4]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028

[5]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations and Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020

[6]

Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control and Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189

[7]

Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175

[8]

Ran Zhuo, Fengquan Li, Boqiang Lv. Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space. Communications on Pure and Applied Analysis, 2014, 13 (3) : 977-990. doi: 10.3934/cpaa.2014.13.977

[9]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791

[10]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations and Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014

[11]

Alhabib Moumni, Jawad Salhi. Exact controllability for a degenerate and singular wave equation with moving boundary. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022001

[12]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

[13]

Bernard Ducomet, Alexander Zlotnik, Ilya Zlotnik. On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation. Kinetic and Related Models, 2009, 2 (1) : 151-179. doi: 10.3934/krm.2009.2.151

[14]

Tatsien Li, Bopeng Rao, Zhiqiang Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 243-257. doi: 10.3934/dcds.2010.28.243

[15]

Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763

[16]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150

[17]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665

[18]

Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3399-3434. doi: 10.3934/dcdsb.2021190

[19]

Jaime Angulo Pava, Nataliia Goloshchapova. On the orbital instability of excited states for the NLS equation with the δ-interaction on a star graph. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5039-5066. doi: 10.3934/dcds.2018221

[20]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations and Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (166)
  • HTML views (94)
  • Cited by (0)

[Back to Top]