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

Roberto de A. Capistrano–Filho; Márcio Cavalcante; Fernando A. Gallego

doi:10.3934/eect.2022019

Article Contents

2023, Volume 12, Issue 1: 1-19. Doi: 10.3934/eect.2022019

This issue Previous Article Next Article Stability properties for a problem of light scattering in a dispersive metallic domain

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
^* Corresponding author: Roberto de A. Capistrano–Filho

Received: December 2021

Revised: March 2022

Early access: April 2022

Published: February 2023

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35R02, 35Q55, 35G30, 93B05, 93B07.

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

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[1]	R. Adami, C. Cacciapuoti, D. 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. Adami, C. Cacciapuoti, D. 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-Artzi, H. 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. Burioni, D. Cassi, M. Rasetti, P. 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-Filho, M. 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. Cerpa, E. 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. Cerpa, E. 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. Fibich, B. 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. Mugnolo, D. 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.