December  2016, 9(6): 2129-2148. doi: 10.3934/dcdss.2016088

Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system

1. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China

2. 

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013

3. 

Nonlinear Scienti c Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received  July 2015 Revised  September 2016 Published  November 2016

In this paper, we discuss two main problems. In the first section, we establish a new global distributed exact controllability of the periodic two-component $\mu\rho$-Hunter-Saxton system on the circle by means of a distributed control. And in the second section, we present corresponding result of the asymptotic stabilization problem about the periodic two-component $\mu\rho$-Hunter-Saxton system. By presenting concrete form of the feedback law, an equivalent system is got.
Citation: Jingqun Wang, Lixin Tian, Weiwei Guo. Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2129-2148. doi: 10.3934/dcdss.2016088
References:
[1]

W. Arendt and S. Bu, Operator-valued fourier multipliers on periodic Besov spaces and applications,, Proc. Edinb. Math. Soc., 47 (2004), 15. doi: 10.1017/S0013091502000378. Google Scholar

[2]

S. P. Banks, Exact boundary controllability and optimal control for a generalised Korteweg de Vries equation,, Proceedings of the Third World Congress of Nonlinear Analysts, 47 (2001), 5537. doi: 10.1016/S0362-546X(01)00657-5. Google Scholar

[3]

R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation,, Appl. Anal., 78 (2001), 255. doi: 10.1080/00036810108840938. Google Scholar

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[5]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation,, SIAM J. Math. Anal., 37 (2005), 1044. doi: 10.1137/040616711. Google Scholar

[6]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, J. Diff. Equa., 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar

[7]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar

[8]

A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996. doi: 10.1137/050623036. Google Scholar

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303. Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar

[11]

J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift,, Math. Control Signal Systems, 5 (1992), 295. doi: 10.1007/BF01211563. Google Scholar

[12]

E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution,, Internat. J. Control, 74 (2001), 1096. doi: 10.1080/00207170110052202. Google Scholar

[13]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Diff. Equa., 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar

[14]

C. de Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation,, Comm. Part. Diff. Equa., 32 (2007), 87. doi: 10.1080/03605300601091470. Google Scholar

[15]

J. Escher, M. Kohlmanna and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations,, Journal of Geometry and Physics, 61 (2011), 436. doi: 10.1016/j.geomphys.2010.10.011. Google Scholar

[16]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 19 (2007), 493. doi: 10.3934/dcds.2007.19.493. Google Scholar

[17]

Y. Fu, Y. Liu and C. Qu, On the blow up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, J. Funct. Anal., 262 (2012), 3125. doi: 10.1016/j.jfa.2012.01.009. Google Scholar

[18]

W. Fu and D. J. Zhang, The Hamiltonian structures of $\mu$-equations related to periodic peakons,, Chin. Phys. Lett., 30 (2013). Google Scholar

[19]

O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation,, J. Diff. Equa., 245 (2008), 1584. doi: 10.1016/j.jde.2008.06.016. Google Scholar

[20]

A. Himonas and G. Misiolek, Wellposedness of the Cauchy problem for a shal low water equation on the circle,, J. Diff. Equa., 161 (2000), 479. doi: 10.1006/jdeq.1999.3695. Google Scholar

[21]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075. Google Scholar

[22]

J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation,, Physica D., 79 (1994), 361. doi: 10.1016/S0167-2789(05)80015-6. Google Scholar

[23]

B. Khesin, J. Lenells and G. Misiolk, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617. doi: 10.1007/s00208-008-0250-3. Google Scholar

[24]

J. Lenells, Traveling wave solutions of the Camassa-Holm equation,, J. Diff. Equa., 217 (2005), 393. doi: 10.1016/j.jde.2004.09.007. Google Scholar

[25]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Equa., 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[26]

F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation,, ESAIM Control Optim. Calc. Var., 11 (2005), 204. doi: 10.1051/cocv:2005002. Google Scholar

[27]

S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation,, SIAM J. Control Optim., 39 (2001), 1677. doi: 10.1137/S0363012999362499. Google Scholar

[28]

B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,, J. Diff. Equa., 253 (2012), 319. doi: 10.1016/j.jde.2012.02.011. Google Scholar

[29]

R. E. Showater, Hilbert Space Methods for Partial Differential Equations,, Pitman, (1977). Google Scholar

[30]

M. Wunsch, The generalized Hunter-Saxton system,, SIAM J. Math. Anal., 42 (2010), 1286. doi: 10.1137/090768576. Google Scholar

[31]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

[32]

K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system,, J. Diff. Equa., 252 (2012), 2131. doi: 10.1016/j.jde.2011.08.003. Google Scholar

[33]

Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672. Google Scholar

[34]

Z. Y. Yin and C. Guan, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, J. Diff. Equa., 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002. Google Scholar

[35]

S. Yu, The spatially periodic Cauchy problem for a generalized 2-component $\mu$-Camassa-Holm system,, Nonlinear Anal. Real World Appl., 19 (2014), 117. doi: 10.1016/j.nonrwa.2014.03.006. Google Scholar

[36]

Y. Zhang, Y. Liu and C. Z. Qu, Blow up of solutions and traveling waves to the two-component $\mu$-Camassa-Holm system,, Int. Math. Res. Not., 15 (2013), 3386. Google Scholar

show all references

References:
[1]

W. Arendt and S. Bu, Operator-valued fourier multipliers on periodic Besov spaces and applications,, Proc. Edinb. Math. Soc., 47 (2004), 15. doi: 10.1017/S0013091502000378. Google Scholar

[2]

S. P. Banks, Exact boundary controllability and optimal control for a generalised Korteweg de Vries equation,, Proceedings of the Third World Congress of Nonlinear Analysts, 47 (2001), 5537. doi: 10.1016/S0362-546X(01)00657-5. Google Scholar

[3]

R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation,, Appl. Anal., 78 (2001), 255. doi: 10.1080/00036810108840938. Google Scholar

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar

[5]

G. M. Coclite, H. Holden and K. H. Karlsen, Global weak solutions to a generalized hyperelastic-rod wave equation,, SIAM J. Math. Anal., 37 (2005), 1044. doi: 10.1137/040616711. Google Scholar

[6]

A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation,, J. Diff. Equa., 141 (1997), 218. doi: 10.1006/jdeq.1997.3333. Google Scholar

[7]

A. Constantin, On the inverse spectral problem for the Camassa-Holm equation,, J. Funct. Anal., 155 (1998), 352. doi: 10.1006/jfan.1997.3231. Google Scholar

[8]

A. Constantin, Global solutions of the Hunter-Saxton equation,, SIAM J. Math. Anal., 37 (2005), 996. doi: 10.1137/050623036. Google Scholar

[9]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303. Google Scholar

[10]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586. Google Scholar

[11]

J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift,, Math. Control Signal Systems, 5 (1992), 295. doi: 10.1007/BF01211563. Google Scholar

[12]

E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution,, Internat. J. Control, 74 (2001), 1096. doi: 10.1080/00207170110052202. Google Scholar

[13]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Diff. Equa., 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2. Google Scholar

[14]

C. de Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation,, Comm. Part. Diff. Equa., 32 (2007), 87. doi: 10.1080/03605300601091470. Google Scholar

[15]

J. Escher, M. Kohlmanna and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations,, Journal of Geometry and Physics, 61 (2011), 436. doi: 10.1016/j.geomphys.2010.10.011. Google Scholar

[16]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 19 (2007), 493. doi: 10.3934/dcds.2007.19.493. Google Scholar

[17]

Y. Fu, Y. Liu and C. Qu, On the blow up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,, J. Funct. Anal., 262 (2012), 3125. doi: 10.1016/j.jfa.2012.01.009. Google Scholar

[18]

W. Fu and D. J. Zhang, The Hamiltonian structures of $\mu$-equations related to periodic peakons,, Chin. Phys. Lett., 30 (2013). Google Scholar

[19]

O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation,, J. Diff. Equa., 245 (2008), 1584. doi: 10.1016/j.jde.2008.06.016. Google Scholar

[20]

A. Himonas and G. Misiolek, Wellposedness of the Cauchy problem for a shal low water equation on the circle,, J. Diff. Equa., 161 (2000), 479. doi: 10.1006/jdeq.1999.3695. Google Scholar

[21]

J. K. Hunter and R. Saxton, Dynamics of director fields,, SIAM J. Appl. Math., 51 (1991), 1498. doi: 10.1137/0151075. Google Scholar

[22]

J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation,, Physica D., 79 (1994), 361. doi: 10.1016/S0167-2789(05)80015-6. Google Scholar

[23]

B. Khesin, J. Lenells and G. Misiolk, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,, Math. Ann., 342 (2008), 617. doi: 10.1007/s00208-008-0250-3. Google Scholar

[24]

J. Lenells, Traveling wave solutions of the Camassa-Holm equation,, J. Diff. Equa., 217 (2005), 393. doi: 10.1016/j.jde.2004.09.007. Google Scholar

[25]

Y. A. Li and P. J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,, J. Diff. Equa., 162 (2000), 27. doi: 10.1006/jdeq.1999.3683. Google Scholar

[26]

F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation,, ESAIM Control Optim. Calc. Var., 11 (2005), 204. doi: 10.1051/cocv:2005002. Google Scholar

[27]

S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation,, SIAM J. Control Optim., 39 (2001), 1677. doi: 10.1137/S0363012999362499. Google Scholar

[28]

B. Moon and Y. Liu, Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,, J. Diff. Equa., 253 (2012), 319. doi: 10.1016/j.jde.2012.02.011. Google Scholar

[29]

R. E. Showater, Hilbert Space Methods for Partial Differential Equations,, Pitman, (1977). Google Scholar

[30]

M. Wunsch, The generalized Hunter-Saxton system,, SIAM J. Math. Anal., 42 (2010), 1286. doi: 10.1137/090768576. Google Scholar

[31]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. Google Scholar

[32]

K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system,, J. Diff. Equa., 252 (2012), 2131. doi: 10.1016/j.jde.2011.08.003. Google Scholar

[33]

Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation,, SIAM J. Math. Anal., 36 (2004), 272. doi: 10.1137/S0036141003425672. Google Scholar

[34]

Z. Y. Yin and C. Guan, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, J. Diff. Equa., 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002. Google Scholar

[35]

S. Yu, The spatially periodic Cauchy problem for a generalized 2-component $\mu$-Camassa-Holm system,, Nonlinear Anal. Real World Appl., 19 (2014), 117. doi: 10.1016/j.nonrwa.2014.03.006. Google Scholar

[36]

Y. Zhang, Y. Liu and C. Z. Qu, Blow up of solutions and traveling waves to the two-component $\mu$-Camassa-Holm system,, Int. Math. Res. Not., 15 (2013), 3386. Google Scholar

[1]

Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719

[2]

Alejandro Sarria. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 641-673. doi: 10.3934/dcdsb.2015.20.641

[3]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643

[4]

Jaeho Choi, Nitin Krishna, Nicole Magill, Alejandro Sarria. On the $ L^p $ regularity of solutions to the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6349-6365. doi: 10.3934/dcdsb.2019142

[5]

Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459

[6]

Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084

[7]

Marcus Wunsch. On the Hunter--Saxton system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 647-656. doi: 10.3934/dcdsb.2009.12.647

[8]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[9]

Zeng Zhang, Zhaoyang Yin. Global existence for a two-component Camassa-Holm system with an arbitrary smooth function. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5523-5536. doi: 10.3934/dcds.2018243

[10]

Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134

[11]

Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041

[12]

Yong Chen, Hongjun Gao, Yue Liu. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407

[13]

Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239

[14]

Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613

[15]

Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062

[16]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699

[17]

Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140

[18]

Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263

[19]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

[20]

José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019039

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]