# American Institute of Mathematical Sciences

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 and 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), 15-33. doi: 10.1017/S0013091502000378. [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, Part 8 (Catania, 2000). Nonlinear Anal., 47 (2001), 5537-5546. doi: 10.1016/S0362-546X(01)00657-5. [3] R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal., 78 (2001), 255-269. doi: 10.1080/00036810108840938. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [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), 1044-1069. doi: 10.1137/040616711. [6] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equa., 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [8] A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036. [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), 303-328. [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [11] J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift, Math. Control Signal Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563. [12] E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution, Internat. J. Control, 74 (2001), 1096-1106. doi: 10.1080/00207170110052202. [13] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Diff. Equa., 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2. [14] C. de Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation, Comm. Part. Diff. Equa., 32 (2007), 87-126. doi: 10.1080/03605300601091470. [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), 436-452. doi: 10.1016/j.geomphys.2010.10.011. [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), 493-513. doi: 10.3934/dcds.2007.19.493. [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), 3125-3158. doi: 10.1016/j.jfa.2012.01.009. [18] W. Fu and D. J. Zhang, The Hamiltonian structures of $\mu$-equations related to periodic peakons, Chin. Phys. Lett., 30 (2013), 080201. [19] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, J. Diff. Equa., 245 (2008), 1584-1615. doi: 10.1016/j.jde.2008.06.016. [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), 479-495. doi: 10.1006/jdeq.1999.3695. [21] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075. [22] J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D., 79 (1994), 361-386. doi: 10.1016/S0167-2789(05)80015-6. [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), 617-656. doi: 10.1007/s00208-008-0250-3. [24] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Diff. Equa., 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [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), 27-63. doi: 10.1006/jdeq.1999.3683. [26] F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM Control Optim. Calc. Var., 11 (2005), 204-218. doi: 10.1051/cocv:2005002. [27] S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control Optim., 39 (2001), 1677-1696. doi: 10.1137/S0363012999362499. [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), 319-355. doi: 10.1016/j.jde.2012.02.011. [29] R. E. Showater, Hilbert Space Methods for Partial Differential Equations, Pitman, 1977. [30] M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304. doi: 10.1137/090768576. [31] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [32] K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system, J. Diff. Equa., 252 (2012), 2131-2159. doi: 10.1016/j.jde.2011.08.003. [33] Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283. doi: 10.1137/S0036141003425672. [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), 2003-2014. doi: 10.1016/j.jde.2009.08.002. [35] S. Yu, The spatially periodic Cauchy problem for a generalized 2-component $\mu$-Camassa-Holm system, Nonlinear Anal. Real World Appl., 19 (2014), 117-134. doi: 10.1016/j.nonrwa.2014.03.006. [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), 3386-3419.

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##### References:
 [1] W. Arendt and S. Bu, Operator-valued fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc., 47 (2004), 15-33. doi: 10.1017/S0013091502000378. [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, Part 8 (Catania, 2000). Nonlinear Anal., 47 (2001), 5537-5546. doi: 10.1016/S0362-546X(01)00657-5. [3] R. Beals, D. H. Sattinger and J. Szmigielski, Inverse scattering solutions of the Hunter-Saxton equation, Appl. Anal., 78 (2001), 255-269. doi: 10.1080/00036810108840938. [4] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [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), 1044-1069. doi: 10.1137/040616711. [6] A. Constantin, On the Cauchy problem for the periodic Camassa-Holm equation, J. Diff. Equa., 141 (1997), 218-235. doi: 10.1006/jdeq.1997.3333. [7] A. Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal., 155 (1998), 352-363. doi: 10.1006/jfan.1997.3231. [8] A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036. [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), 303-328. [10] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [11] J.-M. Coron, Global Asymptotic Stabilization for controllable systems without drift, Math. Control Signal Systems, 5 (1992), 295-312. doi: 10.1007/BF01211563. [12] E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution, Internat. J. Control, 74 (2001), 1096-1106. doi: 10.1080/00207170110052202. [13] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Diff. Equa., 192 (2003), 429-444. doi: 10.1016/S0022-0396(03)00096-2. [14] C. de Lellis, T. Kappeler and P. Topalov, Low-regularity solutions of the periodic Camassa-Holm equation, Comm. Part. Diff. Equa., 32 (2007), 87-126. doi: 10.1080/03605300601091470. [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), 436-452. doi: 10.1016/j.geomphys.2010.10.011. [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), 493-513. doi: 10.3934/dcds.2007.19.493. [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), 3125-3158. doi: 10.1016/j.jfa.2012.01.009. [18] W. Fu and D. J. Zhang, The Hamiltonian structures of $\mu$-equations related to periodic peakons, Chin. Phys. Lett., 30 (2013), 080201. [19] O. Glass, Controllability and asymptotic stabilization of the Camassa-Holm equation, J. Diff. Equa., 245 (2008), 1584-1615. doi: 10.1016/j.jde.2008.06.016. [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), 479-495. doi: 10.1006/jdeq.1999.3695. [21] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075. [22] J. K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D., 79 (1994), 361-386. doi: 10.1016/S0167-2789(05)80015-6. [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), 617-656. doi: 10.1007/s00208-008-0250-3. [24] J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Diff. Equa., 217 (2005), 393-430. doi: 10.1016/j.jde.2004.09.007. [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), 27-63. doi: 10.1006/jdeq.1999.3683. [26] F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation, ESAIM Control Optim. Calc. Var., 11 (2005), 204-218. doi: 10.1051/cocv:2005002. [27] S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control Optim., 39 (2001), 1677-1696. doi: 10.1137/S0363012999362499. [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), 319-355. doi: 10.1016/j.jde.2012.02.011. [29] R. E. Showater, Hilbert Space Methods for Partial Differential Equations, Pitman, 1977. [30] M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304. doi: 10.1137/090768576. [31] Z. Xin and P. Zhang, On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 53 (2000), 1411-1433. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5. [32] K. Yan and Z. Yin, On the Cauchy problem for a two-component Degasperis-Procesi system, J. Diff. Equa., 252 (2012), 2131-2159. doi: 10.1016/j.jde.2011.08.003. [33] Z. Yin, On the structure of solutions to the periodic Hunter-Saxton equation, SIAM J. Math. Anal., 36 (2004), 272-283. doi: 10.1137/S0036141003425672. [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), 2003-2014. doi: 10.1016/j.jde.2009.08.002. [35] S. Yu, The spatially periodic Cauchy problem for a generalized 2-component $\mu$-Camassa-Holm system, Nonlinear Anal. Real World Appl., 19 (2014), 117-134. doi: 10.1016/j.nonrwa.2014.03.006. [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), 3386-3419.
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