Global controllability and stabilizability of Kawahara equation on a periodic domain

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June 2015, 5(2): 335-358. doi: 10.3934/mcrf.2015.5.335

Global controllability and stabilizability of Kawahara equation on a periodic domain

Xiangqing Zhao ^1, and Bing-Yu Zhang ^2,

1.	Department of Mathematics, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, China
2.	Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Oh 45221

Received January 2014 Revised July 2014 Published April 2015

Abstract
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In this paper we study controllability and stabilizability of a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain $\mathbb{T}$ with internal control acting on a sub-domain $\omega $ of $\mathbb{T}$. Earlier in [42], aided by Bourgain smoothing property of the system, we showed that the system is locally exactly controllable and exponentially stabilizable. In this paper, helped further by certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system, we show that the system is globally exactly controllable and globally exponentially stabilizable.

Keywords: propagation of compactness, Kawahara equation, global stabilizability, propagation of regularity, global exact controllability, Bourgain space and Bourgain smoothing..

Mathematics Subject Classification: Primary: 93B05, 93D15; Secondary: 35Q5.

Citation: Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control and Related Fields, 2015, 5 (2) : 335-358. doi: 10.3934/mcrf.2015.5.335

References:

[1]	L. A. Abramyan and Yu. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Soviet Phys. JETP, 61 (1985), 963-966.
[2]	J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-Value problem for the Kortewege Vries Equation posed on a finite domain, Communications in Partial Differetial Equations, 28 (2003), 1391-1436. doi: 10.1081/PDE-120024373.
[3]	J. L. Bona, S. M. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Kortewege Vries and the Korteweg de Vries Burgers equations in aquarter plane, Ann. I. H. Poincaré-AN, 25 (2008), 1145-1185. doi: 10.1016/j.anihpc.2007.07.006.
[4]	J. P. Boyd, Weakly non-local solitons for capillary-gravity waves: Fifth degree Korteweg-de Vries equation, Phys. D, 48 (1991), 129-146. doi: 10.1016/0167-2789(91)90056-F.
[5]	J. Bourgain, Fourrier transform restriction phenomena for certain lattice subsets and applications to nonlinear evlution equations, Part I: The Schrödinger equation, Part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.
[6]	E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., 46 (2007), 877-899. doi: 10.1137/06065369X.
[7]	E. Cerpa and E. Crepeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. I.H. Poincaré-AN, 26 (2009), 457-475. doi: 10.1016/j.anihpc.2007.11.003.
[8]	W. Chen and Z. Guo, Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation, Amer. J. Math., 114 (2011), 121-156. doi: 10.1007/s11854-011-0014-y.
[9]	P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. doi: 10.1090/S0894-0347-1988-0928265-0.
[10]	J. M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398.
[11]	S. B. Cui and S. P. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, J. Math. Appl., 304 (2005), 683-702. doi: 10.1016/j.jmaa.2004.09.049.
[12]	S. B. Cui, D. G. Deng and S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data, Acta Math. Sin., 22 (2006), 1457-1466. doi: 10.1007/s10114-005-0710-6.
[13]	B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36 (2003), 525-551. doi: 10.1016/S0012-9593(03)00021-1.
[14]	J. Gorsky and A. A. Himonas, Well-posedness of KdV with higher dispersion, Math. Comput. Simul., 80 (2009), 173-183. doi: 10.1016/j.matcom.2009.06.007.
[15]	A. Grünrock, New applications of the Fourier restriction norm method, Ph.D thesis, Universität Wuppertal, 2002.
[16]	H. Hasimoto, Water waves, Kagaku, 40 (1970), 401-408 [in Japanese].
[17]	H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations, 19 (2012), 677-693. doi: 10.1007/s00030-011-0147-9.
[18]	J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32 (1988), 253-268. doi: 10.1016/0167-2789(88)90054-1.
[19]	Y. Hu, Discrete Fourier Restriction Phenomena Associated with Some Periodic Dispersive Wave Equations, Ph.D thesis, University of Illinois at Urbana-Champaign, 2012.
[20]	T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Japan, 26 (1969), 1305-1318. doi: 10.1143/JPSJ.26.1305.
[21]	V. I. Karpman and V. Yu. Belashov, Dynamics of two-dimensional soliton in weakly dispersive media, Phys. Lett. A, 154 1991, 131-139.
[22]	T. Kato, Low regularity well-posedness for the periodic Kawahara equation,, , ().
[23]	T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264. doi: 10.1143/JPSJ.33.260.
[24]	C. Kenig, G. Ponce and L. Vega, A biliner estimate with applications to the KdV equations, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[25]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.
[26]	C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval, ESAIM-COCV, 16 (2010), 356-379. doi: 10.1051/cocv/2009001.
[27]	C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Communications in Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336.
[28]	J. F. Li and S. G. Shi, Local well-posedness for the dispersion generalized periodic KdV equation, J. Math. Anal. Appl., 379 (2011), 706-718. doi: 10.1016/j.jmaa.2011.01.026.
[29]	L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), 4129-4153. doi: 10.1016/j.jde.2008.11.004.
[30]	D. L. Russell, Computational study of the Korteweg-de Vries equation with localized control action, in Distributed Parameter Control Systems (eds. G. Chen, E. B. Lee, W. Littman and L. Markus), Lecture Notes in Pure and Appl. Math., 128, Dekker, New York, 1991, 195-203.
[31]	D. L. Russell and B.-Y. Zhang, Exact controllability and stabiability of the Korteweg-de Veies equation, Transaction of the American Mathematical Society, 348 (1996), 3643-3672. doi: 10.1090/S0002-9947-96-01672-8.
[32]	D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control and Optimization, 31 (1993), 659-676. doi: 10.1137/0331030.
[33]	P. N. da Silva, Unique Continuation for the Kawahara Equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. doi: 10.5540/tema.2007.08.03.0463.
[34]	M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12 (1974), 500-508. doi: 10.1137/0312038.
[35]	T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106., American Mathematical Society, Providence, RI, 2006.
[36]	L. Tartar, Nonlinear interpolation and regularity, J. Funct. Anal., 9 (1972), 469-489.
[37]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the linear Kawahara equation with localized damping, Asymptotic Analysis, 58 (2008), 229-252.
[38]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with a localized damping, ESAIM: COCV, 17 (2011), 102-116. doi: 10.1051/cocv/2009041.
[39]	H. Wang, S. B. Cui and D. G. Deng, Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices, Acta Math. Sin., 23 (2007), 1435-1446. doi: 10.1007/s10114-007-0959-z.
[40]	Y. Wu and Y. S. Li, The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity, Math. Meth. Appl. Sci., 33 (2010), 1647-1660. doi: 10.1002/mma.1273.
[41]	B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Cont. Optim., 37 (1999), 543-565. doi: 10.1137/S0363012997327501.
[42]	B.-Y. Zhang and X. Q. Zhao, Control and stabilization of the Kawahara equation on a periodic, Communications in Information and Systems, 12 (2012), 77-95. doi: 10.4310/CIS.2012.v12.n1.a4.

show all references

References:

[1]	L. A. Abramyan and Yu. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Soviet Phys. JETP, 61 (1985), 963-966.
[2]	J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-Value problem for the Kortewege Vries Equation posed on a finite domain, Communications in Partial Differetial Equations, 28 (2003), 1391-1436. doi: 10.1081/PDE-120024373.
[3]	J. L. Bona, S. M. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Kortewege Vries and the Korteweg de Vries Burgers equations in aquarter plane, Ann. I. H. Poincaré-AN, 25 (2008), 1145-1185. doi: 10.1016/j.anihpc.2007.07.006.
[4]	J. P. Boyd, Weakly non-local solitons for capillary-gravity waves: Fifth degree Korteweg-de Vries equation, Phys. D, 48 (1991), 129-146. doi: 10.1016/0167-2789(91)90056-F.
[5]	J. Bourgain, Fourrier transform restriction phenomena for certain lattice subsets and applications to nonlinear evlution equations, Part I: The Schrödinger equation, Part II: The KdV equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.
[6]	E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., 46 (2007), 877-899. doi: 10.1137/06065369X.
[7]	E. Cerpa and E. Crepeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. I.H. Poincaré-AN, 26 (2009), 457-475. doi: 10.1016/j.anihpc.2007.11.003.
[8]	W. Chen and Z. Guo, Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation, Amer. J. Math., 114 (2011), 121-156. doi: 10.1007/s11854-011-0014-y.
[9]	P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439. doi: 10.1090/S0894-0347-1988-0928265-0.
[10]	J. M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398.
[11]	S. B. Cui and S. P. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, J. Math. Appl., 304 (2005), 683-702. doi: 10.1016/j.jmaa.2004.09.049.
[12]	S. B. Cui, D. G. Deng and S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data, Acta Math. Sin., 22 (2006), 1457-1466. doi: 10.1007/s10114-005-0710-6.
[13]	B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36 (2003), 525-551. doi: 10.1016/S0012-9593(03)00021-1.
[14]	J. Gorsky and A. A. Himonas, Well-posedness of KdV with higher dispersion, Math. Comput. Simul., 80 (2009), 173-183. doi: 10.1016/j.matcom.2009.06.007.
[15]	A. Grünrock, New applications of the Fourier restriction norm method, Ph.D thesis, Universität Wuppertal, 2002.
[16]	H. Hasimoto, Water waves, Kagaku, 40 (1970), 401-408 [in Japanese].
[17]	H. Hirayama, Local well-posedness for the periodic higher order KdV type equations, NoDEA Nonlinear Differential Equations, 19 (2012), 677-693. doi: 10.1007/s00030-011-0147-9.
[18]	J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32 (1988), 253-268. doi: 10.1016/0167-2789(88)90054-1.
[19]	Y. Hu, Discrete Fourier Restriction Phenomena Associated with Some Periodic Dispersive Wave Equations, Ph.D thesis, University of Illinois at Urbana-Champaign, 2012.
[20]	T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision-free plasma, J. Phys. Soc. Japan, 26 (1969), 1305-1318. doi: 10.1143/JPSJ.26.1305.
[21]	V. I. Karpman and V. Yu. Belashov, Dynamics of two-dimensional soliton in weakly dispersive media, Phys. Lett. A, 154 1991, 131-139.
[22]	T. Kato, Low regularity well-posedness for the periodic Kawahara equation,, , ().
[23]	T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264. doi: 10.1143/JPSJ.33.260.
[24]	C. Kenig, G. Ponce and L. Vega, A biliner estimate with applications to the KdV equations, J. Amer. Math. Soc., 9 (1996), 573-603. doi: 10.1090/S0894-0347-96-00200-7.
[25]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443. doi: 10.1080/14786449508620739.
[26]	C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval, ESAIM-COCV, 16 (2010), 356-379. doi: 10.1051/cocv/2009001.
[27]	C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Communications in Partial Differential Equations, 35 (2010), 707-744. doi: 10.1080/03605300903585336.
[28]	J. F. Li and S. G. Shi, Local well-posedness for the dispersion generalized periodic KdV equation, J. Math. Anal. Appl., 379 (2011), 706-718. doi: 10.1016/j.jmaa.2011.01.026.
[29]	L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), 4129-4153. doi: 10.1016/j.jde.2008.11.004.
[30]	D. L. Russell, Computational study of the Korteweg-de Vries equation with localized control action, in Distributed Parameter Control Systems (eds. G. Chen, E. B. Lee, W. Littman and L. Markus), Lecture Notes in Pure and Appl. Math., 128, Dekker, New York, 1991, 195-203.
[31]	D. L. Russell and B.-Y. Zhang, Exact controllability and stabiability of the Korteweg-de Veies equation, Transaction of the American Mathematical Society, 348 (1996), 3643-3672. doi: 10.1090/S0002-9947-96-01672-8.
[32]	D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain, SIAM J. Control and Optimization, 31 (1993), 659-676. doi: 10.1137/0331030.
[33]	P. N. da Silva, Unique Continuation for the Kawahara Equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. doi: 10.5540/tema.2007.08.03.0463.
[34]	M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space, SIAM J. Control, 12 (1974), 500-508. doi: 10.1137/0312038.
[35]	T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106., American Mathematical Society, Providence, RI, 2006.
[36]	L. Tartar, Nonlinear interpolation and regularity, J. Funct. Anal., 9 (1972), 469-489.
[37]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the linear Kawahara equation with localized damping, Asymptotic Analysis, 58 (2008), 229-252.
[38]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with a localized damping, ESAIM: COCV, 17 (2011), 102-116. doi: 10.1051/cocv/2009041.
[39]	H. Wang, S. B. Cui and D. G. Deng, Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices, Acta Math. Sin., 23 (2007), 1435-1446. doi: 10.1007/s10114-007-0959-z.
[40]	Y. Wu and Y. S. Li, The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity, Math. Meth. Appl. Sci., 33 (2010), 1647-1660. doi: 10.1002/mma.1273.
[41]	B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Cont. Optim., 37 (1999), 543-565. doi: 10.1137/S0363012997327501.
[42]	B.-Y. Zhang and X. Q. Zhao, Control and stabilization of the Kawahara equation on a periodic, Communications in Information and Systems, 12 (2012), 77-95. doi: 10.4310/CIS.2012.v12.n1.a4.

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