Control and stabilization of the Kawahara equation on a periodic domain

In this paper, we study a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain T (a unit circle in the plane) with an internal control acting on an arbitrary small nonempty subdomain of T. Aided by the Bourgain smoothing property of the Kawahara equation on a periodic domain, we show that the system is locally exactly controllable and exponentially stabilizable with an arbitrarily large decay rate.

[1]  Bing-Yu Zhang,et al.  Exact Boundary Controllability of the Korteweg--de Vries Equation , 1999 .

[2]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[3]  John P. Boyd,et al.  Weakly non-local solutions for capillary-gravity waves: fifth-degree Korteweg-de Vries equation , 1991 .

[4]  Takamori Kato,et al.  Low regularity well-posedness for the periodic Kawahara equation , 2012, Differential and Integral Equations.

[5]  Jean-Michel Coron,et al.  Exact boundary controllability of a nonlinear KdV equation with critical lengths , 2004 .

[6]  Hiroaki Ono,et al.  Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma , 1969 .

[7]  Nikolai A. Larkin,et al.  Kawahara equation in a bounded domain , 2008 .

[8]  Eduardo Cerpa,et al.  Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain , 2009 .

[9]  Lionel Rosier,et al.  Control and stabilization of the Korteweg-de Vries equation: recent progresses , 2009, J. Syst. Sci. Complex..

[10]  D. Korteweg,et al.  On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 2011 .

[11]  M. Slemrod.  A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space , 1974 .

[12]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[13]  Roger Grimshaw,et al.  Water Waves , 2021, Mathematics of Wave Propagation.

[14]  Bing-Yu Zhang,et al.  Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation , 2012 .

[15]  John K. Hunter,et al.  Existence of perturbed solitary wave solutions to a model equation for water waves , 1988 .

[16]  Lionel Rosier,et al.  Exact boundary controllability of the nonlinear Schrödinger equation , 2009 .

[17]  Yi Hu,et al.  Local Well-Posedness of Periodic Fifth-Order KdV-Type Equations , 2011 .

[18]  David L. Russell,et al.  Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain , 1993 .

[19]  David L. Russell,et al.  Exact controllability and stabilizability of the Korteweg-de Vries equation , 1996 .

[20]  Eduardo Cerpa,et al.  Exact Controllability of a Nonlinear Korteweg--de Vries Equation on a Critical Spatial Domain , 2007, SIAM J. Control. Optim..

[21]  V. Karpman,et al.  Dynamics of two-dimensional solitons in weakly dispersive media , 1991 .

[22]  Bing-Yu Zhang,et al.  Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation , 2012, Acta Applicandae Mathematicae.

[23]  Carlos Frederico Vasconcellos,et al.  Stabilization of the Kawahara equation with localized damping , 2011 .

[24]  Takuji Kawahara,et al.  Oscillatory Solitary Waves in Dispersive Media , 1972 .

[25]  Luis Vega,et al.  A bilinear estimate with applications to the KdV equation , 1996 .

[26]  A. Alexandrou Himonas,et al.  Well-posedness of KdV with higher dispersion , 2009, Math. Comput. Simul..

[27]  Lionel Rosier,et al.  Control and Stabilization of the Korteweg-de Vries Equation on a Periodic Domain , 2009 .

[28]  R. Miura.  The Korteweg–deVries Equation: A Survey of Results , 1976 .

[29]  Peter Constantin,et al.  Local smoothing properties of dispersive equations , 1988 .

[30]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .