Optimal control of the viscous Camassa–Holm equation

The optimal control problem with necessity condition for a viscous shallow water equation

“…Due to the independence of coefficients m, a and b in (1.4), the nonlinear term uu x does not disappear after using the transformation y = uu xx , which leads to the difficulty of establishing the estimates for term uu x . This is the major improvement in comparison with the results in the literature [5,10,17,32], where the problems studied are special cases of the optimal control problem (1.3) in this paper. Moreover, we obtain the necessity condition and local uniqueness of optimal control to the optimal control problem (1.3) by using the Gâteaux derivative of cost functional.…”

“…For the methods to establish local well-posedness for the Cauchy problem of CamassaHolm equation and global existence of solutions, one may refer to [7][8][9] and the references therein. Tian [10] studied the optimal control problem for a generalized viscous Camassa-Holm equation. They established the existence and uniqueness of local weak solutions by using the Galerkin method.…”

On the Cauchy problem for a two-component b-family system

“…(1.2) becomes the Camassa-Holm equation, modelling the unidirectional propagation of shallow water waves over a flat bottom. The Cauchy problem of the Camassa-Holm equation [6] has been extensively considered, see [7][8][9][10]. For b = 3 and c 0 = ρ = 0, Eq.…”

On the optimal control problem for the Novikov equation with strong viscosity

“…The optimal control of viscous peakon PDEs with square nonlinearity, such as viscous Camassa-Holm equation [30], viscous Degasperis-Procesi equation [31] and viscous Dullin-Gottwalld-Holm equation [32,41] up to a trivial change of variables [22], we intend to investigate an optimal control problem governed by the strongly viscous Novikov equation…”