This paper is concerned with a viscous shallow water equation, which includes both the viscous Camassa-Holm equation and the viscous Degasperis-Procesi equation as its special cases. The optimal control under boundary conditions is given, and the existence of optimal solution to the equation is proved. 1. Introduction Holm and Staley [1] studied the following family of evolutionary 1+1 PDEs: which describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids. Here , , and is chosen to be the Green’s function for the Helmholtz operator on the line. In a recent study of soliton equations, it is found that (1) for and any is included in the family of shallow water equations at quadratic order accuracy that are asymptotically equivalent under Kodama transformations [2]. When , (1) becomes the -family of equations: which describes a one-dimensional version of active fluid transport. It was shown by Degasperis and Procesi [3] that (2) cannot satisfy the asymptotic integrability condition unless or ; compare [2, 4, 5]. For in (2), it becomes the Camassa-Holm (CH) equation: which is a model describing the unidirectional propagation of shallow water waves over a flat bottom [4]. Equation (3) has a bi-Hamiltonian structure [6] and is completely integrable [7, 8]. It admits, in addition to smooth waves, a multitude of traveling wave solutions with singularities: peakons, cuspons, stumpons, and composite waves [4, 9]. Its solitary waves are stable solitons [10, 11], retaining their shape and form after interactions [10]. The Cauchy problem of (3) has been studied extensively. Constantin [12] and Rodríguez-Blanco [13] investigated the locally well-posed for initial data with . More interestingly, it has strong solutions that are global in time [11, 14] as well as solutions that blow up in finite time [11, 15, 16]. On the other hand, Bressan and Constantin [17] and Xin and Zhang [18] showed that (3) has global weak solutions with initial data . For in (2), it becomes the Degasperis-Procesi (DP) equation: which can be used as a model for nonlinear shallow water dynamics, and its asymptotic accuracy is the same as (3). Degasperis et al. [5] presented that (4) has a bi-Hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to (3) peakons [4, 10, 19]. Dullin et al. [20] showed that (4) can be obtained from the shallow water elevation equation by an appropriate Kodama transformation. The numerical stability of solitons and
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