Optimal Control of Solutions to the Sixth Order Boussinesq Equation with Two Dispersions

Abstract

This article investigates the Cauchy–Dirichlet and Showalter-Sidorov–Dirichlet problems for an inhomogeneous nonlinear sixth-order Boussinesq equation with two dispersions, which corresponds to an incomplete semilinear Sobolev-type equation of the second order in Banach spaces. Sobolev–type equations are distinguished by the non-invertibility of the operator at the highest time derivative. The problem posed models waves in a channel between lakes (or seas) with external wind forcing, internal damping, and convection. We use the Galerkin method adapted for this mathematical model to prove the existence and uniqueness of a nonlocal solution. The starting point for justifying the *-weak convergence of the Galerkin approximations is the formal representation of the solution as a Galerkin sum over the eigenfunctions of the operator – Δ. The proof itself is based on the derived a priori estimates. Based on the proven theorem, we demonstrate the existence of a solution to the optimal control problem with a classical balance functional.