Optimal Control Problem with Necessary Optimality Conditions for the Viscous Dullin-Gottwald-Holm Equation

Jinsoo Hwang

doi:10.1155/2014/623129

Abstract

We study the quadratic cost optimal control problems for the viscous Dullin-Gottwald-Holm equation. The main novelty of this paper is to derive the necessary optimality conditions of optimal controls, corresponding to physically meaningful distributive observations. For this we prove the Gâteaux differentiability of nonlinear solution mapping on control variables. Moreover by making use of the second order Gâteaux differentiability of solution mapping on control variables, we prove the local uniqueness of optimal control. This is another novelty of the paper.

Highlights

In the study of shallow water wave, Dullin et al [1] derived a new integrable shallow water wave equation with linear and nonlinear dispersion as follows: yt + 2ωyx + 3yyx − α2 + γyxxx = 0, (1)where y is fluid velocity, α2 and γ/2ω are squares of length scales, and 2ω = √gh is the linear wave speed for undisturbed water at rest at spatial infinity, where y and its spatial derivatives are taken to vanish
In order to apply the variational approach due to Lions [10] to our problem, we propose the quadratic cost functional J(V) as studied in Lions [10] which is to be minimized within Uad; Uad is an admissible set of control variables in U
In order to apply the variational approach due to Lions [10] to our problem, we proposed the quadratic cost functional as studied in Lions [10] which is to be minimized within an admissible set of control variables

Summary

Introduction

In this paper, we propose the quadratic cost functional for y, which is more reasonable than that for m, and establish the necessary optimality conditions of optimal controls due to Lions [10] in Theorems 5 and 7 for the physically meaningful observations z = y(u) and z = m(u), respectively. To this end, we successfully characterize the Gateaux derivative Dy(u)(V − u) of y(V) in the direction V − u ∈ U, where U is a Hilbert space of control variables and u is an optimal control for quadratic cost.

Preliminaries

Quadratic Cost Optimal Control Problems

Conclusions