Optimal Distributed Control Problem for the &lt;i&gt;b&lt;/i&gt;-Equation

Chunyu Shen

doi:10.4236/jamp.2017.56108

Abstract

This paper is concerned with the optimal distributed control problem governed by b-equation. We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition. By contrasting with our previous result, the proof without considering viscous coefficient is a big improvement. Secondly, based on the well-posedness result, we find a unique optimal control for the controlled system with the quadratic cost functional. Moreover, by means of the optimal control theory, we obtain the sufficient and necessary optimality condition of an optimal control, which is another major novelty of this paper. Finally, we also present the optimality conditions corresponding to two physical meaningful distributive observation cases.

Highlights

Escher and Yin [1] studied the following nonlinear dispersive equation (b-equation):= uut(−0,αx2)uxxut 0+(cx0)u,x +( x∈ b+ R, 1)uu= x + Γuxxx α 2, t > 0, x ∈ R, (1.1)where c0, b, Γ and α are arbitrary real constants
This paper is concerned with the optimal distributed control problem governed by b-equation
We firstly investigate the existence and uniqueness of weak solution for the controlled system with appropriate initial value and boundary condition

Summary

Introduction

Escher and Yin [1] studied the following nonlinear dispersive equation (b-equation):. In [1], Escher and Yin studied b-equation on the line for α > 0 and c0 , b, Γ ∈ R They established the local well-posedness, described the precise blow-up scenario, and proved that the equation has strong solutions which exist globally in time and blow up in finite time. In [64], the authors established the local well-posedness for the nonuniform weakly dissipative b-equation which includes both the weakly dissipative CH equation and the weakly dissipative DP equation as its special cases They studied the blow-up phenomena and the long time behavior of the solutions.

The Existence and Uniqueness of Weak Solution

The Existence and Uniqueness of an Optimal Control

The Sufficient and Necessary Optimality Condition

The Two Cases of Distributive Observations