Active control of an improved Boussinesq system

Kenan Yildirim

doi:10.1051/mmnp/2020024

Abstract

In this paper, optimal control of excessive water waves in a canal system, modeled by a nonlinear improved Boussinesq equation, is considered. For this aim, well-posedness and controllability properties of the system is investigated. Suppressing of the waves in the canal system is successfully obtained by means of optimally determining of canal depth control function via maximum principle, which transforms to optimal control problem to solving an nonlinear initial-boundary-terminal value problem. The beauty of the present paper than other studies existing in the literature is that optimal canal depth control function is analytically obtained without linearization of nonlinear term. In order to show effectiveness and robustness of the control actuation, several numerical examples are given by MATLAB in tables and graphical forms.

Highlights

In recent years, nonlinear evolution equations have acquired great attentions due to their properties on modeling of real world problems and explaining some nonlinear phenomena
In [14], Li considered the maximum principle for an optimal control problem governed by Boussinesq equations including integral type state constraints
A more general form of equation (1.3) is considered by taking the nonlinear term as N (u) in stead of u2 and the system is taken into account as an optimal control problem by considering the optimally controlling of excessive water waves in a canal system, modeled by a nonlinear improved Boussinesq equation

Summary

Introduction

Nonlinear evolution equations have acquired great attentions due to their properties on modeling of real world problems and explaining some nonlinear phenomena. A more general form of equation (1.3) is considered by taking the nonlinear term as N (u) in stead of u2 and the system is taken into account as an optimal control problem by considering the optimally controlling of excessive water waves in a canal system, modeled by a nonlinear improved Boussinesq equation. For this aim, under some assumptions on nonlinear term and solution, wellposedness and controllability results of the system is discussed and uniqueness of the solution is given by a lemma. Obtained results are simulated by means of MATLAB and given by tables and graphical forms for indicating the effectiveness of the introduced control actuation

Mathematical formulation of the problem

Optimal control problem

Numerical results and discussions

Conclusion