Control results with overdetermination condition for higher order dispersive system

TL;DR

This paper establishes new controllability results for a fifth order dispersive water wave model, demonstrating boundary and internal control strategies that satisfy overdetermination conditions, advancing control theory for higher order dispersive systems.

Contribution

It introduces a novel controllability framework for a fifth order dispersive equation, addressing open problems and providing new methods for boundary and internal control.

Findings

Boundary control guarantees overdetermination condition satisfaction.

Internal control also achieves the overdetermination condition.

Answers open questions from previous research in the field.

Abstract

In recent years, controllability problems for dispersive systems have been extensively studied. This work is dedicated to proving a new type of controllability for a dispersive fifth order equation that models water waves, what we will now call the overdetermination control problem. Precisely, we are able to find a control acting at the boundary that guarantees that the solutions of the problem under consideration satisfy an overdetermination integral condition. In addition, when we make the control act internally in the system, instead of the boundary, we are also able to prove that this condition is satisfied. These problems give answers that were left open in [6] and present a new way to prove boundary and internal controllability results for a fifth order KdV type equation.