Control of Kawahara equation with overdetermination condition: The unbounded cases

TL;DR

This paper investigates the internal control of the Kawahara equation on unbounded domains, establishing conditions for controllability, existence of control inputs, and minimal control time, with implications for exact controllability related to the equation's mass.

Contribution

It proves the existence of internal controls satisfying overdetermination conditions for the Kawahara equation on unbounded domains, including minimal time and exact controllability results.

Findings

Existence of internal control inputs under certain initial and boundary conditions.

Control can be achieved in finite minimal time.

Established exact controllability related to the mass of the solution.

Abstract

In this manuscript we consider the internal control problem for the fifth order KdV type equation, commonly called the Kawahara equation, on unbounded domains. Precisely, under certain hypotheses over the initial and boundary data, we are able to prove that there exists an internal control input such that solutions of the Kawahara equation satisfies an integral overdetermination condition. This condition is satisfied when the domain of the Kawahara equation is posed in the real line, left half-line and right half-line. Moreover, we are also able to prove that there exists a minimal time in which the integral overdetermination condition is satisfied. Finally, we show a type of exact controllability associated with the "mass" of the Kawahara equation posed in the half-line.