Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system

TL;DR

This paper investigates the controllability and observability of a simplified, time-discrete version of the stabilized Kuramoto-Sivashinsky system, revealing limitations in exact controllability due to discretization.

Contribution

It introduces new Carleman estimates for the time-discrete fourth-order equation and establishes a relaxed controllability result for the simplified system.

Findings

Proves a $ riangle t$-dependent controllability result.

Develops a new Carleman estimate for the fourth-order discretized equation.

Shows limitations in exact controllability for the discretized system.

Abstract

In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- and second-order parabolic equations. This system can be regarded as a simplification of the well-known stabilized Kumamoto-Sivashinsky equation. Unlike the continuous case, we can prove only a relaxed observability inequality which yields a $\phi(\triangle t)$-controllability result. This result tells that we cannot reach exactly zero but rather a small target whose size goes to 0 as the discretization parameter $\triangle t$ goes to 0. The proof relies on a known Carleman estimate for second-order time-discrete parabolic operators and a new Carleman estimate for the time-discrete fourth-order equation.