Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky 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 φ(△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 △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.

Null controllability of the linear Stabilized Kuramoto-Sivashinsky system using moment method

“…Let us also mention the most recent works [10] and [37] which deals with an insensitizing control problem and controllability of stochastic stabilized KS system, respectively. Lastly, we mention the work [35] which deals with the controllability issues of stabilized KS system in numerical setup by discretizing the time variable.…”