CONTROLLABILITY OF A SIMPLIFIED TIME-DISCRETE STABILIZED KURAMOTO-SIVASHINSKY SYSTEM

In this paper, we study some controllability and observability prop-erties for a coupled system of time-discrete fourth-and second-order para-bolic 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(At)-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 At 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.