An insensitizing control problem for a linear Stabilized Kuramoto-Sivashinsky system

Abstract

In this work, we address the existence of insensitizing controls for a coupled system of fourth-and second-order parabolic equations known as the stabilized Kuramoto-Sivashinsky model. The main idea is to look for control functions such that some functional of the state is locally insensitive to the perturbations of initial data. Let O be a nonempty observation set for the solution component(s) w.r.t. the L 2 -norm(s) and ω be another nonempty set where the interior controls are acting. Then, we study the associated insensitizing control problem under the assumption O ∩ ω = ∅ and as usual, it can be shown that this is equivalent to a null-controllability problem for a cascade system where the number of equations are doubled. This new problem is studied by means of the classical duality arguments and Carleman estimates, but unlike other insensitizing problems for scalar systems, the election of the Carleman tools depends on the number of components of the system to be insensitized.

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

“…Using the source term method, this has been extended to the corresponding nonlinear system (i.e., with uu x term) in [36], which gives null controllability of the system under the condition of √ ν being quadratic irrational. 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.…”