Insensitizing control problems for the stabilized Kuramoto–Sivashinsky system

In this work, we address the existence of insensitizing controls for a nonlinear coupled system of fourth- and second-order parabolic equations known as the stabilized Kuramoto–Sivashinsky model. The main idea is to look for controls such that some functional of the states (the so-called sentinel) is locally insensitive to the perturbations of the initial data. Since the underlying model is coupled, we shall consider a sentinel in which we may observe one or two components of the system in a localized observation set. By some classical arguments, the insensitizing problem can be reduced to a null-controllability one for a cascade system where the number of equations is doubled. Upon linearization, the null-controllability for this new system is studied by means of Carleman estimates but unlike other insensitizing problems for scalar models, the election of the Carleman tools and the overall control strategy depends on the initial choice of the sentinel due to the (lack of) couplings arising in the extended system. Finally, the local null-controllability of the extended (nonlinear) system (and thus the insensitizing property) is obtained by applying the inverse mapping theorem.