  Complex systems are challenging to control because the system responds to the
controller in a nonlinear fashion, often incorporating feedback mechanisms.
Interdependence of systems poses additional difficulties, as cross-system
connections enable malicious activity to spread between layers, increasing
systemic risk. In this paper we explore the conditions for an optimal control
of cascading failures in a system of interdependent networks. Specifically, we
study the Bak-Tang-Wiesenfeld sandpile model incorporating a control mechanism,
which affects the frequency of cascades occurring in individual layers. This
modification allows us to explore sandpile-like dynamics near the critical
state, with supercritical region corresponding to infrequent large cascades and
subcritical zone being characterized by frequent small avalanches. Topological
coupling between networks introduces dependence of control settings adopted in
respective layers, causing the control strategy of a given layer to be
influenced by choices made in other connected networks. We find that the
optimal control strategy for a layer operating in a supercritical regime is to
be coupled to a layer operating in a subcritical zone, since such condition
corresponds to reduced probability of inflicted avalanches. However this
condition describes a parasitic relation, in which only one layer benefits.
Second optimal configuration is a mutualistic one, where both layers adopt the
same control strategy. This work demonstrates that control protocols in systems
of interdependent networks need to take into account higher-order organization
of the system and cannot be designed independently, maximizing benefits only
for their individual layers.
