In this paper, we consider a system that is under the effect of multiple stealthy attackers, whose inputs we design using perfectly and imperfectly rational game-theoretic approaches. The goal of the attackers is to steer the state of the system as far as possible from the origin, so as to disrupt the nominal objective of system regulation. However, to remain stealthy, the attackers must ensure that the total magnitude of their inputs remains below a certain threshold, otherwise they are at risk of being exposed to a detection mechanism that monitors the system. To derive the optimal attack policies for the attackers, we interpret the aforementioned setup as a constrained game, and we solve it in two cases: in the first case, we assume that the attackers are perfectly rational and operate on the Nash equilibrium, which we derive in closed-form; and in the second case, we assume that the attackers are imperfectly rational, and we design two models of bounded rationality as a means to capture their different levels of rationality. Under certain conditions, it is proved that the corresponding bounded rationality models converge to a Nash equilibrium as the levels of rationality increase. Simulations demonstrate the efficiency of the derived attack policies in both the perfectly and the imperfectly rational case.
Fotiadis et al. (2023) studied this question.