Opacity Enforcement by Insertion Functions under Energy Constraints

We investigate the enforcement of opacity by insertion functions, when the system has a certain amount of initial-credit energy for its operation and defense of secrets. The problem is formulated as a two-player game between the insertion function and the system (or environment) with an energy objective and asymmetric partial information. The insertion function must defend the secrets by inserting fictitious output events while guaranteeing that the energy level never drops below zero, under the worst-case scenario of system operation. The insertion function has only partial information about the system, due to unobservable events that affect the energy level. To resolve the partial observation, we construct a game structure called the Energy Insertion Structure (or EIS) that provably embeds insertion functions solving the proposed problem.