# Optimal Opacity-Enforcing Supervisory Control of Discrete Event Systems on Choosing Cost
## 1 Introduction  
In modern society, when important information about enterprises is released online, it generally needs to be as complete as possible. In order to keep the security and opacity of important data transmission, it is necessary to use some unimportant data to confuse and make it difficult to distinguish from an adversary. The more information is used to confuse important data, the better it is. Therefore, some papers (e.g., [1, 2, 3, 4]) discussed the issue of maximum opaque sublanguages. However, in reality, due to the cost involved in using data transmission, less unimportant data and lower transmission costs are preferred. Therefore, this paper proposes an optimization mathematical model to find the most cost-effective information to confuse the important data to be released and proposes an algorithm to obtain the optimal control strategy. In 2004, opacity was first introduced to analyze cryptographic protocols in [5]. In 2005, the research modeled as Petri nets in [6] brought opacity to the field of discrete event systems (DES). Afterward, the research on opacity in DES boomed up. In DES models, the definition of opacity was divided into two cases: language-based opacity [1, 7, 8] (e.g., strong opacity [7], weak opacity [7], non-opacity [7]) and state-based opacity [6, 9, 10, 11, 12, 13] (e.g., current-state opacity [6, 9], initial-state opacity [6], initial-and-final-state opacity [13], \(K\)-step opacity [9, 10, 11], infinite-step opacity [12, 14]). In ref. [15], the works of [13] were extended, and it was shown that the various notions of opacity can be transformable to each other. Then, in Ref. [16], the existing notions of opacity were unified and a general framework was provided. With the definition of opacity, the verification approach was investigated in the previous works. Once a system was not opaque, supervisory control theory [1, 4, 17] or the enforcement approach [18, 19, 20, 21, 22] were presented to ensure opacity in the system. In general, supervisory control theory restricts the behavior of the system to ensure opacity, whereas the enforcement approach does not restrict but modifies the output of the system to ensure opacity. For example, in Ref. [1], fix-point theory was used to check the opacity of the closed system at every iteration to achieve the maximalpermissive supervisor on condensed state estimates. In Ref. [4], the maximal permissive supervisor was obtained using the refining of the plant and observer instead of condensed state estimates. In Ref. [17], strong infinite- and \(k\)-step opacity was transformed into a language and an algorithm was presented to enforce infinite- and \(k\)-step opacity by a supervisor. In Ref. [18], the synthesis of insertion function was extended from current-state opacity [23] to infinite-step opacity and \(K\)-step opacity. In Ref. [19], fictitious events were inserted at the output of the systems to enforce the opacity of the system. And some works [20; 21] extended the method of [19], where the authors of Ref. [20] discussed the problem of opacity enforcement under energy constraints and the authors of Ref. [21] studied supervisory control under local mean payoff constraints. In Ref. [22], current-state opacity was verified and enforced based on an algebraic state space approach for partially observed finite automaton. To ensure opacity, in Ref. [24], some algorithms were proposed to design a controller to control information released to the public. And then, in Ref. [25], the work of [24] was extended, and then an algorithm was presented for finding minimal information release policies for non-opacity. In Ref. [26], a reward was assigned for revealing the information of each transition and the maximum guaranteed payoff policy was found for the weighted DES. In Ref. [27], a dynamic information release mechanism was proposed to verify current-state opacity where information is partially released and state-dependent. To ensure opacity, the secret is preserved by enabling/disabling some events to restrict the behavior of the system.