This note addresses the problem of evaluating the impact of an attack on discrete-time nonlinear stochastic control systems. The problem is formulated as an optimal control problem with a joint chance constraint that forces the adversary to avoid detection throughout a given time period. Due to the joint constraint, the optimal control policy depends not only on the current state, but also on the entire history, leading to an explosion of the search space and making the problem generally intractable. However, we discover that the current state and whether an alarm has been triggered, or not, is sufficient for specifying the optimal decision at each time step. This information, which we refer to as the alarm flag, can be added to the state space to create an equivalent optimal control problem that can be solved with existing numerical approaches using a Markov policy. Additionally, we note that the formulation results in a policy that does not avoid detection once an alarm has been triggered. We extend the formulation to handle multi-alarm avoidance policies for more reasonable attack impact evaluations, and show that the idea of augmenting the state space with an alarm flag is valid in this extended formulation as well.

翻译：本文研究了离散时间非线性随机控制系统中攻击影响的评估问题。该问题被表述为一个带有联合机会约束的最优控制问题，迫使攻击者在给定时间段内避免被检测。由于联合约束的存在，最优控制策略不仅依赖于当前状态，还依赖于整个历史过程，导致搜索空间急剧膨胀，使得该问题通常难以处理。然而，我们发现当前状态以及警报是否已被触发这一信息足以在每个时间步指定最优决策。我们将这一信息称为警报标志，并将其加入状态空间以构建一个等价的最优控制问题，从而可通过现有数值方法利用马尔可夫策略求解。此外，我们注意到该公式推导出的策略会在警报触发后不再规避检测。针对更合理的攻击影响评估，我们将该公式扩展至处理多警报规避策略，并证明在扩展公式中，用警报标志扩充状态空间的思想同样成立。