In this paper, we study age of information (AoI) optimization for status updating in an integrated sensing and communication (ISAC) system. We consider a discrete-time architecture in which a base station interacts with a physical environment and a remote monitor, and at each time slot can operate in one of three modes: sensing, communication, or joint sensing and communication. Each mode is unreliable and incurs a different operational cost. The objective is to minimize a discounted infinite-horizon cost that combines the AoI at the monitor with action-dependent sensing and communication costs. For the single source scenario, we formulate the problem as a Markov decision process with a two-dimensional AoI state and prove that the optimal stationary policy admits an ordered threshold structure in the AoI state space. Since the AoI evolves over an infinite space, we truncate the state space to reduce complexity and rigorously bound the resulting error. The analysis analytically determines the truncation size needed to keep the error below a given threshold. For the multi-source scenario, we formulate the scheduling problem as a restless multi-armed bandit. We develop both a Whittle index policy and an approximate Whittle index policy for scheduling under two different regimes, one where indexability is guaranteed, and one where it is not. Numerical results illustrate the structure of the optimal policy in the single-source case and show that the proposed approximate Whittle index policy performs comparably to the Whittle index policy in the indexable regime, while remaining effective beyond it.

翻译：本文研究了集成感知与通信（ISAC）系统中状态更新的信息时效性（AoI）优化问题。我们考虑一个离散时间架构，其中基站与物理环境和远程监控器交互，每个时隙可在三种模式之一中运行：感知、通信或联合感知与通信。每种模式均不可靠且产生不同的运行成本。目标是最小化折现无限时域成本，该成本将监控器处的AoI与依赖于动作的感知及通信成本相结合。针对单源场景，我们将该问题建模为具有二维AoI状态的马尔可夫决策过程，并证明最优平稳策略在AoI状态空间上呈现有序阈值结构。由于AoI在无限状态空间上演变，我们截断状态空间以降低复杂度，并严格限定由此产生的误差。分析解析确定了保持误差低于给定阈值所需的截断尺寸。针对多源场景，我们将调度问题建模为多臂赌博机问题。我们开发了Whittle索引策略和近似Whittle索引策略，用于两种不同机制下的调度：一种保证可索引性，另一种则不能。数值结果展示了单源场景中最优策略的结构，并表明所提出的近似Whittle索引策略在可索引机制下与Whittle索引策略性能相当，同时在该机制之外仍保持有效性。