A canonical abstraction is the agent-environment loop. At time $t$, the agent observes $o_t$, updates internal state $h_t$, chooses action $a_t$, and receives reward or feedback $r_t$. The environment then transitions to a new state according to its dynamics.

Many agentic systems optimize expected cumulative utility. In reinforcement learning, this is often written as $$J(\pi)=\mathbb{E}_{\pi}\left[\sum_{t=0}^{T} \gamma^t r_t\right]$$ where $\pi$ is the policy and $\gamma \in [0,1)$ is a discount factor.

Agentic AI often blends reactive policies with deliberative planning. The system may use a policy $\pi(a_t \mid h_t)$ for rapid action selection and a planner that searches over future trajectories to satisfy constraints or maximize long-term value.