# How to update the state of this optimal control problem?

Consider a continuous-time optimal control problem where we store all available information up to time $$t$$ as our state ($$S_t=\mathcal{I}_t$$). At each time $$t$$, there are possibilities for updating the state. 1) A new piece of information arrives at time $$t$$ and we should add $$\{(t,a_t)\}$$ to $$\mathcal{I}_t$$, 2) A currently stored pice of information should be removed from $$\mathcal{I}_t$$, that is, a piece of information like $${(s,a_s)}$$ is removed from $$\mathcal{I}_t$$; 3) or there is no change.

In discrete-time, it is possible to write $$\mathcal{I}_{t+1}$$ based on $$\mathcal{I}_{t}$$ and three possibilities. However, I am wondering how to write it in continuous time.

Since there is no rate for arrival of new information or removal of a piece of information, I am confused about how to write it.

$$\mathcal{I}_{t+\delta t}= \mathcal{I}_{t} \cup \{(t,a_t)\} \text{ in case of a new arrival in } [t,t+\delta t]$$?

I would be thankful if you can share a reference that has used the same modeling assumptions.