Logics for timed resource control I'm studying proof theory and I've seen that linear logic can be used as a way to control resource usage, since by the propositions-as-types it is equivalent to the linear lambda calculus.

Is there a logic that allows resource control (like linear logic) and can express properties that vary over a notion of time like linear time temporal logics (LTL)?

Any reference or point to literature is highly appreciated.
As time can be a resource, it is a bit unclear to me what you seek.

Nevertheless, you might want to look at weighted extensions of LTL, like Metric Temporal Logic first defined here. (Specifying real-time properties with metric temporal logic)