Exhausted what you call a Lightning Network payment channel that has restrictions for one way payments (i.e. from Alice to Bob). When a channel is exhausted, the person controlling that side of the channel has no downside to fraudulently trying to commit to a non-exhausted state.
For this reason, it is not preferable to have exhausted channels and they are something to be wary of if you are running a full-node. However, most Lightning Network wallets, and OpenNode, will ensure that channels do not become exhausted.
- Lightning Network – Bitcoin Wiki