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After looking over your request the following solution comes to mind. Using a Off-Delay relay would accomplish what you would like. A signal would activate the relay, it would stay on as long as the signal was present. When the signal activating the coil is deactivated, the Delay portion of the circuit would keep the circuit going for the defined amount of time before deactivating.
If the circuit would be activated again before the delay finishes, it would simply disable the timer until the signal was deactivated again.
I was not able to narrow these results as I do not have any specifications regarding voltages and currents but this link will get you close.
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I hope this helps!