Apply voltage for X seconds

Minimum and maximum time range?

You need to furthre explain what you mean by "certain period of time", seconds/minutes/hours/days? What do you want to happen if the "trigger" event occurs again during the timing cycle? Should it be ignored or should it restart the timing cycle? Perhaps a NE555 timer, 74hc123, 74hc221 or a PIC depending.

You need a One shot timer.. Look on the net for 555 timer examples. I assume you need it to only close once and release to not close again until you remove the signal and recycle it?

Take a look at the monostable 555 circuit here- get rid of the relay/ diode if you need less than 200mA :

maybe feed the multivibrator into something like a 4106 bilateral switch?

Then the bilateral switch can complete a circuit and apply how much voltage you need to. Depending on what type of circuit needs the voltage and/or how much voltage you may end up using a diac or triac or scr.

Have you ever actually USED a bilateral switch? They are low leve, high impedance devices. They are not intnded to switch any current. The on resistance is high, and the part number was 4016, not 4106. Also, the 4016, and its sucessor, the 4066 are QUAD BILATERAL SWITCHES. Early CMOS devices lock up if the voltage is outside the secified range, or just quit working

Ive used them with op amps as a semiconductor replacement for an atari joystick that worked quite well, with a potentiometer joysticks hooked to them. That was when Radio Shack sold such stuff.

That is a completely different application. You were working with low signal levels, and high impedance circuits.