I think the way I would approach this is to use two transmission gates driver by a square wave oscillator. A transmission gate is an electronic switch, which can pass a voltage (within certain limits) or block it. A square wave goes to a voltage for a certain period then returns to zero for the same time, then repeats its self. One gate is driven in the opposite direction to the other. So calling one square wave A and the other B, and their gates GA and GB.
So when the voltage from the sensor is rising, GB always has a more positive voltage coming out of it the GA. Likewise when the voltage is falling its the other way around. The trouble is that the outputs do not occur at the same time, rather one after another, so both the voltages are stored on a capacitor. So now the voltages occur at the same time, so taking one to the negative input of an op amp and the other to the positive input. It can be arranged that the output of the opamp goes from earth to positive for a brief period. This fast pulse can be used to trigger a timer to extend the pulse and hence the operating time of the relay.
Running the multivibrator at1 KHZ would allow you to sample the rise/fall of the sensor output every 1mS or so. You must have a serious think about the voltage fall time of the sensor. i.e. if the voltage drops .5V in .5 seconds or it falls .1V in .1 mS. This will determine the size of the storage capacitors, so they can charge up enough during the .5 mS and yet hold their voltage over the next .5 mS, then be discharged by the next pair of voltage pulses from the gates.
Frank