Measure the maximum value of an exponential analog signal and provide a constant supply having that same value
I need to detect the exact time for which an exponentially decreasing
analog signal is equal to the half of its maximum value.

Use a peak detector circuit to detect (and hold) the peak of the exponentially decaying signal then, use an equal resistor potential divider to produce a voltage level that is half the peak value. Finally, use a comparator with peak/2 fed to one input and the ddecaying signal to the other.
Here's a schematic of a peak detector but newer op-amps will perform a tad better: -

(source: eevblog.com)
Capacitor &quot;C&quot; &quot;holds&quot; the peak voltage and this voltage is buffered by A2. The output of A2 can be reduced by 2 using equal value resistors forming a potential divider.
If the peak of the exponential signal is slightly different on each cycle, then when it returns back to form a peak, a positive edge detector can be used to reset the voltage stored on the capacitor before the peak is reached.
This is basically a BJT or FET that rapidly discharges the voltage. This allows for the peak of each cycle to be &quot;recorded&quot; as opposed to the highest peak amongst several cycles.