Impedance matching for maximum power transfer
Unfortunately no, there is no switching sequence that will make Vth constant or even roughly so.

With the switch open, the voltage will be V1.

When the switch closes, the voltage across the inductor will cause the current though it to increase, slowly increasing the voltage across R2, causing Vth to drop.

However, when the switch opens again, there is no path that the current can continue to flow, and the voltage will revert to V1.

You have almost drawn a buck converter. What it needs to finish it off is a diode from VD to V1. Then, when the switch opens, the current is allowed to continue to flow.

In the limit of a diode with zero drop, a 50% duty cycle will get you V1/2. The higher the frequency, the lower the ripple at Vth. As real diodes have a drop, the duty cycle will need to be offset a bit to compensate.