Induction heating circuit also stable with forward or reversed DC current through work coil?
All high powered commercial induction heaters use an H bridge.

Use a series LC resonant tank circuit. Which you would model as a series RLC (includes resistance of the load).

Drive the series load at a frequency just ABOVE resonance. This reduces OFF switching to near zero. ON switching losses are also near zero. 

Remember to use anti-parallel diodes on each switching device.

Get a circuit simulator to observe that switch OFF losses reduce as you approach resonant frequency.

You want to run slightly above resonance to allow the diodes to forward bias just after switching off. By doing so, the voltage across the NEXT switching device will be near zero during turn ON.

Keep in mind that the coil voltage (and capacitor voltage) will be higher than the output voltage of the H bridge.  Coil volts will be Q times the output voltage of the H bridge. (Q of the coil).

EDIT 1 :  For parallel tank, the H bridge needs to be designed as a current source, (as opposed to a voltage source used with series tuned circuit). 

You don't just apply a square wave to a parallel tank. An inductor is used in series with the output of the H bridge to create a pseudo current source. And control circuit is more complicated.

In series LC circuit the voltage applied (H bridge) is square wave and current is sinusoidal at resonance. Current is triangular above resonance and slowly morphs to sinusoidal as frequency is lowered to near resonance.