Todays F-14s are powered by two different engines (not two different engines in one F-14, but two types in the F-14 fleet), the old TF30 and the new and more reliable General Electric F110. The new engine has increased power and enables the Tomcat to launch from a catapult without the use of afterburners and even allows to climb away after catapult launch when one engine is lost. The F110 was considered so reliable that it entered service even without clearance for afterburner take-offs! In 1981 an alternative engine was tested in the first F-14B, then the designation for a Grumman prototype (BuNo. 157986) with a pair of Pratt & Whitney F401-PW-400 engines. But after extensive tests with this engine technical difficulties occured and the engine was rejected. Today the F-14's two engine-variants can easily be distinguished comparing the exhaust nozzles.

The TF30's afterburner nozzles are regulated open or closed individually by each engine's afterburner fuel control system. The afterburner fuel control pumps are individually operated using the hydraulic pressure produced by the combined and flight hydraulic systems. The port engine powers the combined hydrualic system and the starboard engine powers the flight hydraulic system.
The Tomcat has a "weight on wheels" switch and a "weight off wheels" switch. When weight is "on wheels" (on deck) the nozzles are commanded open to reduce thrust produced by the engines to keep from blowing over ground personnel. When the weight is "off wheels" (airborne) the nozzles will close any time when being in "basic engine" from idle to military power (highest thrust without selecting afterburner) and will open up as afterburner is staged. This feature gives added thrust at idle when being airborne. The switches are mechanically operated by the main landing gear scissors assemblies, but need electrical power to function. When electrical power is taken off the jet completely, the aircraft will default to the weight off wheels condition where the nozzles close so that if there happened to be a total electrical failure airborne you would still have the same amount of thrust at idle that you always do airborne.
When the engine is shut down, the starboard engine is always shut down first. The reason why is because there is a bidirectional pump that has to be checked to ensure that if the starboard engine is lost in flight, the flight side hydraulics will still operate. Once it is ensured that it is working the bidirectional pump will be secured and the flight side hydraulics will go to zero and the afterburner fuel control on that side will no longer be powered because of the lack of hydraulics. Since the weight on wheels switch is still receiving power from the left generator, the starboard nozzle will be trapped open. When the port engine is secured the left generator drops off-line at approximately 55%, however the combined side hydraulics will still operate at 3000 psi for a short period of time as the engine continues to wind down. When the port engine is secured (power off) the nozzle closes because of loss of electrical power and residual fuel pressure through the fuel control to the nozzle actuator (see thumbnail below left).
Basically the F110 is the same. Nozzle position is controlled by the engine AFTC (Augmentor Fan Temperature Controller) based on throttle position, fan discharge pressure and weight on/off wheels. This is automatically computed when the engine is operating in Primary mode (normal electronically controlled). If a failure of a sensor or system causes the engine to revert to Secondary mode (hydro-mechanical control) then the nozzles go to the full closed position and remain there even after weight on wheels. Afterburner operation is prohibited in Secondary mode.
An interesting sub-mode of the nozzle system on F110 engines is known as "RATS" (Reduced Arrestment Thrust System). Basically with weight on wheels and the arresting hook down RATS will cause the engine military thrust to be reduced by approximately 10 percent to prevent overstress to the shipboard arresting gear when landing on a ship. This doesn't really effect the nozzles, just the amount of thrust produced by the engine.

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