Figure: G450 elevator HOPS switches, from G450 Maintenance Manual, §27-34-01, figure 503, sheet 3.
I've never heard of a hard over flight control malfunction in a Gulfstream but it is certainly possible. Should that happen, the Hard Over Protection System (HOPS) should remove hydraulic pressure from the offending acutator(s). The airplane is flyable without the hydraulics, but it is certainly not easy. The secret is to get the airplane into trim and fly it onto the runway by flaring without reducing the power until the wheels are the ground. If you pull the power, it will take both pilots with all the force they can muster to pull the elevator. More about this: G450 Abnormal Procedures / Manual Reversion.
The HOPS is electrically powered to isolate the hydraulics, removing the electrons via the circuit breaker restores the hydraulics. Be very careful about turning a flyable airplane into an unflyable one. More about this: Mishaps / Alaska Airlines 261.
People have died as a result of pilot induced oscillations at altitude, in a Falcon 900. More about this: Mishaps / DA-900B SX-ECH.
What follows comes from the references shown below. Where I think it helpful, I've added my own comments in blue.
Unlike the other axis, the elevator HOPS can remove a single or both hydraulic systems. As a result, you may still have hydraulic pitch control and only be presented with the CAS message as an indication of elevator HOPS activation. If both left and right hydraulics are shut off to the elevator, however, you will only have manual reversion to control the elevator.
Elevator Hydraulics Off
Figure: Elevator actuator diagram, from FlightSafety G450 Pilot Training Manual, figure 15-3.
[G450 Airplane Flight Manual, §3-13-40] In the event an elevator actuator is out of phase with pilot input, the hydraulic system to that actuator will be shut off. Elevator HOPS activation is annunciated to the flight crew by an amber Elevator Hydraulics Off message displayed on CAS.
The elevator Hard Over Protection System (HOPS) has four switches that measure the elevator linkage movement for differences in mechanical and hydraulic movement of the linkage, and four switches internal to the actuator that measure pressures. If the switches detect a difference in input versus output for more than a tenth of a second, they electrically shut hydraulic pressure from the effected system. This shutoff is electric and can be reset or prevented by pulling the ELEV HYD S/O circuit breaker. Resetting the breaker restores protection, but it could also restore the problem that caused the HOPS activation in the first place. More about this: G450 Systems / Elevator / HOPS.
[G450 Airplane Flight Manual, §3-13-40]
WHEN THE ELEVATOR HYDRAULIC SHUTOFF CIRCUIT BREAKER IS PULLED, HYDRAULIC POWER FROM THE ASSOCIATED SYSTEM IS RESTORED TO THAT PORTION OF THE ACTUATOR. WHEN THE CIRCUIT BREAKER IS RESET, THE ACTUATOR WILL REMAIN POWERED IF NO HARDOVER IS DETECTED. IF A HARDOVER IS DETECTED, HOWEVER, THE SHUTOFF WILL BE REPEATED.
If the event started with a very large pitch change caused by an actual problem, pulling the circuit breaker will allow that same problem to recur.
Example of a faulty shut off may be that you've just been upset by the wake of a passing heavy and got into a bit of a pilot induced pitch oscillation.
Take a firm grip of the yoke and be wary of making elevator reversals. A pilot induced oscillation could make the situation worse.
With the loss of all hydraulic power to the elevator, high pitch control forces will result. Use of elevator trim is required for pitch control. During landing, use caution not to overuse trim for the landing flair, which can result in a ballooned landing. Consider using copilot to help with the flare if forces are high.
There are two techniques here:
FSI G450 PTM, FlightSafety International Gulfstream G450 Pilot Training Manual, Volume 2, Aircraft Systems, October 2008
Gulfstream G450 Airplane Flight Manual, Revision 36, December 5, 2013
Gulfstream G450 Maintenance Manual, Revision 18, Dec 12, 2013