After ignition, if any problems occur such as fuel leakage, can a rocket motor can be shut off?
Once it is ignited it is impossible to shut down a solid propellent in a rocket because they have all the necessary things ( fire, oxidizer,fuel) to burn .
but in hybrid rocket is possible to have a controlled burn
When thrust is desired, a suitable ignition source is introduced in the combustion chamber and the valve is opened. The liquid propellant (or gas) flows into the combustion chamber where it is vaporized and then reacted with the solid propellant. Combustion occurs in a boundary layer diffusion flame adjacent to the surface of the solid propellant.
so in hybrid rocket burning can be controlled by controlling the valve (or) by controlling the nozzle geometry
Most (almost all, but not all) are configured for both shut-off and restart.
Some early liquid fueled rockets could not be shut off - they were "gravity fed," and once under acceleration, could not be shut off, as the acceleration itself drove the fuel into the ongoing combustion.
Likewise, some military tactical rockets are non-stoppable liquid fuel rockets.
Most solid fuel rockets are not stoppable. This is due to the fuel mass usually being both fuel and oxidizer mixed in a stable but ignitable configuration.
A few theoretical designs use a solid oxidizer mass adjacent to a solid fuel mass, and one or the other can be ejected.
A few theoretical designs use multiple linear cells, and can have the currently burning cell ejected by a mechanical or pyrotechnic method; in theory later cells could have igniters. I've only seen this used in model rocketry; it's theoretically possible to cap an ejection charge such that it ejects motor 1 without igniting motor 2, and then have, past the cap, a remote ignition for motor 2 that, when it fires, ejects the cap. (I saw someone try to launch just such a rig; I didn't see it fly, due to engine 1 failure.)
In hybrid motors, the fuel is usually solid, and the oxidizer is liquid. The motor is able to be shut off by stopping the oxidizer flow.
Some such motors use a hypergolic pair - that is, a pair that ignites on contact. Such motors are almost always able to be restarted.
Other motors include a re-ignition system of some kind, so that they can be restarted.
Solid Rocket Boosters have often been configured such that they can be detached after ignition but before launch; generally, this is similar to a normal separation event. While this does not turn off the engine, it does allow the engine to launch without the rest of the craft, and from a payload perspective, is preferential to losing the payload as well.
Most NASA, ESA, and Russian Space Agency space launches also have charges to destroy the launch vehicle; manned launches also have an escape rocket that pulls the manned portion away prior to the rocket's destruction sequence. Again, this is not ideal, and isn't really a safe shutdown, but is used to protect downrange ground items from a failed launch.
With liquid fuel and hybrid fuel rockets, the first step of abort is to cease fuel flow; if that fails, then the payload eject and detonate protocols are invoked. With mixed launchers, both SRB detach and then escape/eject and if needed, destruction of the launcher are public listed protocols.
A qualified 'Yes'.
More advanced solid rocket motors can not only be throttled but also be extinguished and then re-ignited by controlling the nozzle geometry or through the use of vent ports. Also, pulsed rocket motors that burn in segments and that can be ignited upon command are available.
The page says rockets, not necessarily rockets meant for space-craft (either launch, or for use in space)
Liquid rocket motors (esp. ones that use turbopumps) are usually shut down before fuel/oxidizer depletion. When a turbopump runs dry, it'll overspeed and disintegrate, and that's best avoided.
Modern solid fueled missile systems use so called "blow off ports" near the top of the motor in order to decrease the pressure of the combustion chamber and effectively "shut down" the thrust generation.
The burning rate of a solid propellant is dictated by: rate = a*P**n
So if the P (chamber pressure) could be brought to a near vacuum, the burning rate will decrease to near 0.