If it is a true pendulum, it will be quite accurate in maintaining time... after that is how clocks were done before we had electronic clocks. In fact, you may have a fight on your hands, if the pendulum natural period is not quite the same as the lighting timing, and you try to force it to match.
If it doesn't keep as wide a swring as you waint, for as long as you want, you can use a true clockwork mechanism to keep it swinging. Depending on how big the ball on the end is, you may want to add more weight to help it overcome the wind resistance, and to keep swinging for a more time. If you can do that, then you just manually start it swinging. You can vary the weight or the length of the pendulum to adjust the period, so that its natural period matches your lighting.
Note that if the pendulum is not powered, the width of the arc it sways, will gradually lessen, but the time for each stroke will not. Yes you can speed up or slow down a pendulum by applying force, but again, it will certainly be easier to not do this, and I suspect it is unnecessary.
Another approach is to let the pendulum own swing be use to time lighting. Just divide your lighting into several cues, and press "go" when your pendulum crosses a certain visual mark.
But, to your original request for a linear actuator. It has been many years since I was involved with a company that would have made use of these types of system, and technology has certainly marched on. So, this knowledge is somewhat old.
You can find linear actuators that are just driven by DC motors. Look at some of the robotics web sites for sources. www.robotshop.com is one.
If the actuator incorporate a sensor, they can act as a servo by attaching an appropriate servo board is connected to it. This the typical type of servo system used in industry. I don't have a good source to refer you to, for this type of industrial system. They also are controlled by applying a DC voltage (I don't recall if it was typically 0-10 or 0-12 v, but either would probably work as you described in your first message.
There is another category of servo systems, created for the radio control toy industry. These are also commonly used in the low-end hobby robotics. This type of servo has the servo-board already mounted inside the motor/gearbox. They accept a pulse, where the width of the pulse tells the servo where to position itself. That industry has driven more development in that category, so that you can get pretty sophisticated and powerful servos
For both of the above types, you need something that will interpret the DMX signal, and then give out a signal that the servo will accept. I imagine there are a couple of companies that can provide something, but Blue Point Engineering has been doing it for a while. I think they are over priced, but I don't know how reliable their products are compare to others, so I can't really say for sure their pricing is not warranted.
This page has a number of their DMX interfaces for servos.
Their $325 USD, DMX - 8866 accepts DMX, and seems to directly control a DC motor with feedback.
Their $135 USD, DMX - 2000 accepts DMX and provides the pulse-width signal for up to 8 radio-control type servos.