In 1966, IBM released its new mainframe operating system, OS/360. 56 years later, it is still actively developed, although it's been rebranded a few times – most recently to z/OS, although it is still probably best-known as MVS. I'm very confident it will still be around in 20 years, and being actively maintained – although no doubt its install base will have shrunk a lot further. By then, it will be 76.
Considering how many untold millions of devices run Linux worldwide – servers for myriad business applications, consumer electronics, motor vehicles, industrial machinery, medical devices, lab equipment, aircraft, spacecraft, satellites – if z/OS can make it to 75, why not Linux? Linux will turn 75 in 2066, and then it only need survive another 6 years to make it to 2072.
Even if some new hotness (such as Google's Fuchsia) overtakes it, its existing momentum will be enough sustain it for decades to come – and people will be willing to pay for necessary maintenance (bug fixes, support for new hardware, network protocol enhancements, etc), just like they do with countless other legacy systems today.
> I fully expect an operating system to emerge in the next few decades that is built around a formally verifiable safety and security model
We already have formally verified operating systems – the open source seL4 microkernel was proven correct in 2009. But, few have adopted it – because, whatever the theoretical advantages a formally-verified microkernel may have, they don't amount to enough in practice to justify the switching costs.