I am wondering if it is possible to propulsively land a solid rocket booster after it detaches from the core booster. I know you could use thrust termination ports to stop thrust, but it would drop the pressure inside and the leftover propellant would burn away. Is there any safe way to shut off an SRB, and could you relight it afterward? Alternatively, could you somehow land an SRB without shutting it off? Thanks!
Fundamentally soft landing a solid requires controlling the thrust. This is not totally impossible, as demonstrated by the Nulka system that hovers by adjusting away from the optimal nozzle shape to 'waste' thrust. The burn off of fuel makes the rocket lighter, so to do a hovering descent would involve a very complex grain structure that runs a tapering down thrust on ascent, then had some sort of slow burning layer for the descent and then a ramp up to high thrust again as that burned through shaped to produce a touchdown, assuming you could precisely predict your burn rates (which change with starting ambient temperature, humidity of the grain and pretty much everything other than moon phase).
Precise trimming of solid driven systems is possible by having large numbers of electrically fired chambers, used in the ASM-135 ASAT terminal guidance system , the question at this point is how useful this 'simple' re-usable solid rocket booster is now there is a complex (and probably custom per launch) grain main motor and a very large number of mini trimming rockets all needing reloading for reflight.
It is worth noting that Soyuz in uses solid rockets to cushion touchdown since the presence of the parachute reduces the needed hoverslam precision (steady and known descent rate, stable orientation). It would probably be possible to use a similar system to land a booster, though the lessons of complex solid firings on Credible Sport are probably relevant.
Yes disclaimers apply
Others addressed "classic" SRBs that fire continuously, and the near-impossibility of such an endeavor with them. But there is an experimental (though quite matured) technology of Electric Solid Propellant which allows extinguishing and relighting your SRB at will, and even fine-tuning the thrust mid-flight. You will likely lose some of specific impulse vs classic SRBs and you'll need to haul a pretty good battery (or other high-power electric power source) but with this technology your SRB is fully (or partially, if you wish) throttleable, relightable and so on.